Mechanical Systems, Classical Models: Volume 3: Analytical Mechanics (Mathematical and Analytical Techniques with Applications to Engineering)

Mechanical Systems, Classical Models MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING Series E...

Author: Petre P. Teodorescu

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Mechanical Systems, Classical Models

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING Series Editor

Alan Jeffrey

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, in all areas of today's Physical Sciences and Engineering, is well established. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems in modern Physical Sciences and Engineering would otherwise have been impossible. The purpose of the series is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume in the series will provide a detailed introduction to a specific subject area of current importance, and then will go beyond this by reviewing recent contributions, thereby serving as a valuable reference source.

For further volumes: http://www.springer.com/series/7311

Mechanical Systems, Classical Models Volume III: Analytical Mechanics

by

Petre P. Teodorescu Faculty of Mathematics, University of Bucharest, Romania

123

Prof. Dr. Petre P. Teodorescu Str. Popa Soare 38 023984 Bucuresti 20 Romania

Translated into English, revised and extended by Petre P. Teodorescu All rights reserved © EDITURA TEHNICĂ, 2002 This translation of “Mechanical Systems, Classical Models” (original title: Sisteme mecanice.Modele clasice, Published by: Ed. Tehnicá, Bucuresti, Bucharest, Romania, 1984-2002), First Edition, is published by arrangement with EDITURA TEHNICĂ, Bucharest, ROMANIA

ISSN 1559-7458 e-ISSN 1559-7466 ISBN 978-90-481-2763-4 e-ISBN 978-90-481-2764-1 DOI 10.1007/978-90-481-2764-1 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2007943834 c Springer Science+Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Contents

Preface

ix

18.

Lagrangian Mechanics 1. Preliminary Results 1.1 Introductory Notions 1.2 Differential Principles of Mechanics 2. Lagrange’s Equations 2.1 Space of Configurations 2.2 Lagrange’s Equations of Second Kind 2.3 Transformations. First Integrals 3. Other Problems Concerning Lagrange’s Equations 3.1 New Forms of Lagrange’s Equations 3.2 Applications

1 2 2 20 45 46 57 65 83 83 98

19.

Hamiltonian Mechanics 1. Hamilton’s Equations 1.1 General Results 1.2 Lagrange’s Brackets. Poisson’s Brackets 1.3 Applications 2. The Hamilton–Jacobi Method 2.1 General Results 2.2 Systems of Equations with Separate Variables 2.3 Applications

115 115 115 138 151 160 160 172 191

20.

Variational Principles. Canonical Transformations 1. Variational Principles 1.1 Mathematical Preliminaries 1.2 The General Integral Principle 1.3 Hamilton’s Principle 1.4 Maupertuis’s Principle. Other Variational Principles 1.5 Continuous Mechanical Systems 2. Canonical Transformations 2.1 General Considerations. Conditions of Canonicity 2.2 Structure of Canonical Transformations. Properties 3. Symmetry Transformations. Noether’s Theorem. Conservation Laws 3.1 Symmetry Transformations. Noether’s Theorem

213 213 214 225 231 242 254 265 265 289 300 301 v

MECHANICAL SYSTEMS, CLASSICAL MODELS

vi 3.2 3.3 21.

22.

Lie Groups Space-Time Symmetries. Conservation Laws

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems 1. Integral Invariants. Ergodic Theorems 1.1 Integral Invariants of Order 2s 1.2 Invariants of First Order 1.3 Ergodic Theorems 2. Periodic Motions. Action-Angle Variables 2.1 Periodic Motions. Quasi-Periodic Motions 2.2 Action-Angle Variables 2.3 Adiabatic Invariance 3. Methods of Exterior Differential Calculus. Elements of Invariantive Mechanics 3.1 Methods of Exterior Differential Calculus 3.2 Elements of Invariantive Mechanics 3.3 Applications 4. Formalisms in the Dynamics of Mechanical Systems 4.1 Formalisms in Spaces with s + 1 Dimensions 4.2 Formalism in Spaces with 2s + 1 or with 2s + 2 Dimensions 4.3 Notions on the Inverse Problem of Mechanics and the Birkhoffian Formalism 5. Control Systems 5.1 Control Systems 5.2 Optimal Trajectories Dynamics of Non-holonomic Mechanical Systems 1. Kinematics of Non-holonomic Mechanical Systems 1.1 General Considerations 1.2 Conditions of Holonomy. Quasi-co-ordinates. Non-holonomic Spaces 2. Lagrange’s Equations. Other Equations of Motion 2.1 Motion of a Rigid Solid on a Fixed Surface 2.2 Lagrange’s Equations 2.3 Applications 2.4 Other Equations of Motion 3. Gibbs–Appell Equations 3.1 Gibbs–Appell Equations of Motion 3.2 Applications 4. Other Problems on the Dynamics of Non-holonomic Mechanical Systems 4.1 Collisions 4.2 First Integrals of the Equations of Motion

311 319 335 335 335 341 354 356 356 361 365 368 368 373 385 390 390 394 397 402 402 407 411 411 411 421 430 430 436 443 467 478 478 482 491 491 498

Contents

23.

24.

Stability and Vibrations 1. Stability of Mechanical Systems 1.1 Stability of Equilibrium 1.2 Stability of Motion 1.3 Applications 2. Vibrations of Mechanical Systems 2.1 Small Free Oscillations About a Stable Position of Equilibrium 2.2 Small Forced Oscillations 2.3 Non-linear Vibrations 2.4 Applications Dynamical Systems. Catastrophes and Chaos 1. Continuous and Discrete Dynamical Systems 1.1 Continuous Linear Dynamical Systems 1.2 Non-linear Differential Equations and Systems of Non-linear Differential Equations 1.3 Discrete Linear Dynamical Systems 2. Elements of the Theory of Catastrophes 2.1 Ramifications 2.2 Elementary Catastrophes 3. Periodic Solutions. Global Bifurcations 3.1 Periodic Solutions 3.2 Global Bifurcations 4. Fractals. Chaotic Motions 4.1 Fractals 4.2 Chaotic Motions

vii 505 505 505 537 554 566 566 597 606 612 629 630 630 648 675 682 683 689 697 698 707 712 712 723

Bibliography

739

Subject Index

759

Name Index

765

“This page left intentionally blank.”

Preface

All phenomena in nature are characterized by motion. Mechanics deals with the objective laws of mechanical motion of bodies, the simplest form of motion. In the study of a science of nature, mathematics plays an important rôle. Mechanics is the first science of nature which has been expressed in terms of mathematics, by considering various mathematical models, associated to phenomena of the surrounding nature. Thus, its development was influenced by the use of a strong mathematical tool. As it was already seen in the first two volumes of the present book, its guideline is precisely the mathematical model of mechanics. The classical models which we refer to are in fact models based on the Newtonian model of mechanics, that is on its five principles, i.e.: the inertia, the forces action, the action and reaction, the independence of the forces action and the initial conditions principle, respectively. Other models, e.g., the model of attraction forces between the particles of a discrete mechanical system, are part of the considered Newtonian model. Kepler’s laws brilliantly verify this model in case of velocities much smaller then the light velocity in vacuum. Mechanics has as object of study mechanical systems. The first two volumes of this book dealt with particle dynamics and with discrete and continuous mechanical systems, respectively. The present one deals with analytical mechanics. We put in evidence the Lagrangian and the Hamiltonian mechanics, where the study of first integrals plays a very important rôle. The Hamilton–Jacobi method is widely considered, as well as the study of systems with separate variables. We mention also a thorough study of variational principles and canonical transformations. The symmetry transformations, including Noether’s theorem, lead to conservation laws. Integral invariants and exterior differential calculus are also included. A particular attention has been given to non-holonomic mechanical systems. Problems of stability and vibrations have been also considered in the frame of Lagrangian and Hamiltonian mechanics. The study of dynamical systems leads to catastrophes, bifurcations and chaos. One presents some applications connected to important phenomena of the nature and one gives also the possibility to solve problems presenting interest from technical, engineering point of view. In this form, the book becomes – we dare say – a unique outline of the literature in the field; the author wishes to present the most important aspects related with the study of mechanical systems, mechanics being regarded as a science of nature, as well as its links to other sciences of nature. Implications in technical sciences are not neglected. Concerning the mathematical tool, the five appendices in the first volume give the book an autonomy with respect to other works, special previous mathematical knowledge being not necessary. The numeration of the chapters follows that of the first ix

x


two volumes, to which one makes reference for various results (theorems, formulae etc.). I am grateful to Mărgărit Baubec, Ph.D., for his valuable help in the presentation of this book. The excellent cooperation with the team of Springer, Dordrecht, is gratefully acknowledged. The book covers a wide number of problems (classical or new ones) as one can see from its contents. It uses the known literature, as well as the original results of the author and his more than 50 years experience as Professor of Mechanics at the University of Bucharest. It is devoted to a large circle of readers: mathematicians (especially those involved in applied mathematics), physicists (particularly those interested in mechanics and its connections), chemists, biologists, astronomers, engineers of various specialities (civil, mechanical engineers etc., who are scientific researchers or designers), students in various domains etc. Bucharest, Romania 7 January 2009

P.P. Teodorescu

Chapter 18 Lagrangian Mechanics The analytical methods of calculation have been introduced to can study in a systematic and unitary form the discrete mechanical systems (with a finite number of particles or with a finite number of degrees of freedom); thus, appears the denomination of “analytical mechanics”, besides that of “theoretical mechanics” (in the English, German and Russian literature) and that of “rational mechanics” (in the French or Italian literature). In “physical mechanics” stress is put on the experimental aspect, while in “technical mechanics” on the possibility to apply in technics the results obtained. In all these cases, the object of study is the same: mechanics, as a science of nature. In this order of ideas, we consider a discrete mechanical system S , formed by n particles Pi (or a finite number of rigid solids), of position vectors ri , subjected to the action of given forces Fi , i = 1, 2, ..., n (the resultants of all external and internal given forces which act upon the particle Pi ) and to certain links Lk , k = 1, 2, ..., nL . The basic problem which is put consists in the determination of the motion (position and velocity) of the mechanical system S in the time interval [t0 , t1 ] (eventually, [t0 , ∞ ) ) if Fi = Fi ( rj , rj ; t ) , Lk = Lk ( rj , rj ; t ) , ri (t0 ) = ri0

and

vi (t0 ) = vi0 , i = 1, 2, ..., n , k = 1, 2, ..., nL are given. A distinction between external and internal forces is not necessary; in exchange, the classification of the constraints and their representation in a finite or differential form, as well as their elimination from the calculation (when it is possible) have a particular importance. The elimination of the constraints and of the constraint forces from the calculation allows to obtain equations of motion which contain only the position vectors and their derivatives with respect to time. In general, one admits only frictionless constraints, which – obviously – restrains much the sphere of the problems which can by treated. Euler, in 1736, and Maclaurin, in 1742, have written the first treatises of mechanics in which they pass from purely geometric methods to methods based on differential calculus (Euler, L., 1736). The brothers Jacob I Bernoulli and Jean I Bernoulli, Daniel Bernoulli, the son of the latter one, Alexis-Claude Clairaut, Jacob Herman and PierreLouis-Moreau de Maupertuis have made important steps, considering many problems and enouncing various “principles”, which allowed to put in equation the problems of mechanics and to give their solution in a unitary form. “A special ability was always necessary to put in evidence the forces which have to be taken into consideration, so P.P. Teodorescu, Mechanical Systems, Classical Models, © Springer Science+Business Media B.V. 2009

1

2


that each problem become piquant, exciting the emulation; ... the treatise on dynamics of Jean le Rond d’Alembert, appeared in 1743, has put an end to this kind of competition, offering a direct and general method to solve or – at least – to put in equation all the problems of dynamics which one could imagine”. These affirmations have been made by Joseph-Louis Lagrange in his famous treatise “Mécanique analytique”, which appeared in Paris, in 1788, in which mechanics is presented as an abstract science, as a chapter of mathematical analysis, which can be studied without using any figure (Lagrange,J.L., 1788). Thus, the theory of systems of ordinary differential equations, the variational calculus, the theory of punctual and canonical transformations, the theory of Pfaff forms, the theory of integral invariants etc. have been developed, the impulse given by Lagrange to the study of mechanical systems having a particular importance for the development of analysis, as well as of other chapters of mathematics. We consider, in the frame of this chapter, a discrete mechanical system S , formed – in general – by a finite number of particles and by a finite number of rigid solids (as it has been considered in Sect. 17.1.1.1); hence, the system has a finite number of degrees of freedom. For the sake of simplicity, we assume – so as it is made in any classical study, without loss of generality – that the system S is formed by a finite number of a particles; when it is necessary or when intervene particular data we refer to rigid solids too. Thus, after some preliminary results, we will do a detailed study of Lagrange’s equations of motion, which will be then applied to some important problems (Appell, P., 1941–1953; Goldstein, H., 1956; Lamb, H., 1929; Landau, L.D. and Lifchitz, E., 1960; La Valée-Poussin, Ch.J. de, 1925; Lurie, A.I., 2002; Mercier, A., 1955).

18.1 Preliminary Results First of all, we will present some introductory motions, which allow to pass from the representation of the discrete mechanical system S in the space E 3 to its representation by means of a representative point P in the representative space E 3 n . Passing in review the differential principles of mechanics, we can study the motion of the mechanical system S in the new considered space.

18.1.1 Introductory Notions In what follows we use the notions of displacement for a mathematical representation of the nL constraints Lk , k = 1, 2,..., nL ; this allows to introduce the notion of representative space E 3 n . A particular attention is given to the natural systems and to the particle subjected to non-holonomic constraints. 18.1.1.1 Displacements. Constraints One of the most important notions in the frame of the analytical methods is that of displacement; this notion has been introduced and considered at large in Sect. 3.2.1.1, putting in evidence the real displacements dri , the possible displacements Δri and the

Lagrangian Mechanics

3

virtual displacements δri of the particle Pi , of position vector ri , i = 1, 2,..., n . Starting from the displacements, one can define the real velocities vi = dri / dt , the possible velocities vi = Δri / Δt and the virtual velocities vi∗ = δri / Δt . In case of a particle subjected to stay on a fixed curve C or on a fixed surface S (steady case), the virtual displacement is directed along the tangent to the curve or is contained in the plane tangent to the surface at the point P , respectively; in both cases, the real displacement belongs to the (finite or infinite) set of virtual displacements. If the rigid S , e.g., is movable, then the virtual displacements take place in the tangent plane to this surface at the moment t (we consider the surface to be frozen at this moment), while the real displacement links the particle P situated at the point P of this surface at this moment t to the particle P situated at the point P of the surface at the moment t + dt ; hence, in a non-steady case, the real displacement does no more belong to the set of virtual displacements of the considered particle. This statement can be made in the case of a deformable surface S too. The constraints to which is subjected the discrete mechanical system S can be external or internal; in the following, this distinction is not useful. As well, we make distinction between unilateral (expressed by inequalities) and bilateral (represented in the form of equalities) constraints; in analytical mechanics, we will consider all the constraints to be – in general – bilateral. Another distinction is made between constraints of contact and constraints at distance; both these types of constraints can be expressed by equalities. A large study of the constraints is made in Chap. 3, Sect. 2.2. Another classification of the constraints, after Hertz, puts in evidence the holonomic (finite, of geometric nature) and the non-holonomic constraints (infinitesimal or differential, of kinematic nature). After Lothar Bolzmann, the constraints which do not depend explicitly on time are called scleronomic (stationary) constraints, those which vary in time being rheonomous (non-stationary) constraints. As a matter of fact, taking into account these classifications, one obtains a mathematical representation of the constraints. We mention also the distinction between ideal (perfect, smooth) constraints and constraints with friction (real). Analytical mechanics has been developed for discrete mechanical systems; some results can be adapted to the case with friction too. In the absence of the constraints Lk , k = 1, 2, ..., nL , the discrete mechanical system

S is a free mechanical system. The axiom of liberation from constraints states that there exists always a system of forces Ri , which is applied upon the mechanical system S subjected to constraints, so that each particle Pi can be treated as a free particle subjected to the action of the force Fi + Ri , i = 1, 2,..., n ; consequently, the mechanical system S can be studied as a free one. The forces Fi are given (known) forces, while the forces Ri are constraint (unknown) forces. The real elementary work dW of the given forces Fi is given by the relation (3.2.3) and the corresponding virtual work δW by the relation (3.2.3'); as well, the real elementary work dWR of the constraints forces Ri is given by (3.2.7), the corresponding virtual work δWR being of the form (3.2.7').

4


The p holonomic constraints are expressed in the form (3.2.8) if they are rheonomous and in the form (3.2.33') if they are scleronomic, the number s = 3n − p representing the number of degrees of geometric freedom of the discrete mechanical system S ; the motion takes place only if p < 3n (hence, s > 0 ) . We notice that a free rigid solid, component of a discrete mechanical system S , introduces 6 degrees of freedom. The m non-holonomic constraints are represented – in general – by means of the Pfaff differential forms in the form (3.2.13) (as well in the form (3.2.13')), with the aid of the real displacements, or in the form (3.2.15) (analogously, in the form (3.2.15')), by means of the virtual displacements. In case of scleronomic constraints, the relations (3.2.32) must also take place; otherwise, the constraints are rheonomous. The number r = 3n − m represents the number of kinematic degrees of freedom of the discrete mechanical system S . In case of holonomic constraints, we have r = s . The total number of constraints is, in general, nL = p + m , while the number of degrees of freedom (obviously, kinematic degrees of freedom, because the holonomic constraints involve the same number of geometric and kinematic degrees of freedom) is 3n − nL . The non-holonomic constraints of the form (3.2.12), considered above, have been called non-holonomic constraints of the first kind by G. Hamel; he introduces also non-holonomic constraints of the second kind, where appear the accelerations of the particles too, in the form f (drj , d2 rj ; dt ) = 0 ,

(18.1.1)

the dimensional homogeneity being ensured. A constraint relation of the form (3.2.13) corresponds to a holonomic constraint if it is an integrable differential Pfaff form; otherwise it corresponds to a non-holonomic constraint. The Theorem 3.2.2 of Frobenius gives the necessary and sufficient conditions, so that such a differential form of first degree be complete integrable (does correspond to an exact differential or does admit an integrable factor). We notice that the two classifications of the constraints (holonomic or non-holonomic and scleronomic or rheonomous) are independent; we can have, e.g., non-holonomic and rheonomous constraints (the most general case) or holonomic and scleronomic constraints (the most particular case). A mechanical system subjected to holonomic or non-holonomic constraints is called a holonomic or non-holonomic mechanical system, respectively; as well, a mechanical system subjected to scleronomic or rheonomous constraints is called scleronomic or rheonomous mechanical system, respectively. In case of scleronomic constraints, the real displacements belong to the set of virtual displacements. As a matter of fact, to have this property it is sufficient that the conditions αk 0 = 0, k = 1,2,..., m , be fulfilled in the relations (3.2.13) (it is not necessary that the constraints by scleronomic); such constraints are called catastatic. As examples of mechanical systems subjected to holonomic and scleronomic constraints we mention the particle compelled to stay on a fixed curve or on a fixed surface, as well as the rigid solid (or a non-deformable discrete mechanical system). If the curve or the surface are movable (rigid and movable with respect to a fixed frame of


5

reference or even deformable), then the constraint is holonomic and rheonomous. We notice that a rigid solid behaves – in a certain sense – as a particle which has not three but six degrees of freedom. Non-holonomic constraints are, e.g., those corresponding to the motion of a heavy homogenous sphere (see Sect. 17.1.2.8) or of a heavy circular disc (see Sect. 17.1.2.9) on a horizontal plane; we mention the motion of a rigid skate on the ice field too, case considered in Sect. 3.2.2.6. An important class of constraints, in the frame of which analytical mechanics has been developed, is the class of ideal constraints. We call ideal constraints those ones for which the virtual work of the constraint forces vanishes δWR =

n

∑ Ri

i =1

⋅ δri = 0

(18.1.2)

for any system of virtual displacements of the considered mechanical system S . If the virtual displacements δri are arbitrary (we have not constraints), then the relation (18.1.2) is fulfilled if and only if Ri = 0, i = 1, 2, ..., n . But if p holonomic constraints of the form (3.2.8) and m non-holonomic constraints of the form (3.2.13) take place, then we can use the method of Lagrange’s multipliers, the constraint forces being of the form (3.2.37). In this case, the real elementary work of the constraint forces is given by (3.2.37'); this work vanishes if fl = 0, l = 1, 2, ..., p , and αk 0 = 0, k = 1, 2, ..., m (conditions in which the real displacements belong to the set of virtual displacements, case of catastatic constraints), in particular in case of scleronomic constraints. Among the mechanical systems subjected to ideal constraints we mention the particle subjected to constraints of contact, frictionless (compelled to stay on a fixed or movable curve or surface), or the discrete mechanical system for which the distance between two particles is function only on time (constraints at distance). The classical ideal constraints of the rigid solid are: the simple support, the hinge and the built-in mounting. Ideal constraints are also the constraints of the rigid solid which can slide without friction along an axis (or, in general, along a superposable curve), the constraints of a rigid solid which can slide frictionless on a plane (or on a given surface), as well as the constraints of the rigid solid which is rolling or pivoting without sliding on a plane (or on a arbitrary surface); in the first cases, the virtual displacement is normal to the constraint force and in the last cases the virtual displacement is zero. We mention also the systems of two rigid solids considered in Sect. 3.2.2.9. As well, a mechanism (a discrete mechanical system formed of two rigid solids linked rigidly or by hinges or by perfectly smooth or rough surfaces) can be considered as a mechanical system subjected to ideal constraints. We mention also the constraints by threads, which are not always bilateral. Among the constraints with friction, we consider that one which corresponds to the sliding friction, for which the analytical methods of mechanics could be adapted. We assume that the virtual displacement δr takes place in an interval of time δt (which can be the interval of time Δt in which take place the possible displacements, the difference of which is the virtual displacement). In this case, the virtual displacement of a point of a free rigid solid with respect to the inertial frame of reference R ′ will be given by


6

δr ′ = vO′∗ δt + ( ω∗ δt ) × r ,

(18.1.3)

where vΟ′∗ δt (t ) and ω∗ δt (t ) are two arbitrary vectors, which define the virtual rototranslation at the moment t , r being the position vector with respect to the non-inertial frame R ; the position of this solid depends thus on six scalar parameters. We can express the virtual displacement δr ′ in the form δr ′ = δrO′ + δχ × r

(18.1.3')

too, where δχ represents a virtual rotation of the frame of reference R with respect to the frame R (of components δψ , δθ and δϕ , corresponding to Euler’s angles). In the case in which the rigid solid has a fixed point (e.g., the point O ), we have vO′∗ = 0 , its position depending only on three scalar parameters. As well, if the point O of the rigid solid has a motion given by rO′ = rO′ (t ) , then the real displacement will be given by dr ′ = vO′ dt + ( ωdt ) × r , with vO′ = drO′ / dt ; in exchange, for a virtual displacement (compatible with the constraints at the moment t ) the point O must have the position fixed at the moment t , while δr ′ = ( ω∗ δt ) × r , the rigid solid having three degrees of freedom too. The virtual work of the forces of inertia is expressed in the form δW i = − ∫∫∫ μ ( r ) a ′( r ; t ) ⋅ δr ′dV , V

(18.1.4)

where the virtual displacement δr ′ is given by (18.1.3), vO′∗ and ω∗ being arbitrary functions on time. Noting that a ′( r ; t ) represents the accelerations field in the real motion of the rigid solid, we can write, with respect to the frame of reference R ′ (we have r = r ′ − rO′ ), δW i = − { [ ( vO′∗ − ω∗ × rO′ ) δt ] ⋅ A ′ + ( ω∗ δt ) ⋅ DO′ ′ } ,

(18.1.4')

where A ′ is the dynamic resultant of the rigid solid, given by (14.1.27), while DO′ ′ is the dynamic moment with respect to the pole O ′ , given by (14.1.28). Analogously, in the case of the rigid solid, the virtual work of the constraints forces of torsor { R , MO } , will be expressed in the form

δWR = − [ ( vO′∗ − ω∗ × rO′ ) δt ] ⋅ R + ( ω∗ δt ) ⋅ MO ' = 0.

(18.1.4'')

18.1.1.2 The Representative Space E 3 n A basic aspect of the analytical methods of calculation consists in replacing the study of the motion of a discrete mechanical system S of geometric support Ω in the space E 3 by the study of the motion of a single geometric point, called representative point,


7

in another space, conveniently built up, called representative space. E.g., in Sects. 3.2.2.2 and 11.1.1.5 has been introduced the representative space E 3 n , by passing from the mechanical system S of n particles Pi , i = 1, 2, ..., n , to a representative point P , of generalized co-ordinates X k , k = 1, 2, ..., 3n , with the aid of the relations Xk = x j , k = 3(i − 1) + j , i = 1, 2,..., n , j = 1, 2, 3 , (i )

(18.1.5)

of the form (3.2.9), where x (j i ) is the component along the Ox j -axis of the position vector ri ; the real generalized displacements of this point are dXk , the possible generalized displacements are ΔXk and the virtual generalized displacements are δXk , k = 1, 2, ..., 3n , the respective properties corresponding to those in the space E 3 . In these conditions, the constraints of the mechanical system S become constraints of the representative point P . We use also the notations (i ) bj 0 = α j 0 , bjk = α , k = 3(i − 1) + l , jl

i = 1, 2,..., n , l = 1,2, 3 ,

(18.1.5')

of the form (3.2.9), where α(jli ) are the components along the Oxl -axis of the given vectors α ji , which specify the constraints relations (3.2.13); these relations take the form ωj ≡

corresponding

to

the

3n

∑ bjk dXk

k =1

+ bj 0 dt , j = 1, 2,..., m ,

relations (3.2.13''), where

(18.1.6)

bjk = bjk ( X1, X 2 ,..., X 3 n ; t ),

k = 0,1,..., 3n , or the form 3n

∑ bjkVk

k =1

+ bj 0 = 0, j = 1, 2,..., m ,

(18.1.6')

corresponding to the relations (3.2.13'''), where Vk = X k = dXk / dt , k = 1, 2,..., 3n , are the components of the generalized velocity of the point P in the representative space E 3n . Obviously, we assume that the constraints are linearly independent, hence that the matrices [bjk ] are of rank m . If a Pfaff differential form ω j is not completely integrable, then the respective constraint is non-holonomic; taking into account the form (18.1.6') of these relations, then the constraint is called kinematic too. We assume that, in general, the mechanical system S has m non-holonomic constraints (the representative point P is on a non-holonomic manifold with 3n − m dimensions, at


8

the intersection of m non-holonomic hypersurfaces in the space E 3n ). We can express these constraint relations also in the form 3n

∑ bjk δXk

k =1

= 0, j = 1, 2,..., m ,

(18.1.7)

corresponding to the relations (3.2.15''), by means of the virtual displacements, or in the form 3n

∑ bjkVk*

k =1

= 0, j = 1, 2,..., m ,

(18.1.7')

corresponding to the relations (3.2.15'''), where Vk* = δX k / Δt are the virtual generalized velocities ( Δt is the interval of time in which takes place a possible displacement). If bj 0 = 0, j = 1, 2,..., m , then the mechanical system is called catastatic. In this case, the set of virtual generalized displacements coincides with the set of possible generalized displacements and the constraints are called catastatic; as well, the set of virtual generalized velocities (which coincides with the set of possible generalized velocities) contains also the case in which these velocities vanish, hence the case of rest with respect to an inertial frame of reference (the denomination used above being thus justified). A mechanical system which is not catastatic is called non-catastatic; in this case bj 0 ≠ 0 . If the conditions bj 0 = 0, bjk =

∂bjk ∂t

= 0, j = 1, 2,..., m , k = 1, 2,..., 3n ,

(18.1.8)

are verified, then the constraints do not depend explicitly on time and are called scleronomic; otherwise, the constraints are rheonomous. We notice that a scleronomic mechanical system is catastatic too, but a catastatic one may be rheonomous too; a noncatastatic mechanical system is always rheonomous. If a Pfaff differential form is completely integrable (for a system of the form (18.1.6) we can use the Theorem 3.2.2 of Frobenius), then the corresponding constraint is holonomic, being of the form fl ( Xk ; t ) ≡ fl ( X1 , X 2 ,..., X 3 n ; t ) = 0, l = 1, 2,..., p ,

(18.1.9)

hence of the form (3.2.8''); such a constraint is also called geometric. In general, we assume that the mechanical system S has p holonomic constraints (the representative point P is on a manifold with 3n − p dimensions, at the intersection of p hypersurfaces in the space E 3 n ). These finite relations can be written also in the differential form 3n

∂f

∑ ∂Xlk

k =1

dXk + fl dt = 0, l = 1, 2,..., p ,

(18.1.10)


9

or in the form ∂f

3n

∑ ∂Xlk Vk

k =1

+ fl = 0, l = 1, 2,..., p ,

(18.1.10')

so that ∂fl = blk , l = 1, 2,..., p , k = 1, 2,...3n . ∂Xk

Analogously, using the virtual generalized displacements, we can write 3n

∂f

∑ ∂Xlk

k =1

δXk = 0, l = 1, 2,..., p ,

(18.1.11)

or 3n

∂f

∑ ∂Xlk Vk*

k =1

= 0, l = 1, 2,..., p .

(18.1.11')

We assume that also these constraints are linearly independent, the matrices [ ∂fl / ∂Xk ] being of rank p . If fl = 0, l = 1,2,..., p , the mechanical system S is catastatic; at the same time we have ∂f ∂ ⎛⎜ l ⎞⎟ ∂ ∂f ⎝ Xk ⎠ = l =0 ∂t ∂Xk

too, so that the respective system is also scleronomic. We notice that these two properties of the constraints coincide in case of the holonomic systems, for which fl ( Xk ) ≡ fl ( X1, X2, ..., X 3 n ) = 0 .

(18.1.12)

Let be two neighbouring, possible, simultaneous (at the moment t ) positions P ( Xk ) and P '( Xk + δXk ) , of the same representative point of the holonomic discrete mechanical system S . The constraint relations will be of the form fl ( Xk ) = 0 , fl ( Xk + δXk ) = 0 , l = 1, 2,..., p . Expanding into a Taylor series, we can write fl ( Xk ) +

3n

∂f

∑ ∂Xlk

k =1

δXk + ... = 0 .

Neglecting the terms of higher order and taking into account the constraint relations,


10

we find the conditions (18.1.11). One observes thus that δXk represent those differential generalized displacements which must be effected by the representative point to pass from a position to another one, at the same moment t , being thus virtual displacements (corresponding to their definition, as difference of possible displacements). If Fi = Fj( i ) i j represents the resultant of the given forces which act upon the particle Pi , i = 1, 2,..., n , in the space E 3 , then we can introduce, in the space E 3n , the given

generalized forces (shortly, the generalized forces) Qk , k = 1, 2,..., 3n , which act upon the representative point P , in the form Qk = Fj , k = 3(i − 1) + j , i = 1,2,..., n , j = 1,2, 3, k = 1,2,..., 3n . (i )

(18.1.13)

As well, by using the axiom of liberation from constraints, one introduces the constraint forces Ri = Rj( i ) i j , j = 1, 2,..., n , which act upon the particle Pi in the space E 3 ; we are thus led to constraint generalized forces Rk , k = 1, 2,....3n , which act in the space E 3n upon the representative point P in the form Rk = Rj , k = 3(i − 1) + j , i = 1,2,..., n , j = 1, 2, 3 . (i )

(18.1.13')

In what concerns the mass mi of the particle Pi , it is convenient to denote mi = mi

(1)

= mi

(2 )

= mi

(3)

, i = 1,2,..., n ,

so that M k = m j , k = 3(i − 1) + j , i = 1, 2,..., n , j = 1, 2, 3, k = 1,2,..., 3n , (i )

(18.1.14)

can be considered as 3 n generalized masses of the representative point P in the space E 3n . The real elementary work of the given generalized forces is given by dW =

n

3

∑ ∑ Fj( i ) dx (j i )

i =1 j =1

=

3n

∑ Qk dXk ,

(18.1.15)

k =1

while the real elementary work of the constraint generalized forces is expressed in the form

dWk = Analogously,

n

3

∑ ∑ R(j i ) dx (j i )

i =1 j =1

=

3n

∑ Rk dXk

k =1

.

(18.1.15')


11 n

3

∑ ∑ Fj( i ) δX (j i )

δW =

i =1 j =1

=

3n

∑ Qk δXk

(18.1.16)

k =1

is the virtual work of the given generalized forces, while δWR =

n

3

∑ ∑ R(j i ) δX (j i )

i = 1 j =1

=

3n

∑ Rk δXk

(18.1.16')

k =1

represents the virtual work of the constraint generalized forces.

18.1.1.3 Natural Systems We have seen in Sects. 1.1.1.12 and in 6.1.1.2 that a system of forces (corresponding, e.g., to a field of forces), which admits a simple potential (or quasi-potential) or a generalized potential (or quasi-potential) is called a natural system. A system of forces Fi = ∇ iU = Fj( i ) i j , ∇ i = i j

∂ ∂x (j i )

, i = 1,2,..., n ,

the components of which (i )

Fj

=

∂U ∂x (j i )

, i = 1, 2,..., n , j = 1, 2, 3 ,

(18.1.17)

derive from a simple potential, forms a system of conservative forces; to a simple quasi-potential of the form U = U ( r1 , r2 ,..., rn ; t ) corresponds a system of quasiconservative forces. Analogously, if Fj( i ) =

∂U ∂x (j i )

−

d ∂U , i = 1, 2,..., n , j = 1, 2, 3 , dt ∂x ( i ) j

(18.1.17')

where U = U ( r1 , r2 ,..., rn , r1 , r2 ,..., rn )

is a generalized potential, then the system of forces is conservative; in case of the generalized quasi-potential U = U ( r1 , r2 ,..., rn , r1 , r2 ,..., rn ; t )

the system of forces is quasi-conservative. In the representative space E 3n , the system of generalized forces is of components Qk =

∂U , k = 1, 2,..., 3 n , ∂Xk

(18.1.18)


12

in case of a simple potential (system of conservative generalized forces) U = U ( X1 , X 2 ,..., Xn )

or of a simple quasi-potential (system of quasi-conservative generalized forces) U = U ( X1 , X 2 ,..., X n ; t )

or of components Qk = [U ]k ≡

∂U d ∂U − , k = 1, 2,..., 3n , ∂Xk dt ∂Xk

(18.1.18')

in case of a generalized potential (system of conservative generalized forces) U = U ( X1 , X 2 ,..., X n , X 1 , X 2 ,..., X n )

or in case of a generalized quasi-potential (system of quasi-conservative generalized forces) U = U ( X1 , X 2 ,..., Xn , X 1 , X 2 ,..., X n ; t ) ;

we have used the Euler-Lagrange derivative corresponding to the index k , given by [

]k

=

∂ d ∂ − , k = 1, 2,..., 3n . ∂Xk dt ∂Xk

(18.1.19)

Noting that the generalized forces do not depend explicitly on the generalized accelerations d2 Xk Xk = , k = 1, 2,..., 3n , dt 2

it results that the generalized potential (quasi-potential) is of the form U =

3n

∑U j X j j =1

+ U0 ,

where

U j = U j ( X1 , X 2 ,..., Xn ) is a vector potential, while U 0 = U 0 ( X1 , X 2 ,..., Xn )

(18.1.20)


13

is a scalar potential, or where

U j = U j ( X1 , X 2 ,..., Xn ; t ) is a vector quasi-potential, while U 0 = U 0 ( X1 , X 2 ,..., Xn ; t )

is a scalar quasi-potential in the space E 3 n . Replacing in (18.1.18'), we obtain Qk =

3n

⎛ ∂U j

∑ ⎜⎝ ∂Xk j =1

−

∂U k ∂X j

∂U 0 ⎞ ⎟X j − U k + ∂X , k = 1, 2,..., 3n , k ⎠

(18.1.21)

in case of the quasi-conservative generalized forces; if the forces are conservative, then we have U k = 0 . Replacing the vector potential U j by the vector potential Uj = Uj +

∂ϕ , ϕ = ϕ ( X1 , X 2 ,..., Xn ; t ) , ∂X j

and taking into account that one can invert the order of application of the operators

∂ d d ∂ = ∂Xk dt dt ∂Xk for a function of class C 2 , we get the same generalized function Qk ; hence, the vector quasi-potential is determined leaving aside a field of gradients in the space E 3n . If to this transformation of vector quasi-potential U j → U j we associate the transformation of scalar quasi-potential U 0 → U 0 = U 0 + ϕ , hence if we effect the gauge transformation (of scale) U →U =U +

dϕ , dt

then the form (18.1.21) of the quasi-conservative generalized force remains invariant. A field of conservative generalized forces is stationary, while a field of quasi-conservative generalized forces is non-stationary. The elementary work of the conservative generalized forces which derive from a simple potential is an exact differential dW = dU ,

(18.1.22)

the work depending only on the extreme positions of the representative point P Wq = U ( X11 , X 21 ,..., X 31n 0 1 P P

) − U ( X10 , X20 ,..., X 30n ) .

(18.1.22')


14

If the trajectory of the point P is a closed hypercurve C in E 3 n , then the corresponding work vanishes (WC = 0 ). If these generalized forces result from a generalized potential, then the elementary work is a total differential dW = dU 0

(18.1.22'')

too, the scalar potential U 0 playing the rôle of the simple potential in the frame of the generalized potential (18.1.20), while the vector potential has no contribution concerning this elementary work; all the considerations made in case of the simple potential remain thus valid. In case of quasi-conservative generalized forces, which derive from a simple quasi-potential, the elementary work is no more a total differential, being of the form dW = dU − Udt ,

(18.1.23)

where we put in evidence the partial derivatives with respect to time. If the generalized forces result from a generalized quasi-potential, then we get dW = dU 0 − U 0 dt − ⎛ 3n ⎞ = dU 0 − ⎜ ∑ U kVk + U 0 ⎟ dt = ⎝ k =1 ⎠

3n

∑Uk dXk

k =1 3n

∂U

∑ ⎛⎜⎝ ∂Xk0

k =1

⎞ − U k ⎟ dXk . ⎠

(18.1.23')

We see, easily, that the transformations mentioned above for the generalized potential have no influence on this elementary work. We notice that a system of conservative generalized forces derives always from a generalized potential or from a simple potential as it depends or not explicitly on the generalized velocities of the representative point P in E 3 n . Introducing the potential energy V = −U , in case of a simple potential, or the potential energy V = −U 0 , in case of a generalized potential, we can write Wq = V0 − V1 = − ΔV , V = − Wq + V0 , 0 1 0 1 P P

P P

(18.1.24)

where ΔV is the variation of the potential energy. Consequently, the potential energy of a discrete mechanical system, acted upon by conservative generalized forces (natural mechanical system), is equal to the work with changed sign, effected by the generalized forces, beginning with the initial moment (excepting an arbitrary constant V0 , which represents the potential energy at the initial moment; often, one chooses V0 so that the minimum of the potential energy be equal to zero). The generalized forces which do not derive from a simple quasi-potential can be expressed in the from

Qk =

∂U + Qk , U = U ( X1 , X2 ,..., Xn ; t ) , ∂Xk

the elementary work being given by

(18.1.25)


15 ⎛ 3n ⎞ dW = dU + ⎜ ∑ QkVk − U ⎟ dt . ⎝ k =1 ⎠

(18.1.25')

In the case in which U = 0 (U = U ( X1 , X 2 ,..., Xn ) is a potential), the quantity 3n

∑ QkVk ,

k =1

of the nature of a power, represents the power of the non-potential generalized forces. The non-potential generalized forces Qk of a vanishing power 3n

∑ QkVk

k =1

=0

are called gyroscopic generalized forces, depending – obviously – on the distribution of the generalized velocities. The generalized forces Qk are, in this case, conservative; if the power of the non-potential generalized forces is non-zero, then these forces are non-conservative. The non-conservative generalized forces of negative power 3n

∑ QkVk

k =1

0 , it results that the representative point P moves in the domain V < h . If we represent the arbitrary generalized forces in the form (18.1.25), then it results d (T + V ) =

3n

∑ Qk

k =1

⎛ 3n ⎞ dXk − Udt = ⎜ ∑ QkVk − U ⎟ dt , ⎝ k =1 ⎠

(18.1.50)


24

where V = −U corresponds to a simple quasi-potential; if U = U ( X1 , X 2 ,..., Xn ) is a simple potential, then we can write 3n d (T + V ) = ∑ QkVk . dt k =1

(18.1.50')

In the case of gyroscopic generalized forces we can write further a conservation theorem of mechanical energy (of the form (18.1.49)), while if the forces are dissipative, then the mechanical energy decreases.

18.1.2.2 Principle of Virtual Work. Lagrange’s Equations of the First Kind We have seen that a discrete mechanical system S subjected to ideal constraints verifies the relation (18.1.2); hence, a representative point P in the space E 3 n is subjected to ideal constraints if the relation (considered by P. Appell as relation of definition) δWR =

3n

∑ Rk δXk

k =1

=0

(18.1.51)

is verified. Starting from d’Alembert’s principle (18.1.42), effecting a scalar product by the virtual generalized displacements δXk , summing for all indices and taking into account the relation of definition (18.1.51) of the ideal constraints, we get the relation 3n

∑ Φk δXk =

k =1

3n

∑ (Qk

k =1

− M k Xk ) δXk = 0 ,

(18.1.52)

which represents a necessary condition to describe the motion of the representative point P . Assuming now that the condition (18.1.52) is fulfilled and that p holonomic constraints of the form (18.1.11) and m non-holonomic constraints of the form (18.1.7) take place, we will use the method of Lagrange’s multipliers; we can thus write ⎛ ∑ ⎜ Φk + k =1 ⎝ 3n

p

∑ λI

I =1

∂fI + ∂Xk

m

⎞

j =1

⎠

∑ μj bjk ⎟ δXk

= 0,

where λl , l = 1, 2,..., p , μj , j = 1, 2,..., m , are non-determinate scalars (Lagrange’s multipliers) and where we took into account that one can invert the order of summation, in a finite double sum. The p + m constraints are linear and distinct, the matrix of the coefficients ∂fl / ∂Xk and bjk being of rank p + m ; we express thus the virtual generalized displacements δX1 , δX 2 ,..., δX p + m (we can always choose them so that the determinant Δp + m of the respective coefficients in the constraints relations be non-zero) by means of the other 3n − ( p + m ) virtual generalized displacements, where the latter ones can be considered as independent. We put the conditions that the


25

expressions in the brackets which multiply the first p + m virtual generalized displacements do vanish, determining thus, univocally, p + m multipliers λl and μj (these multipliers are given by a system of p + m linear equations with p + m unknowns, of determinant Δp + m ≠ 0 ). The independent (and arbitrary) virtual generalized displacements δX p + m + 1 , δX p + m + 2 , ..., δX 3 n can be all taken equal to zero, expecting only one, let be δXk ; this

non-zero virtual generalized displacement being arbitrary, it results that the bracket by which it is multiplied must vanish too. Making, successively, k = p + m + 1 , k = p + m + 2 , ..., 3n , and taking into account the preceding considerations, it results that all the brackets which multiply the virtual generalized displacements must vanish. Finally, we can write

Φk +

p

∑ λI

I =1

∂fI + ∂X k

m

∑ μj bjk j =1

= 0, k = 1, 2,..., 3 n .

(18.1.53)

These equations are equivalent with the equations (18.1.42); in case of ideal constraints, we can express the constraint generalized forces in the form Rk =

p

∑ λI

I =1

∂fI + ∂Xk

m

∑ μj bjk , k j =1

= 1, 2,..., 3n ,

(18.1.54)

equivalent to (3.2.37). We can state (the relation (18.1.52) becomes a sufficient condition too) Theorem 18.1.4 (theorem of virtual work; d’Alembert-Lagrange). The motion of the representative point P , subjected to ideal constraints in the space E 3 n , takes place so that the virtual work of the lost generalized forces of d’Alembert, which act upon this point, vanishes for all the systems of virtual generalized displacements of the respective point. Taking into account the equivalence between the relation (18.1.52), which represents the form taken by Newton's equations, it results that the theorem of virtual work can be considered as being a principle (the principle of virtual work or the principle of virtual generalized displacements), because – starting from it – one can solve the fundamental problems of dynamics in case of ideal constraints. The equation (18.1.52) is called also the basic equation (the first form), while the equations (18.1.53) are known as Lagrange’s equations of the first kind. Introducing the virtual generalized velocities Vk∗ , we can write the conditions (18.1.52) in the form 3n

∑ ΦkVk∗

k =1

=0

(18.1.52')

too, the considered principle being thus called also the principle of virtual generalized velocities.


26

If some elements of the discrete mechanical system S are rigid solids, then one uses the formula (14.1.59) for the corresponding part of the virtual work. In the static case, the principle of virtual work becomes 3n

∑ Qk δXk

k =1

= 0,

(18.1.52'')

being stated for the first time in 1717 by Jean I Bernoulli. Let be a mechanical system S of n particles Pi , of weight mi g , i = 1, 2,..., n ; choosing a co-ordinate axis Ox along the descendent vertical, we can write the virtual work of these given forces in the form δW =

n

∑ mi

i =1

⎛ n ⎞ g δx i = δ ⎜ g ∑ mi x i ⎟ = δ ( M gξ ) = M gδξ = 0, ⎝ i =1 ⎠

where ξ is the applicate of the mass centre of the system S . We can thus state Theorem 18.1.5 (E. Torricelli). A mechanical system subjected to ideal constraints and to the action of its own weight is in equilibrium only and only for an extremum of the applicate of its centre of mass. Also this theorem can be considered to be a principle (Torricelli’s principle), which may be applied for a certain sphere of problems (in case of a uniform gravitational field). As we have seen in Sect. 4.1.1.7 (in case of a single particle), the position of equilibrium is stable, labile or indifferent as the applicate of the mass centre has a minimum, a maximum or is constant, respectively. Using the expression (18.1.54) of the constraint generalized forces and the constraint relations (18.1.6), (18.1.10), we can write the real elementary work (18.1.15') of the constraint generalized forces, in case of ideal constraints, in the form ⎛ p dWR = − ⎜ ∑ λl fl + ⎝ l =1

m

⎞

j =1

⎠

∑ μ j b j 0 ⎟ dt .

(18.1.55)

In case of catastatic constraints (we have fl = 0 , l = 1, 2,..., p , and bj 0 = 0 , l = 1, 2,..., m ), the real elementary work of the constraint generalized forces vanishes; indeed, in this case the real generalized displacements belong to the set of virtual generalized displacements, the relation (18.1.51) implying dWR = 0 . The conditions in which take place the relations (18.1.48) and (18.1.49) are thus entirely justified. Starting from the relations (18.1.6), we can write the constraint relations with the aid of the possible generalized displacements ΔX k , k = 1, 2,..., 3n , in the form 3n

∑ bjk ΔXk

k =1

+ bj 0 = 0, j = 1, 2,..., m .

Introducing the possible generalized velocities Vk =

ΔX k , k = 1, 2,..., 3 n , Δt

(18.1.56)


27

we can write 3n

∑ bjkVk

k =1

+ bj 0 = 0, j = 1, 2,..., m ,

(18.1.56')

too. Let be also other possible generalized velocities Vk + ΔVk , for the same position of the representative point P , at the same moment t , so that 3n

∑ bjk (Vk

k =1

+ ΔVk

) + bj 0

= 0, j = 1, 2,..., m .

Subtracting the last two relations one from the other, it results 3n

∑ bjk ΔVk

k =1

= 0, j = 1, 2,..., m .

(18.1.56'')

= 0, l = 1, 2,..., p .

(18.1.56''')

Analogously, we can write 3n

∂f

∑ ∂Xlk ΔVk

k =1

Hence, these finite variations of the possible generalized velocities verify the relations (18.1.7), (18.1.11) for virtual generalized velocities. One can thus write the basic equation (18.1.52) in the second form 3n

∑ Φk ΔVk

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) ΔVk = 0 ,

(18.1.57)

for a given position of the representative point P , at a fixed moment t ; this form of the equation plays an important rôle in case of the phenomenon of collision. If ΔVk are differential quantities, then these variations are virtual generalized velocities (differences of possible generalized velocities) and we find again the form (18.1.52') of the basic equation, as it has been shown by P. E. B. Jourdain, in 1908. The corresponding principle is know also as Jourdain’s principle, being expressed in the form 3n

∑ Φk δX k

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) ΔX k = 0 .

(18.1.57')

By a total differentiation of the constraint relation (18.1.56') with respect to time, we obtain the relations which must be verified by the possible generalized accelerations A = V , k = 1,2,..., 3n , in the form k

k


28 3n

⎡

∑ ⎢⎣bjk Ak

k =1

+

dbjk ⎤ dbjk Vk ⎥ + = 0, j = 1, 2,...m , dt dt ⎦

(18.1.58)

where d = dt

∂

3n

∑Vk ∂Xk

k =1

+

∂ . ∂t

(18.1.59)

We consider also the possible generalized accelerations Ak + ΔAk for the same position and the same velocity of the representative point P , at the same moment t ; as above, we obtain the relations 3n

∑ bjk ΔAk

k =1

= 0, j = 1, 2,...m .

(18.1.58')

Analogously, it results 3n

∂f

∑ ∂Xlk ΔAk

k =1

= 0, l = 1, 2,..., p .

(18.1.58'')

We see that the finite variations of the possible generalized accelerations verify the relations (18.1.7), (18.1.11), corresponding to the virtual generalized displacements. We are thus led to the third form of the basic equation 3n

∑ Φk ΔAk

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) ΔAk = 0 ,

(18.1.60)

for a given position and generalized velocity of the representative point P , at a fixed moment t . If ΔAk are differential quantities, as Gauss and Gibbs have considered, we can write this equation in the form 3n

∑ Φk δXk

k =1

=

3n

∑ (Qk

k =1

− M k Xk ) δXk = 0 ,

(18.1.60')

where these variations are virtual generalized accelerations (differences of possible generalized accelerations) .

18.1.2.3 Gauss’s Principle. Hertz’s Principle Let us suppose that a particle Pi , of mass mi , of the mechanical system S subjected to constraints is, at the moment t , at the position Pi of position vector ri , velocity vi and acceleration ai , i = 1, 2,..., n . If no force would act upon the particle Pi , then this one would effect the real displacement (Fig. 18.1)


29 JJJJJG Pi Pi ′ = vi Δt ,

(18.1.61)

during the time t , while if upon this one would act upon only the given force Fi , then JJJJJG the particle would effect the real displacement Pi Pi ′′ , so that the derivation of the particle Pi in this motion would be given by the formula (5.1.21) in the form JJJJJG 1 F i ( Δt )2 . Pi Pi ′′ = 2 mi

(18.1.62)

Fig. 18.1 Gauss’s principle. Deviations

Finally, if the particle Pi is acted upon both by the given and the constraint forces, JJJJJG then it effects the real displacement Pi Pi ′′′ , in the time interval Δt ; the corresponding deviation of the particle is JJJJJG 1 Pi Pi ′′′ = ai ( Δt )2 , 2

(18.1.62')

JJJJG collinear with the acceleration ai . Let be also a possible displacement PQ i i , different JJJJJG from Pi Pi ′′′ , effected in the same interval of time Δt . JJJJJJG JJJJJG We notice that Pi ′′ Pi ′′′ and Pi ′′Qi represent the deviation of the particle Pi from the

free motion, in the time interval Δt to its real motion (the particle subjected to JJJJJG JJJJJJG JJJJJG constraints). Starting from the relation Pi ′′Qi = Pi ′′ Pi ′′′ + Pi ′′′Qi , we can write n

∑ mi Pi ′′Qi

i =1

2

=

n

n

i =1

i =1

∑ mi Pi ′′ Pi ′′′ 2 + ∑ mi Pi ′′′Qi

2

n JJJJJJG JJJJJJG + 2 ∑ mi Pi ′′ Pi ′′′ ⋅ Pi ′′′ Qi . i =1

JJJJJG JJJJJG JJJJJJG We may also write Pi ′′′Qi = Pi Qi − Pi Pi ′′′ , hence as a difference between a JJJJJG possible and a real displacement (which is a possible displacement too); hence, Pi'''Qi


30

JJJJJJG JJJJJJG JJJJJJG is a virtual displacement δri . On the other hand, Pi ′′ Pi ′′′ = Pi ′ Pi ′′′ − Pi ′ Pi ′′ ; taking into

account (18.1.62), (18.1.62') and introducing the lost forces of d’Alembert of the form (18.1.59), it results n JJJJJJG JJJJJG 2 ∑ mi Pi ′′ Pi ′′′ ⋅ Pi ′′′Qi = − ( Δt )2 i =1

n

∑ ( Fi

− mi ai ) ⋅ δri = − ( Δt )2

i =1

n

∑ Φi

i =1

⋅ δri .

Using the theorem of virtual work (in case of bilateral constraints) in the form (11.1.63), we notice that this sum vanishes; if the constraints are unilateral, then the relation (11.1.63'') takes place, the above sum being positive. As a conclusion, this sum is non-negative in case of ideal constraints, so that n

JJJJJG

∑ mi Pi ′′Qi

i =1

2

≥

JJJJJJG

n

∑ mi Pi ′′ Pi ′′′ 2

,

i =1

the equality taking place only if Qi ≡ Pi ′′′ . Gauss takes as measure of the deviation of the particle Pi from its free motion, in the real or in a possible motion, the constraint Z i = 2mi

Pi ′′ Pi ′′′ 2 ( Δt ) 4

,

or the constraint Z i = 2mi

Pi ′′′Qi

2

( Δt )4

,

respectively; in this case Z ≤ Z, Z =

n

n

i =1

i =1

∑ Zi , Z = ∑ Zi .

(18.1.63)

Hence, among all the possible motions of the discrete mechanical system S , the real motion is that for which the constraint Z is minimal (any other possible constraint Z is greater); but we notice that for all possible displacement must correspond the same positions and the same velocities at the moment t , the accelerations being different. JJJJJJG Taking into account the expression of the displacement Pi ′′ Pi ′′′ , we can express the constraint Z in the form Z =

1 n 1 1 n 1 Φ ι2 = ∑ ( F − mi ai )2 , ∑ 2 i =1 mi 2 i =1 mi i

(18.1.64)


31

while for the representative point P we can write 1 3n 1 1 3n 1 Φ κ2 = ∑ ( Qk − Mk Xk ∑ 2 k =1 M k 2 k =1 M k

Z =

)2 .

(18.1.65)

From the condition of minimum of the constraint in the real motion it results δZ = 0 ; because only the accelerations vary, we get δZ =

3n

1

∑ M k Φk δΦk

k =1

3n

3n

k =1

k =1

= − ∑ Φk δXk = − ∑ ( Qk − M k Xk

) δXk

= 0,

(18.1.66)

equation which can be identified with the equation (18.1.60'). Starting from this equation and from the relations (18.1.58'), (18.1.58''), written in the form 3n

∑ bjk δXk

k =1 3n

∂f

= 0, j = 1, 2,..., m ,

∑ ∂Xlk δXk

k =1

= 0, l = 1, 2,..., p ,

(18.1.67) (18.1.67')

respectively, and introducing Lagrange’s multipliers – on the same way as in the previous subsection – we find the equations (18.1.42). We can thus state Theorem 18.1.6 (theorem of the least constraint; C. F. Gauss). The motion of the representative point P , subjected to ideal constraints in the space E 3 n , takes place so that the constraint Z of this point has a minimum for virtual variations of the generalized accelerations (the generalized co-ordinates and the generalized velocities are considered constant). We can also say that, in the same conditions concerning the position, the generalized velocities and the generalized accelerations, the first variation of the constraint Z of the representative point must vanish (the constraint Z is stationary along the trajectory of the representative point P ). We have seen that the demonstration of this theorem implies the theorem of virtual work, being thus equivalent to Newton’s principle. Also the theorem of the least constraint, stated by Gauss in 1829, can be considered as a principle (the principle of the least constraint), because – starting from it – one can find again Newton’s equations in case of ideal constraints. H. Hertz considers that the given generalized forces, which act upon the mechanical system S , can be replaced by given constraints between this system and neighbouring systems which act upon it; thus, he studies only mechanical systems which are not acted upon by given generalized forces ( Qk = 0 ). As well, to obtain a geometric interpretation, he assumes that the generalized masses M k are equal to unity (hence, M 1 = M 2 = ... = M 3 n ). Thus, it results 2

Z =

1 3 n 2 1 3 n ⎛ d2 Xk ⎞ Xk = ∑ ⎜ . ∑ 2 k =1 2 k =1 ⎝ dt 2 ⎟⎠

(18.1.68)


32

In case of a single particle, of co-ordinates x1 , x 2 , x 3 , we have (we can assume, in this case, that the mass m is not equal to unity) Z =

m 2 ( x + x22 + x32 2 1

)=

m 2 a , 2

(18.1.69)

where a is the acceleration in the considered motion; we obtain thus δZ = a δa . Using Frenet’s frame of reference, we can write a2 =

v4 + v 2 ; ρ2

this sum of squares is minimal if v = 0 (which takes place because the given force is equal to zero and one assumes that the constraints are catastatic, so that dWR = 0 , while the constraint force has no component along the tangent to the trajectory), hence if the motion is uniform and if the curvature 1/ ρ is minimal. Hence, the trajectory of the point must be of minimal curvature (as near as possible to a straight line). In case of a discrete mechanical system S , of representative point P , we can express the kinetic energy in the form (with the above considerations)

( )

2

T =

1 3n 2 1 3 n dX k ⎞ 1 ds Xk = ∑ ⎛⎜ ⎟ = ∑ 2 k =1 2 k = 1 ⎝ dt ⎠ 2 dt

2

=

1 2 s , 2

where ds 2 = dX12 + dX 22 + ... + dX 32n is the square of the arc element in the space E 3 n . Because there are not generalized forces and assuming that to constraints are catastatic ( dWR = 0 ), it results that T = const. As we can notice, 2 3 n dX 3 n dX d2 X dX k k ⎞ k k = 0, X k = s, ∑ ⎛⎜ ⎟ = 1, ∑ ds ds 2 k = 1 ⎝ ds ⎠ k = 1 ds

so that d2 Xk 2 dXk Xk = s + s ; ds ds 2

in the mentioned conditions, we have s = v = 0 , hence the motion of the representative point on its trajectory is uniform. We can thus replace the quantity Z by the quantity K2 =

d2 Xk = 2 k = 1 ds 3n

∑

n

∑

i =1

( ddrs ) i

2

=

n

∑ Ki2 ,

(18.1.70)

i =1

where we define by K the curvature of the path travelled through by the system (in a certain sense, a measure of the curvatures of the paths of all the particles Pi which form


33

the system S ); the minimum of the curvature K corresponds, obviously, to the minimum of the constraint Z . We can thus state Theorem 18.1.7 (theorem of the least curvature; theorem of the most straight path; H. Hetz). The motion of the discrete mechanical system S , subjected to ideal catastatic constraints, in the absence of the given generalized forces (in spontaneous motion) and assuming a mass of equal generalized components, takes place, starting from a given initial uniform motion, so that the curvature K of the path travelled through by the system be minimal, with respect to any other possible path (the path travelled trough be the most straight possible one). As in the preceding cases, we can consider this theorem as a principle (the principle of least curvature or the principle of the most straight path). We mention that, because of the conditions which are imposed, this principle is not so easy to apply; but the connections to Lagrangian mechanics and to relativistic mechanics are interesting. Starting from a constraint relation of the form ϕ j ( X1 , X 2 ,..., Xn , X 1 , X 2 ,..., X n ; t ) = 0, j = 1, 2,..., m ,

which is not necessarily linear in the generalized velocities, and effecting a total derivative with respect to time, we obtain 3n

⎛ ∂ϕj

∑ ⎜⎝ ∂Xk X k

k =1

+

∂ϕ j ⎞ Xk ⎟ + ϕ j = 0, j = 1, 2,..., m . ∂X k ⎠

Varying only the accelerations, we find the relations 3n

∂ϕ j

∑ ∂X

k =1

δXk = 0, j = 1, 2,..., m ,

(18.1.71)

k

which are of the same form as the relations (18.1.67) ( bjk = ∂ϕ j ∂X k ). We can thus state that Gauss’s principle can be applied even if the kinetic constraint relations are not linear in the generalized velocities, having thus a larger sphere of applicability than the principle of virtual work (for which the mentioned relations must be linear in the generalized velocities). We remark that Jourdain’s principle has a position somewhat intermediary between the d’Alembert-Lagrange principle and Gauss's principle.

18.1.2.4 Applications Let be a particle P subjected to move on a deformable surface of equation f (x1 , x 2 , x 3 ; t ) = 0 (we use the notations in the space E 3 ); the constraint relation can be written in the form f, j δx j = 0 . If we use the principle of virtual work in the form

Φj δx j = 0 and if we eliminate, e.g., the virtual displacement δx 3 , then we get

(18.1.72)

34


f,1 f,2 ⎛ ⎞ ⎛ ⎞ ⎜ Φ1 − f Φ3 ⎟ δx1 + ⎜ Φ2 − f Φ3 ⎟ δx 2 = 0 . ,3 ,3 ⎝ ⎠ ⎝ ⎠

To satisfy this relation for any virtual displacement δx1 and δx 2 , the two brackets must vanish, hence f,1 f,2 f,3 = = . Φ1 Φ2 Φ3

(18.1.73)

This result corresponds to that obtained in Sect. 6.2.2.2 by the method of Lagrange’s multipliers.

Fig. 18.2 Motion of a particle on a fixed or movable or deformable sphere

In particular, let be a particle P in motion on a sphere S , fixed or movable and deformable ( r ′ − rO′ )2 = l 2 , of centre rO′ = rO′ (t ) and radius l = l (t ) (Fig. 18.2). Noting that grad[( r ′ − rO′ )2 − l 2 ] = 2( r ′ − rO′ ) , Lagrange’s equation of the first kind is of the form r ′ = λ ( r ′ − rO′ ) , λ indeterminate scalar (we assume that the given force is equal to zero); shifting the origin to the centre of the sphere, we can write the equation of motion in the form r = λr − rO′ , r = r ′ − rO′ , too, with the constraint r 2 = l 2 , putting thus in evidence the complementary force of transportation −rO′ of the movable frame of reference (in general, non-inertial). By a vector product by r and by a scalar product by r , respectively, we get d d 2 d ( r × r ) = rO′ × r , r = λ r2 − 2 rO′ ⋅ r . dt dt dt

If rO′ = 0 (hence, if the movable frame is inertial, the centre of the sphere having a JJJJJG rectilinear and uniform motion), then we have r × r = C , C = const , wherefrom C ⋅ r = 0 and dr 2 dt = 2λ ll ; hence, the trajectory is a great circle of the deformable sphere (the intersection of the plane C ⋅ r = 0 with the sphere). In particular, if the radius l of the sphere is constant, then the trajectory is a geodesic of it. Let now be a particle P which moves on a deformable curve of equations fk (x1 , x 2 , x 3 ; t ) = 0 , k = 1, 2 (in the space E 3 ); the constraint relations can be written


35

in the form fk , j δx j = 0 , k = 1,2 . Associating the condition (18.1.72) and eliminating the virtual displacements, we obtain Φ1

Φ3

Φ3

f1,1

f1,2

f1,3 = 0 ,

f2,1

f2,2

f2,3

(18.1.74)

relation which – together with the constraint relations – allows to determine the motion of the particle (see Sect. 6.2.2.1 too, where Lagrange’s multipliers are used).

Fig. 18.3 Motion of a particle on a movable circle of variable radius

Let us study, in particular, the motion of a particle P on a circle C , movable and of variable radius, represented by the intersection of the above considered sphere S with the plane α ⋅ ( r ′ − rO′ ) = 0 , α = α (t ) , which passes through rO′ and is normal to α (Fig. 18.3). As in the preceding case, in the absence of the given force, we can write the equation of motion in the form r = λ1 r + λ2 α − rO′ , λ1 , λ2 indeterminate scalars, r = r ′ − rO′ , with the constraint relations r 2 = l 2 , α ⋅ r = 0 . By a vector product by r and successive scalar products by r and α , respectively, we get

d ( r × r ) = λ2 r × α + rO′ , dt d 2 d r = λ1 r 2 − 2 rO′ ⋅ r + 2λ2 α ⋅ r , dt dt α ⋅ r = λ2 α2 − α ⋅ rO' . JJJJJG If α = const (hence, if the circle C is in a plane of fixed direction), then we have α ⋅ r = 0 , α ⋅ r = 0 too (according to the constraint relations). Thus, it results

λ2 = α ⋅

rO′ , α2

hence the component λ2 α normal to the plane α ⋅ r = 0 , of the constraint force (the component contained in the plane is λ1 r ). If rO′ = 0 (the movable frame of reference


36

is inertial), then the constraint force is contained in the plane of the circle, which has a fixed orientation. Let be two particles P1 and P2 of masses m1 and m2 and position vectors r1 and r2 , upon which act the given forces F1 and F2 , respectively; we assume a constraint relation at distance of the form ( r1 − r2 )2 = l 2 , l = l (t ) , which can be written as ( r1 − r2 ) ⋅ δr1 − ( r1 − r2 ) ⋅ δr2 = 0 . The principle of virtual work is written in the form Φ1 ⋅ δr1 + Φ2 ⋅ δr2 = 0 , being thus led to Lagrange’s equations of the first kind Φ1 + λ ( r1 − r2 ) = 0, Φ2 − λ ( r1 − r2 ) = 0 .

(18.1.75)

We obtain thus Φ1 + Φ2 = 0 . Assuming that F1 = m1 g + F12 , F2 = m2 g + F21 , g = g (t ) , where F12 and F21 are internal forces ( F12 + F21 = 0 ), it results m1 r1 + m2 r2 = (m1 + m2 )g , wherefrom t

τ

0

0

m1 r1 + m2 r2 = ( m1 + m2 ) ∫ dτ ∫ g ( τ )dτ + C1t + C2 , JJJJJG with C1 , C2 = const ; hence, the centre of mass C has a motion of acceleration g (t ) , being situated on the segment of a line P1P2 , so that

CP1 =

m2 l m1l , CP2 = , l = P1P2 . m1 + m2 m1 + m2

The particles P1 and P2 will be at any moment on two concentric spheres of radii CP1 and CP2 , respectively (hence, remains to be studied the problem of motion of a particle on a given sphere). In the particular case of two heavy particles, the mass centre C will describe a parabola situated in a vertical plane and having a vertical axis; the particles will be in a plane of fixed orientation, their motion with respect to the centre C taking place after the law of areas. The analytic methods of computation have been developed, especially, in case of discrete mechanical systems; however, they can be applied successfully also to problems concerning continuous mechanical systems. We consider thus a perfectly flexible, inextensible and non-torsionable thread, of given length l , fixed at the points P 0 and P 1 , acted upon by forces p (s ) on unit length. The principle of virtual work is written in the form ( Φ ⋅ δr ) ds ∫Pq P 1

2

= 0, Φ = p − μ a ,

(18.1.76)

where μ (s ) is the linear density, while a(s ) is the acceleration of an arbitrary point P . The relation ds 2 = (dr )2 leads to ds 2 δ(ds ) = dr2 ⋅ δ(dr ) ; because the thread is inextensible, we have δ(ds ) = 0 , so that the constraint relation (which takes place for all the elements of arc) is written in the form


37 dr ⋅ δ ( dr ) = 0 . ds

Starting from the points P of the curve C and considering a given system of virtual displacements (at any point P corresponds a virtual displacement δr at the moment t ), we obtain the points P , which form the curve C . If P ( r ) and P ′( r + dr ) are two neighbouring points on the curve C and P ( r + δr ) is a corresponding point on the JJJG JJJJG curve C , one can reach the point P ′ on the path PP + PP ′ or on the path JJJJG JJJJG PP ′ + P ′P ′ (Fig. 18.4); hence, the point P ′ can be specified by r + δr + d( r + δr ) = r + dr + δ( r + dr ) ,

wherefrom d( δr ) = δ(dr ) , so that the two operators are linked by the relation dδ = δd .

(18.1.77)

Obviously, the considerations made above involve, at the most, holonomic constraints; the relation of permutation (18.1.77) takes place just in such a case.

Fig. 18.4 Displacements of a particle laying on a curve

The constraint relation becomes dr ⋅ d ( δr ) = 0 , ds

so that Lagrange’s equation of the first kind will be ⎡

dr

⎤

Φ ⋅ δrds + λ ⋅ d ( δr ) = 0 . ∫Pq ⎥⎦ P ⎢ ds ⎣ 0

Integrating by parts

1


38 dr

dr

⋅ d ( δr ) = λ λ ∫Pq P ds ds 0

1

⋅ δr

P1 P0

−

δr ⋅ d ( ∫Pq P 0

1

λ dr ds

)

and noting that δr vanishes at the fixed points P 0 and P 1 , it results Φd s + d ( ∫Pq P ⎢ ⎣

⎡

0

1

)

λ dr ⎤ ⋅ δr = 0 . ds ⎥⎦

Because this relation must take place for any virtual displacement δr , we obtain Φd s − λ

dr = 0; ds

introducing the tension in the thread T =T

dr ,T = − λ , ds

we can write Φ+

dT = 0, ds

(18.1.78)

finding thus again the equation (12.2.10) of motion of the thread. We present now some simple applications too, eventually with a practical character.

Fig. 18.5 Motion of a coach on a railway in a curved line

Let use consider a railway which, on a certain length, follows a curve of curvature radius ρ . D’Alembert’s principle, written for the mass centre C of a coach, in the normal plane to the curve (Fig. 18.5), leads to G + N + Ft = 0 , where G is the weight of the coach, N is the normal reaction of the ground, while Ft is the transportation force of a non-inertial frame of reference (the centrifugal force), of magnitude


39 mv 2 . ρ

Ft =

To can verify the equation of dynamic equilibrium, one must give to the ground an inclination of angle α , so that N sin α − Ft = 0 , N cos α − G = 0 , whence tan α =

Ft v2 = ; G ρg

the exterior rail must be superelevate to a height h = l sin α ≅ l tan α ( l is the clearance of the railway), so that h ≅

v 2l , gρ

(18.1.79)

at an arbitrary point of the curve.

Fig. 18.6 Belidor’s crane bridge

Let be now a rigid bar OA of weight G , which is rotating in a vertical plane, about a fixed point O . A perfectly flexible, inextensible and non-torsionable thread, which passes over a fixed pulley B , situated on the vertical line of O , connects the point A to a point P , where is suspended a weight Q . One must find the locus of the point P (a curve Γ ), so that the bar AO be in indifferent equilibrium (Fig. 18.6). The bar AO models Belidor’s crane bridge. We denote n = θ , OBP n =ϕ ; OA = 2l , AB = a , BP = r , OB = h , h > 0, BOA

we choose OB as Ox 2 -axis, the Ox1 -axis being horizontal, so that x C2 = l cos θ ,

x 2P = h − r cos ϕ . The constraint relations are a + r = L , r = r (ϕ ) , where L is the length of the thread. The mechanical system has only one degree of freedom and its


40

position can be specified by the angle ϕ . The principle of virtual work is written in the form −G δx C2 − Q δx 2P = 0 . We have δa + δr = 0 , δr = r ′(ϕ ) δϕ , a 2 = 4l 2 + h 2 − 4lh cos θ ,

hence a δa = 2lh sin θ δθ ; in this case δx C2 = −l sin θ δθ = −

a a a δa = δr = r ′(ϕ )δϕ . 2h 2h 2h

On the other hand,

δx 2P = − cos ϕ δr + r sin ϕ δϕ = [ r (ϕ ) sin ϕ − r ′(ϕ ) cos ϕ ] δϕ . We obtain thus the condition of equilibrium −

Ga r ′(ϕ ) + Q [ r ′(ϕ ) cos ϕ − r (ϕ ) sin ϕ ] = 0 , 2h

which determines the corresponding angles ϕ . We get thus the differential equation

{

}

G [ L − r (ϕ )] − Q cos ϕ r ′(ϕ ) + Qr (ϕ ) sin ϕ = 0 . 2h

Fig. 18.7 Equilibrium of three particles acted upon by three forces in a particular case

By integration, we obtain G 2 PL r (ϕ ) − r (ϕ ) + Qr (ϕ ) cos ϕ = const. 4h 2h

Hence, the equation of the curve r is of the form


41

r2 − ( α + β cos ϕ )r + γ = 0, α =

2 Pl 4Qh , γ = arbitrary, ,β = − G G

(18.1.80)

representing – in general – ovals of Descartes; for γ = 0 it results a Pascal’s limaçon. C. Neumann considered three particles P1 , P2 , P3 , linked so that the area of the triangle P1P2 P3 be constant, and upon which act the given forces F1 , F2 , F3 , respectively (Fig. 18.7). If r1 , r2 , r3 are position vectors of the particles P1 , P2 and P3 , respectively, and if we denote JJJJJG rjk = Pj Pk = rk − rj , j , k = 1, 2, 3, r12 × r13 = C u ,

where u = u(t ) is a vector normal to the plane P1P2 P3 , while C = const , then the constraint relation is of the form ( r12 × r13 )2 = C 2 = const . Taking into account the properties of the mixed product of these vectors, we notice that the constraint relation can be written in the form

[ ( r12 × r13 ) × r23 ] ⋅ δr1 + [ ( r12 × r13 ) × r31 ] ⋅ δr2 + [ ( r12 × r13 ) × r12 ] ⋅ δr3 = 0 ; associating the principle of virtual work, given by F1 ⋅ δr1 + F2 ⋅ δr2 + F3 ⋅ δr3 = 0 ,

and introducing Lagrange’s multiplier λ , we can write the equations of equilibrium (corresponding to Lagrange’s equations of the first kind) in the form F1 = − λC u × r23 , F2 = − λC u × r31 , F3 = − λC u × r12 ,

(18.1.81)

where we have used the notation introduced above too. It results that, for the equilibrium, the forces F1 , F2 and F3 must be in a plane normal to the unit vector u , hence in the plane of the triangle P1P2 P3 ; they must be normal to the sides opposite to the vertices at which they act (hence, their supports must pierce at the orthocentre H of the triangle, while their magnitudes must be in direct proportion to these sides). As a simple application of Gauss's principle, let us consider Atwood’s machine (see Sect. 11.1.2.2 and Fig 11.1), where the weights G1 and G 2 are modeled as particles and where we neglect the influence of the wheel of weight G . The constraint will be Z =

1⎡ 1 1 ( m1g − m1a )2 + ( m2 g − m2a )2 ⎤⎥ , ⎢ 2 ⎣ m1 m2 ⎦

where we assume that m1 > m2 ; in this case δZ = [ − m1 (g − a ) + m2 (g + a ) ] δa = 0 ,


42 being thus led to a =

m1 − m2 g, m1 + m2

(18.1.82)

a formula which corresponds to (11.1.21).

18.1.2.5 General Theorems. Conservation Theorems Let be discrete a mechanical system S of n particles Pi , acted upon by the given forces Fi , i = 1, 2,..., n ; assuming that the constraints allow an arbitrary virtual rigid displacement (with respect to an inertial frame of reference) δri′ = vO′∗ δt + ( ω∗ δt ) × ri , ri = ri′ − rO′ , i = 1, 2,..., n ,

(18.1.83)

where O is the pole of a non-inertial frame, rigidly linked to the mechanical system S in its rigid displacement (corresponding to the formula (18.1.3)). The principle of virtual work is thus written in the form n

∑ Φi

i =1

⋅ [ vO′∗ δt + ( ω∗ δt ) × ri ] = 0 .

Because the vectors vO′∗ and ω∗ are arbitrary, it results n

∑ Φi

i =1

n

= 0, ∑ ri′ × Φi = 0 , i =1

(18.1.84)

where we took into account the second relation (18.1.83). We can also write τO ′ { Φi } = 0 ,

(18.1.84')

the torsor of d’Alembert’s lost forces, with respect to the pole of an inertial frame of reference, vanishing. Starting from this result, we find again the general theorems of mechanics (the theorem of momentum and the theorem of moment of momentum). Comparing with the relation (11.1.61), written with respect to the same pole O ′ , where the lost forces of d’Alembert equilibrate the constraint forces, we see that in this new form of the theorem of torsor do not appear the constraint forces; this happens because the relation (11.1.61) can be written for any mechanical system, while the relation (18.1.84') takes place only for the mechanical systems which admit an arbitrary rigid displacement. In case a free rigid solid which, obviously, admits an arbitrary rigid displacement, the theorem of torsor becomes (V is the volume of the rigid solid, μ is the density, while F is a volumic force, which – eventually – can be reduced to a superficial one)


43

{

τO ' Φ dV

} = 0, Φ =

F − μ r ,

(18.1.85)

wherefrom

∫V ( F − μ r ) dV

= 0,

∫V r × ( F − μ r ) dV

= 0,

(18.1.85')

resulting the theorem of momentum and the theorem of moment of momentum, respectively. If the solid would have a fixed point, we could admit that this one is just the point O , so that rO′ = 0 and vO′∗ = 0 ; we obtain

∫V Φ ⋅ ⎡⎣ ( ω

∗

)

δt × r ⎤⎦ dV =

∫V ( r × Φ ) ⋅ ( ω δt ) dV ∗

= 0.

The virtual angular velocity ω∗ being arbitrary, it results

∫V r × Φ dV

= 0,

(18.1.86)

which leads to the theorem of moment of momentum. In the case in which the rigid solid has two fixed points, hence a fixed axis, we can take this axis as the O ′x 3′ -axis, O ′ ≡ O , after the vector ω∗ ; in projection on the axis O ′x 3′ ≡ Ox 3 , we can write

∫V ( x1 Φ2

− x 2 Φ1 ) ( ω ∗ δt ) dV = 0 ,

∫V ( x1 Φ2

− x 2 Φ1 ) dV = 0 ,

wherefrom (18.1.87)

being thus led to the projection of the theorem of moment of momentum on a fixed axis. If, for a discrete mechanical system S , the constraints admit as virtual displacement a translation of constant direction, specified by the constant unit vector u and of arbitrary magnitude δr ′ = k u δt , vO′∗ = k u , and if ω∗ = 0 , then we get

∫V Φ ⋅ u dV

= 0,

(18.1.88)

hence the theorem of momentum projected on the direction of unit vector u ; this allows to write a scalar first integral, component of a conservation theorem of momentum (corresponding to the theorem expressed by the first formula (18.1.84)). As well, if the mechanical system S has constraints which allow a rotation about an axis of fixed direction, specified by the unit vector u , then we can write δr ′ = ( ω∗ uδt ) × r , vO′ = 0 ; we obtain thus (if we take the point O on the fixed axis, then we have rO′ = 0 , O ≡ O ′ , hence r ′ = r )

44


∫V ( r ′ × Φ ) ⋅ udV

= 0,

(18.1.89)

that is the theorem of moment of momentum with respect to the pole O ′ , projected on the fixed axis of unit vector u . We obtain thus a scalar first integral too, component of a first integral of moment of momentum (corresponding to the theorem expressed by the second formula (18.1.84)). As in case of the theorem (18.1.84), in these results intervene only given forces; in exchange, they take place only for the mentioned particular motions of the rigid solid. One can make analogous considerations concerning the theorem of kinetic energy for the conservation of the mechanical energy, assuming that the constraints of the mechanical system S are catastatic.

18.1.2.6 Influence of the Forces of Friction Let us consider a particle P subjected to the action of a given force F and to holonomic constraints, which lead to a constraint force R . In case of a single holonomic an rheonomous constraint ϕ (x1 , x 2 , x 3 ; t ) = 0 , we can write R = N + T , where N is the normal constraint force and T is the tangential constraint force. If the constraint would be scleronomic, hence the force T would correspond to the sliding friction on the fixed rigid surface S (of equation ϕ ( x1 , x 2 , x 3 ) = 0 ). We can consider, e.g., a hydrodynamic friction, of the form T = − k v , k = k ( r , r ; t ) > 0 , which leads to the replacement of the given force F = F ( r , r ; t ) by a new force F = F − k v , the problem remaining the same from a mathematical point of view. In case of a Coulombian friction, of the form T = fN v / v , f = f ( r , r ; t ) , the problem is put in a different from, due to the unknown factor N , corresponding to the normal constraint force; we remain now to this second case. We assume a constraint ϕ ( r ; t ) = 0 , which may correspond to a family of rigid

surfaces St in motion, depending on the parameter t . The velocity of the particle is

v = vt + vr , where vt is the velocity of transportation and vr is the sliding velocity on the surface St (on the nature of a relative velocity); in this case, in the expression of the constraint force T , the velocity v is replaced by the velocity vr . The constraints are no more ideal, so that the condition of ideal constraint takes no more place; this condition is replaced by an analogous condition of the form R × W = 0,

(18.1.90)

where the vector W plays the rôle of the virtual displacement δr . The principle of virtual work is thus replaced by Φ⋅W = 0.

(18.1.91)

Noting that R = ± N n − fN τ = N ( ± n − f τ ) , where τ and n are unit vectors in the frame of Darboux, we can write the condition (18.1.90) also in the form


45 (n ± fτ) ⋅ W = 0 ,

(18.1.90')

neglecting the scalar N ≠ 0 . To express the unit vectors τ and n as functions of the data of the problem, we use the curvilinear co-ordinates u and v on the surface St so that v = ru′u + rv′v + r ; putting in evidence the partial derivatives with respect to these co-ordinates, it results vr = ru′u + rv′v and then τ = vers vr . As well, n = vers gradϕ = vers ( ru′ × rv′ )

(see Sect. 4.1.1.4 too). Eliminating W between the relation (18.1.90'), (18.1.91), which do not contain the unknown scalar N , by means of Lagrange’s multiplier λ , we obtain Φ + λ(n ± fτ) = 0.

(18.1.92)

If we associate also the scalar constraint relation ϕ = 0 to this vector equation, then we can determine the unknowns r = r (t ) and λ = λ (t ) . In case of two holonomic constraints ϕk ( r ; t ) = 0 , k = 1, 2 , which can correspond to a family of rigid curves C t in motion, depending on the parameter t , we proceed as above and obtain the equations Φ ⋅ W = 0, ( nk ± f τ ) ⋅ W = 0, k = 1, 2 .

(18.1.93)

We are thus led to Lagrange’s equation of the first kind Φ + λ1 ( n1 ± f τ ) + λ2 ( n2 ± f τ ) = 0 .

(18.1.94)

In the scleronomic case, this equation – to which we associate the constraint relations – is sufficient to solve the problem. In the rheonomous case, we use a representation in curvilinear co-ordinates r = r (q ; t ) (see Sect. 4.1.1.3 too) and representations r = r (q , qk ; t ) , k = 1, 2 , of the surface, so that the position of an arbitrary point of these ones be specified by the same values of the parameters q and qk , no matter its position in space. We have nk = vers ( rq′ × rq′k ) and τ = vers vr = q vers rq′ , this being the unit vector of the tangent to the curve C t ; in this case too, the problem is entirely determined.

18.2 Lagrange’s Equations A decisive step ahead has been made by Lagrange, by eliminating the holonomic constraints from the computation and by introducing independent generalized co-ordinates, in case of mechanical systems subjected only to such constraints; thus one passes from the representative space E 3n to a new representative space Λs , called the


46

space of configurations. The equations of motion of the representative point are thus Lagrange’s equations of second kind, which will be studied in this paragraph.

18.2.1 Space of Configurations By introducing the Lagrangian generalized co-ordinates, in a minimal parameterization, one passes from the space E 3 (or from the representative space E 3n ) to the space of configurations Λs . In what follows, we put in evidence the form taken by various mechanical quantities (displacement, velocity, acceleration, kinetic energy, work etc.), obtaining a new formulation of the fundamental problem of mechanics in the new representative space. 18.2.1.1 Generalized Co-ordinates. Representative Space Λs Let us consider a discrete mechanical system S of n a particles Pi , i = 1, 2,..., n , subjected to p holonomic constraints of the form (3.2.8), the corresponding representative point P in the space E 3n being subjected to p constraints of the form (18.1.9). In this case, the system S has 3n − p = s degrees of geometric freedom, its position being thus specified, at a moment t , by means of s independent parameters. To have motion, it is necessary that s ≥ 1 ; if s = 1 , then we say that the mechanical system is with complete constraints. Let be q1 , q2 ,..., qs a set of such parameters, which will be called Lagrangian generalized co-ordinates (shortly, generalized co-ordinates). The position of the mechanical system S will be thus given by relations of the form ri = ri (q1 , q2 ,..., qs ; t ) ≡ ri (q j ; t ), i = 1, 2,..., n ;

(18.2.1)

correspondingly, the position of the representative point P in the space E 3n is specified by Xk ≡ Xk (q j ; t ), k = 1,2,..., 3n .

(18.2.2)

Obviously, the relations (18.2.1) verify identically the constraint relations (3.2.8), while the relations (18.2.2) verify identically the constraint relations (18.1.9). If the constraints are scleronomic, hence of the form (3.2.33'), then the relations (18.2.1) take the form ri = ri (q1 , q1 ,..., qs ), i = 1, 2,..., n .

(18.2.1')

As well, for constraints of the form (18.1.12), the relations (18.2.2) become Xk ≡ X k (q1 , q2 ,..., qs ), k = 1, 2,..., 3n .

(18.2.2')

Indeed, if for the relations (18.1.12), e.g., we assume that det ⎡⎣ ∂fl ∂Xk ⎤⎦ ≠ 0 , l , k = 1, 2,..., p (the matrix ⎡⎣ ∂fl ∂Xk ⎤⎦ is of rank p ), then it results, from the theorem of implicit functions, that Xk ≡ Xk ( X p + 1 , X p + 2 ,..., X 3 n ), k = 1, 2,..., p . We take


47

X p + 1 = q1 , X p + 2 = q2 ,..., X 3 n = qs , s = 3n − p ; one obtains thus relations of the

form (18.2.2'). Obviously, in case of relations of the form (18.2.1') or of the form (18.2.2') one can have only holonomic and scleronomic constraints. In particular, if no one holonomic constraint takes place (the mechanical system S is free) or if we ignore these constraints, we can take Xk = qk , k = 1, 2,..., 3n , corresponding to the representative space E 3n . But if we take into account all p holonomic constraints, the mechanical system S having only s = 3n − p degrees of freedom, then the parameterization qk , k = 1,2,..., s considered above is minimal. Otherwise, if we do not take into account all the holonomic constraints or if, during the motion, appear supplementary holonomic constraints of the form fl (qk ; t ) ≡ fl (q1 , q2 ,..., qs ; t ) = 0, l = 1,2,..., h ,

(18.2.3)

then the mechanical system S remains with s − h degrees of freedom. Obviously, these constraints are independent too, so that the matrix ⎡⎣ ∂fl ∂q j ⎤⎦ is of rank h ; as well, we suppose that no one of these constraints is a consequence of the equations of motion (it is not a first integral of these equations). The constraints of the form (18.2.3) are rheonomous; we can consider, in particular, also scleronomic constraints of the form fl (q j ) ≡ fl (q1 , q2 ,..., qs ), l = 1,2,..., p .

(18.2.3')

We mention that one can have also unilateral constraints, expressed by means of inequalities; but, in what follows, we will consider only bilateral constraints. The space of generalized co-ordinates q1 , q2 ,..., qs is, as well, a representative space with s dimensions, called the space of configurations (figurative space or Lagrange’s space), which will be denoted by Λs (in the honour of Lagrange). A point P (q1 , q2 ,..., qs ) is a representative point, its position in the space Λs specifying univocally the position of the representative point P in the space E 3 n , as well as the position of the discrete mechanical system S in the space E 3 ; obviously, we must assume that the matrix ⎡⎣ ∂Xk ∂q j ⎤⎦ is of rank s , so that – on the basis of the theorem of implicit functions – to a given discrete mechanical system S corresponds only one representative point P (we notice that one can have several non-zero determinants of order s , so that we can express the minimal parameterization q j in various modes, depending on the generalized co-ordinates Xk ; if one takes into account the constraint relations (18.1.9) too, one sees that the respective expression are equivalent). The succession of value taken by the generalized co-ordinates during time leads to the functions qk = qk (t ), t ∈ [t0 , t1 ], k = 1,2,..., s , which represent the parametric equations of the trajectory of the representative point P , characterizing – at the same time – the motion of the mechanical system S .


48

In case of a particle constrained to stay on a curve one uses only one generalized co-ordinates q ( x i = x i (q , t ), i = 1,2, 3 ; see Sect. 6.2.2.1) in the space Λ1 ; e.g., in case of the simple pendulum this co-ordinate is the angle at the centre θ (see Sect. 7.1.3.1). In case of a particle constrained to stay on a surface we use two generalized co-ordinates ( r = r (q1 , q2 ; t ) ; see Sect. 6.2.2.2) in the space Λ2 ; e.g., in case of the spherical pendulum these co-ordinates are the applicate z and the angle at the centre θ (see Sect. 7.1.3.7). We notice that the particles of the mechanical system S can be replaced by rigid solids. E.g., in case of a physical pendulum we use also a generalized co-ordinate (the angle θ ; see Sect. 14.2.1.2), in the space Λ1 , while in case of a double pendulum one introduces two generalized co-ordinates (the angles θ1 and θ2 ; see Sect. 17.1.1.2), in the space Λ2 . As it can be seen, the generalized co-ordinates are not necessarily lengths; they can be angles too. Moreover, in the frame of the same Lagrangian space, some co-ordinates can be lengths, the other ones can be angles or even other quantities; it is not necessary that all generalized co-ordinates have all the same physical dimension. Practically, one chooses as generalized co-ordinates the parameters which can specify the configuration (position) of the discrete mechanical system S at any moment t . 18.2.1.2 Generalized Displacements. Generalized Velocities. Constraints. Generalized Accelerations Starting from the relation (18.2.1), we can express the real displacements in the form (we denote ri = ∂ri ∂t ) dri =

∂ri dq j + ri dt , i = 1, 2,..., n , ∂q j

(18.2.4)

putting thus in evidence the real generalized displacements dq j . Here and in what follows we use the convention of summation of dummy indices (from 1 to s ) in the space Λs ; analogously, we express the possible displacements in the form Δri =

∂ri Δq j + ri Δt , i = 1, 2,..., n , ∂q j

(18.2.5)

the virtual displacements being given by δri =

∂ri δq , i = 1, 2,..., n . ∂q j j

(18.2.6)

One introduces thus the possible generalized displacements Δq j and the virtual generalized displacements δq j , which have the same properties as those corresponding to the spaces E 3 and E 3 n . E.g., the virtual generalized displacements are differences of possible generalized displacements (as one sees from (18.2.5), (18.2.6)); as well, the


49

real generalized displacements belong to the set of possible generalized displacements, which – in general – does not coincide with the set of virtual generalized displacements. Let be two virtual displacements δ1 ri and δ2 ri of the same particle Pi , written in the form δ1 ri =

∂ri ∂r δ1q j , δ2 ri = i δ2q j , ∂q j ∂q j

where δ1q j and δ2q j are corresponding virtual generalized displacements; we may write δ1 ri + δ2 ri =

∂ri ( δ q + δ2q j ∂q j 1 j

)

too. Hence, by composition of two virtual displacements one obtains a new virtual displacement (the same property for the virtual generalized displacements). Let be the virtual generalized displacements δq j ; if −δq j is, as well, a virtual generalized displacement of the representative point P at a given moment t , then the displacement δq j is a reversible virtual generalized displacements. From the relations (18.2.6), it results that to reversible virtual displacements correspond reversible virtual generalized displacements. The velocities of the particles of the mechanical system S will be given by (we denote q j = dq j dt ) vi =

∂ri q + ri , i = 1, 2,..., n , ∂q j j

(18.2.4')

putting in evidence the real generalized velocities q j (shortly, the generalized velocities). As well, we express the possible velocities in the form (we denote q j = Δq j Δt ) vi =

∂ri q + ri , i = 1, 2,..., n , ∂q j j

(18.2.5')

the virtual displacements being given by (we denote q∗j = δq j Δt ) vi∗ =

∂ri ∗ q , i = 1, 2,..., n . ∂q j j

(18.2.6')

We introduce thus the possible generalized velocities q j and the virtual generalized velocities q∗j . We mention that also these quantities can have different dimensions (can be linear velocities, angular velocities etc.).


50

Using the generalized displacements introduced above, we can express the constraint relation (18.2.3) in one of the forms ∂fl dq j + fl dt = 0, l = 1, 2,..., p , ∂q j

(18.2.7)

∂fl δq = 0, l = 1, 2,..., p . ∂q j j

(18.2.8)

(

If fl = 0 , then it results ∂ ∂fl ∂q j

)

∂t = ∂fl ∂q j = 0, j = 1, 2,..., s , too, so that the

constraints are scleronomous; in this case, the set of virtual generalized displacements coincides with the set of possible generalized displacements, while the real generalized displacements belong to both sets (in case of holonomic constraints). Obviously, we can express the relations (18.2.7), (18.2.8) in one of the forms ∂fl q + fl = 0, l = 1, 2,..., p , ∂q j j

(18.2.7')

∂fl ∗ q = 0, l = 1, 2,..., p , ∂q j j

(18.2.8')

where we have introduced the generalized velocities and the virtual generalized velocities, respectively. If we replace in (3.2.13) the real displacements by the relations (18.2.4) or (18.2.4'), then we can express – in general – the constraint relations in one of the forms akj dq j + ak 0 dt = 0, k = 1, 2,..., m ,

(18.2.9)

akj q j + ak 0 = 0, k = 1, 2,..., m ,

(18.2.9')

where akj = ak 0 =

n

∑ αkl

i =1

n

⋅

∂ri 0 , j = 1, 2,..., s , ∂q j

∑ αkl ⋅ ri + αk 0 , k = 1,2,..., m .

(18.2.9'')

i =1

We notice, from (18.2.4') – (18.2.6'), that a necessary condition to have catastatic constraints is ri = 0, i = 1, 2,..., n ; indeed, in this case, to q j = 0 corresponds vi = 0 (hence, rest with respect to an inertial frame of reference), the inverse implication taking, as well, place (the matrix ⎡⎣ ∂Xk ∂q j ⎤⎦ is of rank s ). In case of holonomic constraints, the respective condition in sufficient too (as a mater of fact, in this case the constraints are scleronomic). In case of non-holonomic constraints one must have also αk 0 = 0 ; hence, from (18.2.9'') it results that these constraints are catastatic if and only if ak 0 = 0, k = 1,2,..., m . We notice that


51 akj =

n

∑ α ki

i =1

⋅

∂ri + ∂q j

n

∑ αki

i =1

⋅

∂ri . ∂q j

Because ri = 0 , while the matrix ⎡⎣ ∂Xk ∂q j ⎤⎦ is of rank s , it results that akj = 0 , j = 1,2,..., s , ⇔ α ki = 0, i = 1, 2,..., n , k = 1, 2,..., m . Hence, if the conditions akj = 0 are fulfilled, then the constraints (18.2.9'') are scleronomic.

We can express the non-holonomic constraint relations with the aid of the virtual generalized displacements too in one of the forms akj δq j = 0, k = 1,2,..., m ,

(18.2.10)

akj q ∗j

(18.2.10')

= 0, k = 1,2,..., m .

By means of the possible generalized displacements, it results akj Δq j = 0, k = 1, 2,..., m ,

(18.2.11)

akj q j∗ = 0, k = 1, 2,..., m ,

(18.2.11')

too. We can thus that, in case of catastatic constraints (holonomic or non-holonomic), the set of virtual generalized displacements coincides with the set of possible generalized displacements and the real generalized displacements belong to these sets. We notice, from (18.2.10), that – in case of bilateral constraints – the virtual generalized displacements are reversible; in case of non-strict unilateral constraints, only some virtual generalized displacements are reversible (these which are equated to zero in the constraint relations). In case of a scalar function ϕ (q1 , q2 ,..., qs ; t ) or of a vector function W (q1 , q2 ,..., qs ; t ) , the operator of total differentiation with respect to time is of the form d ∂ ∂ . = q j + dt ∂q j ∂t

(18.2.12)

This operator contains the operator of space differentiation q j ∂ ∂q j and the operator of time differentiation ∂ ∂t . Noting that ∂ d ∂2 ∂2 d ∂ ∂2 ∂2 = q j + , = q j + ∂qk dt ∂qk ∂q j ∂qk ∂t dt ∂qk ∂q j ∂qk ∂t ∂qk

and assuming that the considered functions are of class C 2 with respect to the corresponding variables (in this case, according to Schwartz’s theorem, the mixed derivatives are independent on the order of differentiation), we have the operator relation


52

∂ d d ∂ . = ∂qk dt dt ∂qk

(18.2.13)

Starting from (18.2.4') and taking into account (18.2.12), (18.2.13), we can calculate the acceleration of the particle Pi in the form ai =

dvi d ⎛ ∂ri = dt dt ⎜⎝ ∂q j

∂ri ∂ri dri ∂ ⎞ ⎟ q j + ∂q qj + dt = ∂q ( vi + ri ) q j + ∂q qj + ri , j j j ⎠

wherefrom

ai =

∂ri ∂2 ri ∂r qj + q j qk + 2 i q j + ri , i = 1, 2,..., n . ∂q j ∂q j ∂qk ∂q j

(18.2.14)

We have thus put in evidence the generalized accelerations qj = d2q j dt 2 , j = 1,2,..., s .

18.2.1.3 Kinetic Energy. Work. Natural Systems Starting from the expression (18.2.14') of the velocity of the particle Pi , we can calculate the kinetic energy of the discrete mechanical system S in the form T =

1 n 1 n ⎛ ∂r ⎞ ⎛ ∂r ⎞ mi vi2 = ∑ mi ⎜ i q j + ri ⎟ ⋅ ⎜ i q j + ri ⎟ . ∑ q ∂ 2 i =1 2 i =1 ⎝ ∂q j j ⎠ ⎝ ⎠

It results T = T2 + T1 + T0 ,

(18.2.15)

where T2 =

1 g q q , g = gkj = 2 jk j k jk

∂r

n

∑ mi ∂qij

i =1

⋅

∂ri , j , k = 1, 2,...s , ∂qk

(18.2.15')

is a quadratic form in the generalized velocities, T1 = g j q j , g j =

n

∂r

∑ mi ∂qij

i =1

⋅ ri , j = 1, 2,...s ,

(18.2.15'')

is a linear form in the generalized velocities, while

T0 = g 0 , g 0 =

1 n mi ri , 2 i∑ =1

is a constant with respect to these velocities.

(18.2.15''')


53

We notice that T0 ≥ 0 , being thus a non-negative quantity; the equality to zero takes place in case of catastatic constraints (if the constraints are holonomic, then they are scleronomic too). The linear form T1 can have any sign; also this form vanishes in case of catastatic constraints. Hence, in case of such constraints we have T = T2 =

1 n mi 2 i∑ =1

2

⎛ ∂ri ⎞ ⎜ ∂q q j ⎟ ⎝ j ⎠

≥ 0.

(18.2.16)

This quadratic form vanishes only if each bracket vanishes, hence for ∂ri q = 0, i = 1, 2,...n . ∂q j j

We notice that the matrix [ ∂Xk ∂q j ] of the coefficients of this system of linear equations is of rank s ; it results q j = 0, j = 1,2,..., s , the quadratic form T2 being positive definite. We remark, as well, that – in case of catastatic constraints – we have T2 = 0 too, the kinetic energy non-depending explicitly on time. One can state that g ≠ 0, g = det ⎡⎣ g jk ⎦⎤ .

(18.2.17)

Indeed, otherwise, the system of linear equations g jk qk = 0, j = 1,2,..., s , would have non-trivial solutions; thus, a linear combination of these equations could lead to 2T2 = g jk q j qk = 0 , without having all generalized velocities equal to zero. But the quadratic form T2 is positive definite, obtaining thus a contradiction. We mention also that T2 is a non-singular quadratic form (eventually, excepting the points (isolated) in which the correspondence (18.2.1) (or (18.2.2)) is not one-to-one). In general, the s (s + 1) 2 coefficients of the quadratic form T2 are of the form

g jk = g jk (q1 , q 2 ,..., qs ; t ),

j , k = 1, 2,..., s ;

as

well,

g j = g j (q1 , q 2 ,..., qs ; t ),

j = 1, 2,..., s . In case of catastatic constraints we have g jk = g jk (q1 , q 2 ,..., qs ),

j , k = 1, 2,..., s , g j = g j (q1 , q2 ,..., qs ), j = 1,2,..., s . The real elementary work of the given forces (external and internal) is given by dW =

n

∑ Fi

i =1

⋅ dri =

n

∑ Fi

i =1

⋅

∂ri dq j , ∂q j

where we have taken into account (18.2.4); we can also write d W = Q j dq j , Q j =

n

∑ Fi

i =1

⋅

∂ri , j = 1,2,..., s . ∂q j

(18.2.18)


54

By analogy, Q j are called generalized forces (by convention, we can assume that they act upon the representative point P ). The virtual work of these forces is expressed in the form δW = Q j δq j .

(18.2.19)

For the practical computation of the components (co-ordinates) of the generalized force, we can fix all co-ordinates excepting one of them (let be qk , to which corresponds the virtual displacement δqk ); calculating δW in E 3 , one can obtain Qk , making then – successively – k = 1, 2,..., s . We notice that a component Q j of the generalized force has the dimension of a force only if the corresponding generalized co-ordinate q j has the dimension of a length. Supposing that the given forces Fi , i = 1, 2,..., n , derive from a simple quasi-potential U ( ri ; t ) = U ( r1 , r2 ,..., rn ; t ) , then we can write U [ ri (q j ; t ); t ] = U (q1 , q2 ,..., qs ; t ) ,

obtaining thus a simple quasi-potential in generalized co-ordinates; indeed we can write Qj =

n

∑ Fi ⋅

i =1

∂ri = ∂q j

n

∑ ∇iU ⋅

i =1

∂ri = ∂q j

(i ) ∂U ∂x k ∑ ∑ ( i ) ∂q j , i = 1 k = 1 ∂x n

3

k

wherefrom Qj =

∂U , j = 1, 2,..., s , ∂q j

(18.2.20)

the generalized forces being thus quasi-conservative. If U = U (q1 , q1 ,..., qs ) is a simple potential, then the corresponding generalized forces are conservative. But we must notice that the existence of a quasi-potential for the given generalized forces does not imply the existence of a quasi-potential for the given corresponding forces in the space E 3 . If Q j = [U ]j =

∂U d ∂U − , j = 1,2,..., s , dt ∂q j ∂q j

(18.2.21)

then we say that the generalized forces are quasi-conservative, deriving from a generalized quasi-potential U = U (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) . We notice that the generalized forces cannot depend on the generalized accelerations (because the given forces cannot depend on the accelerations), so that the quasi-potential is of the form U = U k qk + U 0 ,

where U j = U j (q1 , q2 ,..., qs ; t ), j = 0,1, 2,..., s . Hence,

(18.2.22)


55

∂U j ⎞ ∂U 0 ⎛ ∂U k Qj = ⎜ − ⎟ qk − U j + ∂q , j = 1,2,..., s , q q ∂ ∂ j k ⎠ ⎝ j

(18.2.21')

in case of quasi-conservative generalized forces; if the generalized forces are conservative, corresponding to a generalized potential, where U j = U j (q1 , q2 ,..., qs ) , j = 0,1,2,..., s , then we have U j = 0 . By following gauge transformation U k → U k = U k + ∂ϕ ∂qk , k = 1, 2,..., s , U 0 → U 0 = U 0 + ϕ , where

have

ϕ = ϕ (q1 , q2 ,..., qs ; t ) is a function of class C 2 (to ( ∂ ∂qk )(d dt ) = (d dt )( ∂ ∂qk ) ), hence by a transformation of the form

U → U = U + dϕ dt , we obtain the same generalized forces Q j . Hence, the genera-

lized quasi-potential from which derives a quasi-conservative generalized force is determined excepting the total derivative with respect to time of a function ϕ of class C 2 . A field of quasi-conservative generalized forces is non-stationary, while a field of conservative generalized forces is stationary. The elementary work of conservative generalized forces is a total differential, being of the form (18.1.22), in case of a simple potential, or of the form (18.1.22''), in case of a generalized potential. If the generalized forces are quasi-conservative, deriving from a simple quasi-potential, then the elementary work is of the form (18.1.23); if these generalized forces derive from a generalized quasi-potential, then we get ⎛ ∂U 0 ⎞ − U j ⎟ dq j . dW = dU 0 − U 0 dt − U j dq j = dU 0 − (U j q j + U 0 ) dt = ⎜ ∂ q ⎝ j ⎠

(18.2.23)

We notice that the gauge transformation mentioned above for the generalized potential does not affect this elementary work. The mechanical systems which admit a simple or generalized potential (quasipotential) are called natural systems. The generalized forces which do not derive from a simple quasi-potential can be always expressed in the form Qj =

∂U + Q j , U = U (q1 , q2 ,..., qs ; t ) , ∂q j

(18.2.24)

the elementary work being given by dW = dU + ( Q j q j − U ) dt .

(18.2.24')

If U = U (q1 , q2 ,..., qs ) is a potential, hence if U = 0 , then the quantity P = Q j q j represents the power of the non-potential generalized forces. By analogy with the definitions given in E 3 and E 3 n , the non-potential generalized forces Q j of zero power ( P = 0 ) are called gyroscopic generalized forces and depend, obviously, on the


56

distribution of the generalized velocities; in this case, the generalized forces Q j are conservative. If the power of the non-potential generalized forces does not vanish, then these forces are non-conservative. If P < 0 , then the corresponding generalized forces are dissipative. The quasi-conservative forces (18.2.21') which derive from a generalized quasi-potential are of the form (18.2.24), so that (we take U = U 0 ) ∂U j ⎛ ∂U k − Qj = ⎜ ∂qk ⎝ ∂q j

⎞ ⎟ qk − U j , j = 1, 2,..., s . ⎠

(18.2.24'')

If U j = 0, j = 1,2,..., s then it results Q j q j = 0 , the non-potential generalized forces being thus gyroscopic. Hence, any generalized force which admits a generalized potential is the sum of a generalized force which admits a simple potential and a gyroscopic generalized force. Let be, in general, a non-potential generalized force of the form Q j = − αij qi , j = 1, 2,..., s .

(18.2.25)

By a decomposition in symmetric and skew-symmetric parts of the tensor quantity αij , we observe that the non-potential generalized forces Q j = − α( ij )qi , j = 1, 2,..., s ,

(18.2.25')

where the quadratic form R =

1 α q q , 2 ( ij ) i j

(18.2.26)

called the Rayleigh dissipative function, is positive, are dissipative; indeed, we have Q j q j = − α( ij )qi q j ≤ 0 .

We notice that Qj = −

∂R , j = 1, 2,..., s . ∂q j

(18.2.26')

Analogously, the non-potential generalized forces Q j = − α[ ij ]qi = α[ ji ]qi , j = 1, 2,..., s ,

(18.2.25'')

are gyroscopic, because Q j q j = α[ ji ]qi q j = 0 . We observe that the notion of power of the non-potential generalized forces has a signification larger than the classical one, coinciding with the latter one in case of a potential (U = 0 ).


57

18.2.2 Lagrange’s Equations of Second Kind The basic problem of motion of a discrete mechanical system S in the space E 3 is reduced to the problem of motion of the corresponding representative point P , subjected to the action of the generalized forces Q j , j = 1, 2,..., s , and to m non-holonomic constraints (18.2.9) (or (18.2.9')) in the space Λs (the representative point P is on a non-holonomic manifold with s − m dimensions, at the intersection of m non-holonomic hypersurfaces). In what follows, we deduce Lagrange’s equations of second kind in case of holonomic or non-holonomic constraints; a special attention will be given to the case of quasi-conservative forces, by introducing Lagrange’s kinetic potential.

18.2.2.1 Lagrange’s Equations in Case of Holonomic Constraints Let be a discrete mechanical system S subjected to holonomic ideal constraints; the motion of this mechanical system, hence the motion of the representative point P in the space Λs will be governed by the theorem of virtual work, written in the form n

∑ mi

i =1

d2 ri ⋅ δri = δW = dt 2

n

∑ Fi

i =1

⋅ δri ,

(18.2.27)

corresponding to the formulae (11.1.59), (11.1.63) and to the formula (18.1.52), respectively. We calculate n

∑ mi

i =1

=

n

d2 ri ⋅ δri = dt 2

⎡ d ⎛ dr

∑ mi ⎢⎣ dt ⎝⎜ dti

i =1

⋅

n

∑ mi

i =1

d2 ri ∂ri ⋅ δq j dt 2 ∂q j

∂ri ⎞ dri d ⎛ ∂ri ⎞ ⎤ − ⋅ δq . ∂q j ⎠⎟ dt dt ⎝⎜ ∂q j ⎠⎟ ⎦⎥ j

The formula (18.2.4') allows to write ∂vi ∂r = i , i = 1, 2,..., n , j = 1, 2,..., s . ∂q j ∂q j

(18.2.28)

Using the operator relation (18.2.13) too, it results n

∑ mi

i =1

=

1 ⎡d ∂ ( mi vi2 ⎢ 2 i∑ = 1 ⎣ dt ∂q j n

∂vi ⎞ ∂v ⎤ − vi ⋅ i ⎥ δq j ⎟ ∂q j ⎠ ∂q j ⎦ i =1 ∂ ⎤ ⎡ d ⎛ ∂T ⎞ ∂T ⎤ − ( mi vi2 ) ⎥ δq j = ⎢ dt ⎜ ∂q ⎟ − ∂q ⎥ δq j , ∂q j j ⎦ ⎦ ⎣ ⎝ j ⎠

d2 ri ⋅ δri = dt 2

)

n

⎡d⎛

∑ mi ⎢⎣ dt ⎝⎜ vi

⋅

where we have introduced the kinetic energy T . Taking into account also the formula (18.2.19), we can write the equation (18.2.27) in the form ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q − Q j ⎥ δq j = 0 . j ⎣ ⎝ j ⎠ ⎦

(18.2.27')

58


As a matter of fact, this represents the fourth form of the basic equation. Because the constraints are holonomic, it results that the virtual generalized displacements δq j are independent; hence, we may write d ⎛ ∂T ⎞ ∂T − = Q j , j = 1,2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.29)

getting thus the equation of motion of the representative point P , that is Lagrange’s equations of second kind (shortly, Lagrange’s equations), obtained be Lagrange in 1760. This system of differential equations of second order (the kinetic energy depends on the generalized co-ordinates, the generalized velocities and time) allows to determine the unknown functions, q j = q j (t ), j = 1, 2,..., s , t ∈ [ t0 , t1 ] . The fundamental problem of mechanics remains, further, deterministic; starting from the initial conditions (11.1.9), written for a discrete mechanical system, and taking into account the relations (18.2.1) or (18.2.1') and (18.2.4'), we associate the initial conditions of Cauchy type

q j (t0 ) = q j0 , q j (t0 ) = q j0 .

(18.2.30)

These conditions are univocally determinate (as we have seen in Sects. 18.2.1.1 and 18.2.1.2, in the general case). If q j0 = 0, q j0 = 0, j = 1,2,..., s , then we say that we have to do with homogeneous initial conditions. Thus, the problem is entirely formulated; the functions q j (t ) must be of class C 2 .

18.2.2.2 Case of Non-holonomic Constraints. Natural Systems. Kinetic Potential. Conditions of Equilibrium Let us suppose that the discrete mechanical system S , hence also the representative point P in the space Λs , is subjected to m < s non-holonomic constraints, which are expressed in the form (18.2.10), by means of the generalized displacements. In this case the problem is reduced to the determination of the functions q j = q j (t ), j = 1,2,..., s , so that, besides Lagrange’s equations (18.2.29), the relations (18.2.27') be also satisfied, for any virtual generalized displacements δq j , which verify the constraint relations (18.2.10); applying the method of Lagrange’s multipliers, used several times till now, we can write m ⎡ d ⎛ ∂T ⎞ ∂T ⎤ λk akj ⎥ δq j = 0 . − − Q − j ∑ ⎜ ⎟ ⎢ dt ∂q ⎣ ⎝ j ⎠ ∂q j ⎦ k =1

(18.2.31)

Assuming that the non-holonomic constraints are distinct, the matrix [akj ] is of rank m ; we can denote the generalized co-ordinates so that det[akj ] ≠ 0, j , k = 1,2,..., m ,

which allows to determine univocally the multipliers λk , by annulling the first m


59

square brackets. From the constraint relations we obtain the virtual generalized displacements δql , l = 1, 2,..., m , as functions of the virtual generalized displacements δqm + 1 , δqm + 2 ,..., δqs , which can be considered to be independent; following the usual reasoning, it follows that all square brackets vanish, so that we obtain Lagrange’s equations in case of non-holonomic constraints (Lagrange’s equations with multipliers) in the form

d ⎛ ∂T ⎞ ∂T − = Q j + Rj , j = 1, 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.32)

where Rj =

m

∑ λk akj , j

k =1

= 1, 2,..., s ,

(18.2.33)

are the components of the constraint generalized force. To determine the s + m unknowns (the functions q j = q j (t ), j = 1,2,..., s , and the parameters λk , k = 1,2,..., m ) we have thus at our disposal s equations of Lagrange with multipliers and m constraint relations. If the mechanical system S has h holonomic constraints (18.2.3), which have not been eliminated, then the constraint forces are Rj =

m

h

k =1

l =1

∂f

∑ λk akj + ∑ μl ∂qlj

,

(18.2.33')

where μl , l = 1, 2,..., h , are also Lagrange’s multipliers which must be determined, taking into account supplementary constraint relations too. In case of a natural system, the generalized forces derive from a simple or generalized quasi-potential, being of the form (18.2.20) or of the form (18.2.21). Introducing Lagrange’s kinetic potential (the Lagrangian, quantity of the nature of the energy)

L = T +U ,

(18.2.34)

we can write the equations (18.2.32) in the form d ⎛ ∂L ⎞ ∂L − = Rj , j = 1, 2,..., s . dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.35)

If we express the generalized forces in the form (18.2.24), the kinetic potential being, further, given by (18.2.34), then the considered Lagrange equations become

d ⎛ ∂L ⎞ ∂L − = Q j + Rj , j = 1, 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.36)


60

putting in evidence the non-potential generalized forces. If the constraints of the discrete mechanical system equations become

S are holonomic, then these

d ⎛ ∂L ⎞ ∂L − = Q j , j = 1,2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.37)

while in case of a natural system it results

d ⎛ ∂L ⎞ ∂L − = 0, j = 1, 2,..., s . dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.38)

Using the Euler–Lagrange operator, we can formally write [T ]j + Q j + Rj = 0, j = 1, 2,..., s ,

(18.2.32')

[T ]j + Q j = 0, j = 1, 2,..., s ,

(18.2.29')

[ L ]j + Rj = 0, j = 1, 2,..., s ,

(18.2.35')

[ L ]j + Q j + Rj = 0, j = 1,2,..., s ,

(18.2.36')

[ L ]j + Q j = 0, j = 1,2,..., s ,

(18.2.37')

[ L ]j = 0, j = 1, 2,..., s .

(18.2.38')

as well as

Thus, Lagrange’s equations take a simpler form, which is easier to handle in case of natural systems; moreover, one obtains – in this case – interesting results concerning the integration of the respective system of equations. We notice that

L = L2 + L1 + L0 ,

(18.2.34')

where we take into account (18.2.15), so that

L2 = T2 , L1 = T1 + U − U 0 , L0 = T0 + U 0 ,

(18.2.34'')

in case of a generalized quasi-potential; if the quasi-potential is a simple one, then U = U 0 . The mechanical systems which admit a kinetic potential are called mechanical systems of first kind. In general, we can consider a mechanical system the motion of which is specified by a Lagrangian L = L (q1 , q1 ,..., qs , q1 , q2 ,..., qs ; t ) , even in the case in which the system is non-natural, with the condition that the Hessian of this function with respect to the generalized velocities be non-zero


61 ⎡ ∂2 L ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂q j ∂qk ⎦

(18.2.34''')

As a matter of fact, if the system is natural, then we have ∂2 L ∂2T = = g jk , ∂q j ∂qk ∂q j ∂qk

so that the considered Hessian is given by det[ g jk ] = g ≠ 0 , as it has been shown in Sect. 18.2.1.3. To the state of rest of the discrete mechanical system S with respect to an inertial frame of reference corresponds a position of rest of the representative point P in the space Λs . From (18.2.32) it results that (we make q j = 0, qj = 0, j = 1,2,..., s ) Q j + Rj = 0, j = 1, 2,..., s .

(18.2.39)

Obviously, these relations must be verified also by the initial conditions, where q j0 = 0, j = 1, 2,..., s . More precisely, the conditions (18.2.39) should be written in the form Q j (q10 , q20 ,..., qs0 , 0, 0,..., 0; t ) + Rj (q10 , q20 ,..., qs0 , 0, 0,..., 0; t ) = 0, t ∈ [t0 , t1 ],

(18.2.39')

where we have put in evidence also the position of equilibrium (corresponding to the initial conditions). If the discrete mechanical system S is subjected only to holonomic constraints, then the conditions of equilibrium are written in the form Q j = 0, j = 1, 2,..., s .

(18.2.40)

If the generalized forces derive from a simple quasi-potential, we obtain ∂U = 0, j = 1, 2,..., s , ∂q j

(18.2.40')

hence a necessary condition for the quasi-potential U to admit an isolated extremum. Concerning the state of rest, one puts important problems of stability, which will be considered afterwards.

18.2.2.3 Normal Form of Lagrange’s Equations Taking into account the expression (18.2.15), (18.2.15'), (18.2.15''), and (18.2.15''') of the kinetic energy, we can find an explicit form of Lagrange’s equations (18.2.32). We calculate thus ∂T ∂q j = g jk qk + g j and then


62

∂g jk d ⎛ ∂T ⎞ ⎛ ∂g j ⎞ = g jk qk + q q + + g jk ⎟ qk + g j , ⎜ ⎟ ∂ql k l ⎜⎝ ∂qk dt ⎝ ∂q j ⎠ ⎠ ∂gk ∂q 0 ∂T 1 ∂gkl = q q + q + ; ∂q j 2 ∂q j k l ∂q j k ∂q j

noting that the product qk ql is symmetric in the indices k and l , it results ∂g jk ∂ql

qk ql =

∂g jl ⎞ 1 ⎛ ∂g jk q q , + ∂qk ⎟⎠ k l 2 ⎜⎝ ∂ql

so that we write the equations (18.2.32) in the form g jk qk + [ kl , j ]qk ql + (2 γ jk + g jk )qk −

∂g 0 + g j = Q j + Rj , j = 1, 2,..., s , ∂q j

(18.2.41)

where we have introduces Christoffel’s symbols of the first kind [ kl , j ] =

∂g jl ∂g 1 ⎛ ∂g jk + − kl ⎜ 2 ⎝ ∂ql ∂qk ∂q j

⎞ ⎟ , j , k , l = 1, 2,..., s , ⎠

(18.2.42)

⎞ ⎟ , j , k = 1, 2,..., s . ⎠

(18.2.43)

as well as the skew-symmetric quantities γ jk = − γkj =

∂g 1 ⎛ ∂g j − k ⎜ 2 ⎝ ∂qk ∂q j

We have thus obtained a system of linear equations in the generalized accelerations qk , k = 1, 2,..., s . Let g jk = g kj be the algebraic complement of the element g jk in the determinant g , which is definite by (18.2.17); the normalized algebraic complement will be g jk = g kj = g jk g . We notice that the relations det[ g jk ] =

1 , g g mj = δkm , g jk

(18.2.44)

where δkm is Kronecker’s symbol (equal to unity for k = m and equal to zero for k ≠ m ), take place. Multiplying the system of equations (18.2.41) by g mj and summing with respect to j , it results (we replace then the free index m by j ) ⎧⎪ j qj + ⎨ ⎩⎪ k

⎫⎪ ∗ ∗ ⎬ q q + α jk qk + β j = Q j + Rj , j = 1, 2,..., s , l ⎭⎪ k l

(18.2.45)


63

where we have introduced Christoffel’s symbols of second kind, defined by (the relations of definitions for Christoffel’s symbols correspond to the relations of definition (A.1.44) and (A.1.45) from E 3 ) ⎧⎪ j ⎨ ⎩⎪ k

⎫⎪ mj ⎬ = [ kl , m ]g , j , k , l = 1, 2,..., s , l ⎭⎪

(18.2.42')

as well as the notations (as everywhere in this work, we make not considerations on covariance or on contravariance, the position of the indices – up or down – having no one signification) ⎛ ∂g ⎞ αmk = (2 γ jk + g jk )g mj , βm = ⎜ − 0 + q j ⎟ g mj , ∂ q j ⎝ ⎠ mj mj ∗ ∗ Qm = Q j g , Rm = Rj g , m = 1, 2,..., s .

(18.2.46)

The system of equations (18.2.45) represents thus the normal form of Lagrange’s equations, in which the generalized accelerations are expressed as functions of the generalized velocities and of the generalized co-ordinates; this form of the equations can be always obtained, because g ≠ 0 . In the case of catastatic constraints we have T = T2 , and the equations (18.2.45) take the form ⎧⎪ j qj + ⎨ ⎩⎪ k

⎫⎪ ∗ ∗ ⎬ q q = Q j + Rj , j = 1, 2,..., s . l ⎭⎪ k l

(18.2.47)

If the constraints are holonomic, then the equations (18.2.45) become ⎧⎪ j qj + ⎨ ⎪⎩ k

⎫⎪ ∗ ⎬ q q + α jk qk + β j = Q j , j = 1, 2,..., s , l ⎪⎭ k l

(18.2.45')

while the equations (18.2.47) read (in this case, the constraints are scleronomic too) ⎧⎪ j qj + ⎨ ⎪⎩ k

⎫⎪ ∗ ⎬ q q = Q j , j = 1, 2,..., s . l ⎪⎭ k l

(18.2.47')

In case of a single particle, the equations (18.2.47') become (6.1.24''').

18.2.2.4 Theorems of Existence and Uniqueness We can replace the system of s differential equations of second order (18.2.45') by a system of 2s differential equations of first order in the normal form ⎪⎧ j q j = u j , u j = ϕ j , ϕ j = Q j∗ − ⎨ ⎪⎩ k

⎪⎫ ⎬ u u − α jk uk − β j , j = 1,2,..., s , l ⎪⎭ k l

(18.2.48)


64

where u j = u j (q1 , q2 ,..., qs ; t ), ϕ j = ϕ j (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) . This system is autonomous; if the time does not intervene explicitly in u j and ϕ j , then the system is autonomous (or dynamical). We associate to this system the initial conditions (18.2.30) in the form q j (t0 ) = q j0 , u j (t0 ) = q j0 , j = 1, 2,..., s ,

(18.2.48')

the problem at the limit (18.2.48), (18.2.48') being thus a problem of Cauchy type. The problem (18.2.45'), (18.2.30) is equivalent to the problem (18.2.48), (18.2.48'); for the latter one we can state Theorem 18.2.1 (of existence and uniqueness; Cauchy–Lipschitz). If the functions u j and ϕ j , j = 1,2,..., s , are continuous on the interval (2s + 1) -dimensional D , specified

q j0 − Q j0 ≤ q j ≤ q j0 + Q j0 , q j0 − U j0 ≤ u j ≤ q j0 + U j0 ,

by

t0 − T0 ≤ t ≤ t0 + T0 , Q j0 , U j0 , T0 = const, j = 1, 2,..., s , and definite on the space

Cartesian product of the phase space (of canonical co-ordinates q1 , q2 ,..., qs , u1 , u2 ,..., us ) by the time space (of co-ordinate t ) and if Lipschitz’s conditions u j (q1 , q2 ,..., qs ; t ) − u j (q1 , q2 ,..., qs ; t ) ≤

1 λ q − qk , μj T k k

ϕ j (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) − ϕ j (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) ≤

1 1 λ ⎡ q − qk + uk − uk ⎤ ν j T k ⎢⎣ τ k ⎦⎥

are satisfied for j = 1,2,..., s , where T > 0 is a time constant independent on q j , u j and t , τ is a time constant equal to unity, while λk , μj , ν j are constants equal to unity (dimensionless if the corresponding generalized co-ordinate is a length and having the dimension of a length if the corresponding generalized co-ordinate is nondimensional), then there exists a unique solution q j = q j (t ), u j = u j (t ) of the system (18.2.48), which satisfies the initial conditions (18.2.48') and is definite on the interval t0 − T ≤ t ≤ t0 + T , where Q j0 U j0 ⎡ ( ! ) , τμj ( ! ), T T ≤ min ⎢T0 , λj V V ⎣

⎤ ⎥ , V = max μj u j ( ! ) , τν j ϕ j ( ! ) in D , ⎦

(

)

the sign (!) indicating “without summation”. The continuity of the functions u j and ϕ j on the interval D ensures the existence of the solution, according to Peano’s theorem. The condition of Lipschitz must be fulfilled too for the uniqueness of the solution; these latter conditions may be replaced by other ones less restrictive, in accordance with which the partial derivatives of the first order of the functions u j and ϕ j , j = 1,2,..., s , must exist and be bounded in absolute value


65

on the interval D . Besides, the conditions in the Theorem 18.2.1 are sufficient conditions of existence and uniqueness, which are not necessary too. The existence and the uniqueness of the solution have been put in evidence only on the interval [t0 − T , t0 + T ] , in the neighbourhood of the initial moment t0 (as a matter of fact, the moment t0 must not be the initial moment, but can be a moment arbitrarily chosen); taking, e.g., the moment t0 + T as initial moment, it is possible – repeating the above reasoning – to extend the solution on a interval of length 2T1 if, naturally, the sufficient conditions of existence and uniqueness of the theorem are verified in the neighbourhood of the new initial moment. Thus, we can prolong the solution for t ∈ [t1 , t2 ] , corresponding to a certain interval of time in which the considered mechanical phenomenon takes place, or even for t ∈ [t1 , ∞ ) or t ∈ ( −∞, t1 ] or t ∈ ( −∞, ∞ ) . We can state also theorems on the continuous or analytical dependence of the solution on a parameter, analogous to the Theorems 6.1.3 and 6.1.4; as well, we can state Theorem 18.2.2 (on the differentiability of the solutions). If, in the neighbourhood of a point P (q1 ,q2 ,..., qs , u1 , u2 ,..., us ; t ) ∈ D , the functions u j (q1 ,q2 ,..., qs ; t ) and ϕ j (q1 ,q2 ,..., qs , u1 , u2 ,..., us ; t ), j = 1,2,..., s , are of the class C k , then the solutions

q j (t ) and u j (t ) of the system (18.2.48), which satisfy the initial conditions (18.2.48'), are of class C k + 1 in a neighbourhood of the point P . In particular, if q j0 = 0 and Q j (t0 ) = 0 , which leads also to ϕ j (t0 ) = 0 ,

j = 1,2,..., s , the constraints being scleronomic, then we obtain q j (t ) = q j0 , j = 1,2,..., s , the solution being unique, according to the Theorem 18.2.1; hence, the representative point is at rest with respect to a given fixed frame of reference.

18.2.3 Transformations. First Integrals In the study of Lagrange’s equations an important rôle is played by the transformations of co-ordinates and by the gauge transformations which will be considered in what follows. As well, to integrate this system of equations, the problem is put to find first integrals; in this order of ideas, we will put in evidence – especially – the first integral of energy.

18.2.3.1 Point Transformations We call point transformation a transformation of generalized co-ordinates of the form q j = q j (q1 ,q2 ,..., qs ; t ), j = 1, 2,..., s ,

(18.2.49)

where q j are functions of class C 2 with respect to the new generalized co-ordinates; we assume that


66

⎡ ∂q j ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂qk ⎦

(18.2.49')

the correspondence between the two systems of co-ordinates being one-to-one. In this section, we will denote dq j dt = q j and will have q j =

∂q j ∂q j qk + , j = 1, 2,..., s , ∂qk ∂t

(18.2.49'')

hence, q j = q j (q1 ,q2 ,..., qs , q1 , q2 ,..., qs ; t ) , so that ∂q j ∂q j = , j , k = 1,2,..., s . ∂qk ∂qk

(18.2.50)

One the other hand

∂q j ∂ 2q j ∂ 2q j ∂ ⎛ ∂q j ⎞ ∂ ⎛ ∂q j ⎞ d ⎛ ∂q j ⎞ , = = = ql + q + ∂qk ∂qk ∂ql ∂qk ∂t ∂ql ⎜⎝ ∂qk ⎟⎠ l ∂t ⎜⎝ ∂qk ⎟⎠ dt ⎜⎝ ∂qk ⎟⎠ wherefrom it result the relation between operators

( ) = ddt ⎛⎜⎝ ∂∂q

∂ d ∂qk dt

k

⎞ ⎟ , k = 1, 2,..., l . ⎠

(18.2.51)

By the transformation (18.2.49), (18.2.49'), and (18.2.49'') we obtain T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) = T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) , where T is a new function which represents the kinetic energy in the generalized co-ordinates. We can write ∂T ∂T ∂q j ∂T ∂q j ∂T ∂q j ∂T d ⎛ ∂q j = + = + ∂qk ∂q j ∂qk ∂q j ∂qk ∂q j ∂qk ∂q j dt ⎜⎝ ∂qk ∂T ∂T ∂q j ∂T ∂q j d ⎛ ∂T ⎞ ∂T d ⎛ ∂T ⎞ ∂q j = = = + , ∂q j ∂qk ∂q j ∂qk dt ⎜⎝ ∂qk ⎟⎠ dt ⎜⎝ ∂q j ⎟⎠ ∂qk ∂q j ∂qk

⎞ ⎟, ⎠

d ⎛ ∂q j ⎞ , dt ⎜⎝ ∂qk ⎟⎠

where we took into account (18.2.50), (18.2.51); analogously, the generalized forces read Qk =

n

∑ Fi

i =1

⋅

∂ri = ∂qk

n

∑ Fi

i =1

⋅

∂ri ∂q j , ∂q j ∂qk

wherefrom Qk = Q j

∂q j , k = 1, 2,..., s , ∂qk

(18.2.52)


67

Thus, it results

d ⎛ ∂T ⎞ ∂T ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ∂q j − − Qk = ⎢ ⎜ − − Qj ⎥ . ⎜ ⎟ ⎟ dt ⎝ ∂qk ⎠ ∂qk ⎣ dt ⎝ ∂q j ⎠ ∂q j ⎦ ∂qk

(18.2.53)

Hence, if the equations (18.2.29) take place, then we will have d ⎛ ∂T ⎞ ∂T − = Qk , k = 1, 2,..., s , dt ⎜⎝ ∂qk ⎟⎠ ∂qk

(18.2.54)

too. On the other hand, taking into account the condition (18.2.49'), the relations (18.2.54) entail the equations (18.2.29). We can thus state Theorem 18.2.3 Lagrange’s equations are invariant to a point transformation of the form (18.2.49), (18.2.49') in case of a holonomic discrete mechanical system.

18.2.3.2 Gauge Transformations We have seen in Sect. 18.2.2.2 that the motion of a natural mechanical system is governed be Lagrange’s equations (18.2.38), for which it is necessary to determine Lagrange’s kinetic potential L . We notice that the Lagrangian K L , K = const , leads to the same equations. One puts the problem to find a kinetic potential L ′ = L + L , if this one exists, to obtain the same equations (18.2.38); in this case, one must have

d ⎛ ∂L ⎞ ∂L − ≡ 0, dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.55)

for any q j = q j (t ), j = 1, 2,..., s . For this, it is necessary that

L = a j q j + a 0 , a j = a j (q1 , q2 ,..., qs ; t ), j = 1, 2,.., s , a 0 = a 0 (q1 , q2 ,..., qs ; t ), otherwise the coefficients of the generalized accelerations qj cannot vanish, Replacing in (18.2.25), it results (the point corresponds to a partial derivative with respect to time) ∂ak ⎛ ∂a j ⎜ ∂q − ∂q j ⎝ k

∂a 0 ⎞ ⎟ qk + a j − ∂q ≡ 0, j = 1,2,..., s , j ⎠

so that we obtain the conditions ∂a j ∂a ∂a − k = 0, a j − 0 = 0, j , k = 1,2,..., s . ∂qk ∂q j ∂q j

Hence, it exists a function ϕ = ϕ (q1 , q2 ,..., qs ; t ) , so that


68 aj =

∂ϕ , j = 1, 2,..., s , a 0 = ϕ . ∂q j

We find thus

L =

∂ϕ dϕ . q + ϕ = ∂q j j dt

(18.2.56)

We can state Theorem 18.2.4 In case of a discrete mechanical system which admits a kinetic potential L , this one is determined excepting the total derivative with respect to time of an arbitrary function of class C 2 , of generalized co-ordinates and time. In particular, the Lagrangian can be of the form (18.2.34). This gauge transformation corresponds to the analogous transformation put in evidence in Sect. 18.2.1.3 for a generalized quasi-potential.

18.2.3.3 First Integrals of Lagrange’s Equations A function f (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) which is reduced to a constant along the integral curves of Lagrange’s system of equations (18.2.29) is a first interval of this system. If we determine l ≤ 2s first integrals fk (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) = ck , ck = const, k = 1, 2,..., l ,

(18.2.57)

the matrix ∂ ( f1 , f2 ,..., fl ) ⎡ ⎤ M = ⎢ ⎥⎦ ( q , q ,..., q , q , q ,..., q ) ∂ s s ⎣ 1 2 1 2

being of rank l , then all these integrals are functionally independent (independent first integrals) and we can express l unknown functions of the system (18.2.29) by means of the other ones; replacing in this system of equations, the problem is reduced to the integration of a system of equations with 2s − l unknowns (hence, a smaller number of unknowns). If l = 2s , then all the first integrals are independent, so that the system (18.2.57) of first integrals determines all the unknown functions. We notice that the first integrals (18.2.57) are no more independent for l > 2s ; hence we can set up at the most 2s independent first integrals. Solving the system (18.2.57) for l = 2s , we obtain (the matrix M is a square matrix of order 2s , for which det M ≠ 0 ) q j = q j (t ; c1 , c2 ,..., c2 s ), q j = q j (t ; c1 , c2 ,..., c2 s ), j = 1, 2,..., s ,

(18.2.58)

hence the general integral of the system of equations (18.2.29). Thus, one puts in evidence 2s scalar constants of integration. Imposing the initial conditions (18.2.30), we get


69

q j (t0 ; c1 , c2 ,..., c2 s ) = q j0 , q j (t0 ; c1 , c2 ,..., c2 s ) = q j0 , j = 1, 2,..., s ;

the conditions (18.2.30) being independent, we can write ⎡ ∂ (q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ) ⎤ det ⎢ ⎥ ≠ 0 ∂ (c1 , c2 ,..., c2 s ) ⎣ ⎦

and – on the basis of the theorem of implicit functions – we deduce

ck = ck (t0 ; q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ), k = 1, 2,..., 2s . Thus, it results, finally, q j = q j (t ; t0 , q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ) ,

q j = q j (t ; t0 , q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ) .

(18.2.59)

Hence, in the frame of the conditions of the theorem of existence and uniqueness, the principle of virtual work and the principle of initial conditions determine, univocally, the motion of the discrete mechanical system S , subjected to ideal and holonomic constraints, in a finite interval of time; by prolongation, the statement can become valid for any t . Thus, the deterministic aspect of mechanics is put in evidence in its Lagrangian presentation too. If f1 = c1 and f2 = c2 are two distinct first integrals, then f ( f1 , f2 ) = c will be also a first integral. Any other first integral is expressed, in general, as a function of the 2s distinct first integrals of Lagrange’s equations. From (18.2.57) it results dfk dt = 0 ; hence dfk df q j + k qj + f = 0, k = 1,2,.., l , dq j dq j

(18.2.57')

relations which must be identically satisfied if one takes into account the equations of motion (or the general integral of these equations, for any constants of integration).

18.2.3.4 First Integrals of Painlevé and Jacobi. First Integral of Mechanical Energy We can find a first integral particularly important and useful of the system of equations (18.2.29), multiplying these equations by the generalized velocities q j , j = 1, 2,..., s , and summing for all indices j ; we obtain

d ⎛ ∂T dt ⎜⎝ ∂q j We notice that

∂T ⎞ ⎟ q j − ∂q q j = Q j q j . j ⎠


70

d ⎛ ∂T ⎞ d ⎛ ∂T ⎞ ∂T q = q − q . dt ⎜⎝ ∂q j ⎟⎠ j dt ⎜⎝ ∂q j j ⎟⎠ ∂q j j

Because T = T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) , we have, as well, ∂T ∂T dT = q j + q + T , dt ∂q j ∂q j j

so that

d ⎛ ∂T ⎞ q − T ⎟ = Q j q j − T . dt ⎜⎝ ∂q j j ⎠

(18.2.60)

Taking into account (18.2.15) and noting that ∂T2 ∂T q = 2T2 , 1 q j = T1 , ∂q j j ∂q j

according to Euler’s theorem concerning the homogeneous functions, we can also write d ( T − T0 ) = Q j q j − T . dt 2

(18.2.60')

If it exists a function W = W (q1 , q2 ,..., qs ; t ) , so that Q j q j − T = dW dt (the function W cannot depend on the generalized velocities too, because dW dt would introduce also generalized accelerations), then we state Painlevé’s first integral T2 − T0 − W = h , h = const .

(18.2.61)

We notice that one must have −T0 = W , because they are terms which do not depend on the generalized velocities q j ; as well, if the given forces Fi , i = 1, 2,..., n , do not depend on the velocities, hence if the generalized forces Q j , j = 1, 2,..., s , do not depend on the generalized velocities, then cannot appear quadratic terms in the generalized velocities, resulting T2 = 0 . In these conditions, we state that Painlevé’s first integral does not contain explicitly the time. If the generalized forces admit a simple quasi-potential of the form (18.2.20), then we can introduce the kinetic potential (18.2.34), so that the relation (18.2.60) reads d ⎛ ∂L ⎞ q − L ⎟ + L = 0 ; dt ⎜⎝ ∂q j j ⎠

(18.2.62)

assuming that the kinetic potential L does not depend explicitly on time (L = 0) , we get Jacobi’s first integral (Jacobi,C.G.J., 1882)


71 ∂L q − L = h , h = const , ∂q j j

(18.2.63)

which can be expressed also in the form (hence, a particular case of Painlevé’s first integral) T2 − T0 − U = h , h = const .

(18.2.63')

In particular, if the mechanical system is catastatic, hence scleronomic (being holonomic), then we have T = 0 , so that L = 0 if U = 0 , hence if the forces are conservative, deriving from a simple potential; we obtain the first integral of mechanical energy E = T − U = h , T = T2 , h = const .

(18.2.64)

The relation (18.2.62) with L = 0 can be written in the form

⎡ d ⎛ ∂L ⎞ ∂L ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q ⎥ q j = 0 j ⎦ ⎣ ⎝ j ⎠

(18.2.62')

too. Starting from this result, some authors try to obtain Lagrange’s equations, which is not possible because the generalized velocities q j cannot by arbitrarily chosen (they correspond to the motion in the actual state). If the generalized forces admit a generalized quasi-potential of the form (18.2.21), with (18.2.22), then we can express them by the relations (18.2.3); we obtain ∂U j ⎞ ∂U 0 ⎛ ∂U k − Q j q j = ⎜ ⎟ q j qk − U j q j + ∂q q j ∂ ∂ q q j j k ⎝ ⎠ ∂U 0 dU 0 dU 0 = − U 0 − U j q j = − U . q − U j q j = ∂q j j dt dt

The relation (18.2.60') becomes d (T − T0 − U 0 ) + L = 0 . dt 2

(18.2.65)

Supposing that the kinetic potential L = T + U does not depend explicitly on time (L = 0) , it results a first integral of Jacobi type (particular case of the first integral of Painlevé) T2 − T0 − U 0 = h , h = const .

(18.2.66)

As well, in case of scleronomic constraints we have T0 = 0 (and T0 = 0 too), being thus sufficient that U 0 = 0 , hence that the forces be conservative, deriving from a generalized potential, to have a first integral of mechanical energy in the form

72


E = T − U 0 = h , T = T2 , h = const .

(18.2.67)

The mechanical systems which admit a first integral of Jacobi or a first integral of Jacobi type are called generalized conservative systems; we have seen that these systems are natural systems for which the kinetic potential L does not depend explicitly on time (L = 0) . The quasi-potential U or the scalar potential U 0 are quantities of energetical nature; hence, the potential energy is V = −U , in case of a simple quasi-potential, or V = −U 0 , in case of a generalized quasi-potential. We introduce thus the generalized mechanical energy

E = T2 − T0 + V ,

(18.2.68)

which is constant in case of Jacobi’s first integral or of a first integral of Jacobi type. In particular, if the forces are conservative (we have V = 0 ) and the constraints are scleronomic (we have T0 = 0 and T = T2 ), then the generalized mechanical energy is equal to the mechanical energy, obtaining thus the first integral of mechanical energy. In case of generalized forces of the form (18.2.24), the relation (18.2.60) reads d ⎛ ∂L ⎞ q j − L ⎟ + L = Q j q j . ⎜ dt ⎝ ∂q j ⎠

(18.2.69)

If the kinetic potential does not depend explicitly on time ( L = 0 ), then it results

dE d ⎛ ∂L ⎞ = q − L ⎟ = Q j q j . dt dt ⎜⎝ ∂q j j ⎠

(18.2.69')

We can thus state that the mechanical system S admits a Jacobi’s first integral (written in the form (18.2.63) or in the form (18.2.63')) for generalized non-potential forces too. The relation (18.2.69') allows also to notice that, in case of dissipative nonpotential forces, the energy of the mechanical system diminishes. We observe that in none of the above considered first integrals does not intervene the component T1 of the kinetic energy; this is explained because the kinetic energy is determined excepting the total derivative with respect to time of a function ϕ (q1 , q2 ,..., qs ; t ) of class C 2 , hence excepting dϕ dt = ( ∂ϕ ∂q j )q j + ϕ . In case of non-holonomic constraints, Lagrange’s equations have the form (18.2.32), so that the relation (18.2.60) becomes d ⎛ ∂T ⎞ q − T ⎟ = ( Q j + R ) q j − T . dt ⎜⎝ ∂q j j ⎠

Taking into account (18.2.29') and (18.2.33), we can also write


73

m d ⎛ ∂T ⎞ q j − T ⎟ = Q j q j − T − ∑ λk ak 0 . ⎜ dt ⎝ ∂q j ⎠ k =1

(18.2.70)

If the constraints are catastatic (we have ak 0 = 0, k = 1, 2,..., m ), then the relation (18.2.60) becomes (we have T = 0 too)

d ⎛ ∂T ⎞ q − T ⎟ = Q j q j , dt ⎜⎝ ∂q j j ⎠

(18.2.71)

so that the influence of these constraints is not seen; written in the form (18.2.60'), this relation becomes (we have also T0 = T1 = 0 ) dT2 = Q j q j . dt

Expressing the generalized ( E = T2 + V , V = −U )

forces

in

(18.2.71') the

form

dE = Q j q j − U , dt

(18.2.25),

we

can

write

(18.2.72)

so that, in the case in which the potential part does not depend explicitly on time (U = 0 ), while the non-potential forces are gyroscopic, we get a first integral. Hence, in general, a natural and catastatic, non-holonomic mechanical system, for which the generalized forces derive from a simple or generalized potential, has a first integral of energy of the form E = T + V = h , h = const .

(18.2.73)

Such a mechanical system is called a conservative system. If these generalized forces have only the non-potential part (U = 0 ), which is gyroscopic, even the kinetic energy T is conserved during the motion. We notice that this first integral of energy differs from that obtained for a discrete mechanical system S in E 3 because the holonomic constraint forces do not intervene any more. As a matter of fact, in case of catastatic constraints no essential difference appear, the real elementary work of the corresponding constraints forces vanishing. We notice also that, if we take into account (18.2.25'), (18.2.26) in case of a potential (U = 0 ), then the relation (18.2.72) leads to dE = −2 R , dt

(18.2.74)

where R is Rayleigh’s dissipative function, which characterizes the decrease of the mechanical energy E .


74

18.2.3.5 Reduction of Lagrange’s System of Equations by Means of Jacobi’s First Integral One can state Theorem 18.2.5. Jacobi’s first integral allows the reduction of Lagrange’s system of equations (18.2.38), corresponding to a holonomic and natural mechanical system with s degrees of freedom, to a system of equations corresponding to a mechanical system with s − 1 degrees of freedom (Jacobi,C.G.J., 1882). In particular, in case of a mechanical system with a single degree of freedom (a mechanical system with complete constraints), the first integral ∂L q − L = h , h = const , ∂q

(18.2.75)

reduces the problem to a quadrature; indeed, from (18.2.75) we can get q = f (q ) , wherefrom

t − t0 =

q

∫q

0

dq . f (q )

(18.2.75')

In case of a mechanical system with s degrees of freedom, we make the change of generalized velocities q1 → q1 , q2 → q1q2′ ,..., qs → q1qs′ , with qk′ = dqk dq1 = qk q1 ,

k = 1,2,..., s , so that L (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) = Λ(q1 , q2 ,..., qs , q1 , q2′ ,..., qs′ ) . Differentiating, we can write too ∂L ∂Λ ∂L ∂Λ ∂qk′ 1 ∂Λ = = = , j = 1, 2,..., s , ( ! ) , k = 2, 3,..., s , ∂q j ∂q j ∂qk ∂qk′ ∂qk q1 ∂qk′ ∂Λ ∂L = + ∂q1 ∂q1

∂L ∂qk ∂L = + ∂ ∂q1 q 1 k =2 s

∑ ∂qk

s

∂L

∑ ∂qk

k =2

qk′ =

∂L + ∂q1

∂L qk . q1 k =2 s

∑ ∂qk

By the above mentioned change of variables in the first integral (18.2.63) and by taking into account the preceding results, we can write

q1

∂Λ −Λ = h, ∂q1

(18.2.76)

wherefrom q1 = q1 (q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) . We denote L ′ = ∂Λ ∂q1 , so that

L ′ = L ′(q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) . Using the notations thus introduced and taking into account the form (18.2.15) of the kinetic energy, we can write T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) = T (q1 , q2 ,..., qs , q1 , q2′ ,..., qs′ ) ,


75

so that T = q12T , where T = T (q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) ; on the other hand, assuming the existence of a simple quasi-potential, we have T = U + h , h = const , while L ′ = ∂T ∂q1 = 2q1T , whence L ′ = 2 T (U + h ) . We can calculate ∂L ′ ∂2 Λ ∂ 2 Λ ∂q = + 2 1 , j = 1, 2,..., s , ∂q j ∂q j ∂q1 ∂q1 ∂q j ∂L ′ ∂2 Λ ∂2 Λ ∂q = + 2 1 , k = 2, 3,..., s ; ∂qk′ ∂qk′ ∂q1 ∂q1 ∂qk′

as well, by a partial differentiation of the relation (18.2.76) with respect to the generalized co-ordinates and the new generalized velocities, we obtain q1

∂ 2 Λ ∂q1 ∂2 Λ ∂Λ + q1 = , j = 1, 2,..., s , 2 ∂q ∂ ∂ ∂ q q qj ∂q1 j j 1

q1

∂2 Λ ∂q1 ∂2 Λ ∂Λ + q1 = , k = 2, 3,..., s . 2 ∂q ′ ∂qk′ ∂q1 ∂qk′ ∂q1 k

Comparing with the previous results, we may write ∂L ′ 1 ∂Λ ∂L ′ 1 ∂Λ = , j = 1, 2,..., s , = , k = 2, 3,..., s , ∂q j ∂qk′ q1 ∂q j q1 ∂qk′

wherefrom, using the above obtained relations, we get ∂L ′ 1 ∂L ∂L ′ ∂L = , j = 1, 2,..., s , = , k = 2, 3,..., s . ∂q j ∂qk′ ∂qk q1 ∂q j

Lagrange’s equations (18.2.38) lead to d ⎛ ∂L ′ ⎞ ∂L ′ − q1 = 0, k = 2, 3,..., s , ⎜ ⎟ ′ dt ⎝ ∂qk ⎠ ∂qk

or to d ⎛ ∂L ′ ⎞ ∂L ′ − = 0, k = 2, 3,..., s , dq1 ⎜⎝ ∂qk′ ⎟⎠ ∂qk

(18.2.77)

so that the functions qk (q1 ), k = 2, 3,..., s are determined by a system of s − 1 equations of second order of Lagrange type (the equations of Jacobi, who has put in evidence this method of calculation) in a (s − 1) -dimensional space. Thus, the number of degrees of freedom of the initial mechanical system has been reduced by a unity.


76

From the expression of the function L ′ of Lagrangian type, one observes that – in general – the new independent variable q1 intervenes explicitly, so that it is no more possible to obtain a first integral of the type of Jacobi’s first integral. By means of the notations thus introduced, we notice that q1 = t − t0 =

q1

∫q

0 1

T dq1 , U +h

(U + h ) T , so that

(18.2.77')

completing thus the information given by the equations (18.2.77) about the dependence with respect to time.

18.2.3.6 Hidden Co-ordinates. Ignorable Co-ordinates We call, after Helmholtz, hidden co-ordinate qk that co-ordinate which does not intervene explicitly in the expression of the kinetic energy T , hence for which ∂T ∂qk = 0 ; because we can have h ≤ s hidden co-ordinates, it is convenient to denote them q1 , q2 ,..., qh . If the co-ordinate qk does not intervene explicitly in the expression of the kinetic potential L = T + U , corresponding to a natural system which admits a simple or generalized quasi-potential, hence if ∂L ∂qk = 0 , then this co-ordinate is called an ignorable co-ordinate (after W. Thomson) or a kinostenic coordinate (after J. J. Thomson); this denomination is used also in the case of a nonnatural mechanical system which admits a Lagrangian, with the condition (18.2.34'''). We can have r ≤ s ignorable co-ordinates q1 , q2 ,..., qr . Sometimes only the denomination of hidden co-ordinates is used; but we will use both the denominations of hidden and ignorable co-ordinates, to can make distinction in various situations. The mechanical systems in which appear ignorable co-ordinates are called – by some authors – gyroscopic systems (because such situations appear often in case of the gyroscope). If ∂U ∂qk = 0 , where U is a simple or generalized quasi-potential, then an eventually corresponding ignorable co-ordinate is also a hidden one. We notice that we can have ∂ (T + U ) ∂qk = 0 , without having necessarily ∂T ∂qk = 0 (Pars, L., 1965). In the case of h hidden co-ordinates q1 , q2 ,..., qh , the first h equations of Lagrange for holonomic constraints (18.2.29) read d ⎛ ∂T ⎞ = Qk , k = 1, 2,..., h . dt ⎜⎝ ∂qk ⎟⎠

(18.2.78)

If we have Qk = 0 too, then there result h first integrals ∂T = ck , k = 1, 2,..., h , ∂qk

(18.2.78')


77

each first integral corresponding to a hidden co-ordinate. Assuming the existence of r ignorable co-ordinates q1 , q2 ,..., qr , Lagrange’s equations (18.2.38) take the form d ⎛ ∂L ⎞ = 0, k = 1, 2,..., r , dt ⎜⎝ ∂qk ⎟⎠

(18.2.79)

wherefrom the first integrals ∂L = C k , k = 1, 2,..., r . ∂qk

(18.2.79')

By convention, we denote pj =

∂L , j = 1, 2,..., s . ∂q j

(18.2.80)

In the case of a simple quasi-potential (or in the absence of a quasi-potential), it results pj =

∂T , j = 1, 2,..., s . ∂q j

(18.2.80')

If, in particular, the mechanical system is free or none holonomic constraint has been eliminated, the space of Lagrange being just the space E 3n , then – starting from the kinetic energy corresponding to the n particles – we can write ∂T ∂ ⎛1 n ⎞ = mi vi2 ⎟ = m j v j . ∂v j ∂v j ⎜⎝ 2 i∑ ⎠ =1

We notice that this represents just the magnitude of the momentum of the particle corresponding to the velocity v j . Starting from this observation, we will call the magnitudes p j , j = 1, 2,..., s , components of the generalized momentum; the first integrals (18.2.78') will be written in the form pk = ck , k = 1, 2,..., h ,

(18.2.78'')

while the first integrals (18.2.79') read pk = C k , k = 1, 2,..., r .

(18.2.79'')

If U is a generalized quasi-potential of the form (18.2.22), then it results ∂T = p j − U j , j = 1, 2,..., s , ∂q j

(18.2.81)


78

and in case of hidden co-ordinates we obtain d ( p − U k ) = Qk . dt k

If we have Qk = 0 too, then we can write the first integrals pk − U k = ck , k = 1, 2,..., h .

(18.2.81')

We notice also that the notion of generalized momentum has a larger sense than the classical one, corresponding to the case in which the generalized co-ordinate is a length. Let us consider, e.g., a particle for which the kinetic energy is expressed in cylindrical coordinates ( T = (m 2)(r 2 + r 2 θ2 + z 2 ) ); in this case, pθ = ∂L ∂θ = ∂T ∂θ = mr 2 θ = m ( r × r )z . Hence, to a generalized co-ordinate which is an angle corresponds a

generalized momentum which is a component of a moment of momentum. If the relation (18.2.80') takes place, then the relation (18.2.78) reads pk =

dpk = Qk , k = 1, 2,..., h . dt

(18.2.82)

Hence, in case of a hidden co-ordinate and in the absence of a quasi-potential, we can write an universal theorem of mechanics. If the generalized co-ordinate is a length, then the generalized force is of the nature of a force (so that the product Qk δqk be a work), obtaining thus a theorem of momentum corresponding to the respective linear coordinate; but if the generalized co-ordinate is an angle, then the generalized force – from the same reasons – is of the nature of a moment, resulting a theorem of moment of momentum corresponding to the respective angular co-ordinate. As a matter of fact, we can state Theorem 18.2.6 (theorem of generalized momentum). In case of several hidden coordinates and in the absence of a quasi-potential, in the free motion of the representative point in the space of configurations, a theorem of generalized momentum of the form (18.2.82) is verified. In the case in which Qk = 0, k = 1, 2,..., h , we can write a conservation theorem of generalized momentum, which is – in fact – a first integral of the form (18.2.78''). As well, in case of generalized forces of the form (18.2.24) and of several ignorable co-ordinates, Lagrange’s equations lead to the relations pk =

dpk = Qk , k = 1, 2,..., r , dt

(18.2.82')

where Qk are non-potential generalized forces; hence, we can state

Theorem 18.2.6' (theorem of generalized momentum; second form). In case of several ignorable co-ordinates, in the free motion of the representative point in the space of configurations, a theorem of generalized momentum of the form (18.2.82') is verified.


79

If Qk = 0, k = 1, 2,..., r , then we get a conservation theorem of generalized momentum, hence a first integral of the form (18.2.79'').

18.2.3.7 The Routh–Helmholtz Theorem We can state Theorem 18.2.7 (Routh–Helmholtz). If a discrete mechanical system S , with s degrees of freedom, has r ignorable co-ordinates, then the determination of the free motion of the representative point in the space of configurations is reduced to the integration of a system of differential equations of Lagrange type, corresponding to a mechanical system with s − r degrees of freedom and to the calculation of r quadratures. Indeed, let us perform the transformation of Routh, E.J. 1892, 1898]

R =

r

∂L

∑ ∂qk

qk − L .

k =1

(18.2.83)

The first integrals (18.2.79'), where L is of the form (18.2.34'), can be seen as a system of r linear equations in the generalized velocities qk , k = 1, 2,..., r ; the condition (18.2.34''') being fulfilled, we determine the functions qk = qk (qr + 1 , qr + 2 ,..., qs , qr + 1 , qr + 2 ,..., qs ;C 1 ,C 2 ,...,C r ; t ), k = 1, 2,..., r ,

where the generalized co-ordinates qr + 1 , qr + 2 ,..., qs are called palpable co-ordinates (unlike the ignorable co-ordinates). Replacing in (18.2.83), we get

R = R (qr +1 , qr + 2 ,..., qs , qr +1 , qr + 2 ,..., qs ;C 1 ,C 2 ,...,C r ; t ) . In this case, applying a variation δ to the function R of Routh, we obtain δR =

∂R δq j + j = r + 1 ∂q j s

∑

∂R δq j + j = r + 1 ∂q j s

∑

∂R

r

∑ ∂C k

k =1

δC k .

Noting that δL =

∂L δq j + j = r + 1 ∂q j s

∑

∂L qk = k = 1 ∂qk r

δ∑

r

∂L

∑ ∂qk

k =1

s

∂L

∑ ∂q j j =1

δqk +

δq j ,

r

∑ qk δC k

k =1

and taking into account the relation of definition (18.2.83), we get, as well, δR = −

∂L δq j − j = r + 1 ∂q j s

∑

∂L δq j + j = r + 1 ∂q j s

∑

r

∑ qk δC k .

k =1


80

The variation δ being arbitrary, by comparison of the two relations, it results −

∂L ∂R ∂L ∂R = = ,− , j = r + 1, r + 2,..., s , ∂q j ∂q j ∂q j ∂q j qk =

∂R , k = 1, 2,..., r , ∂C k

so that, replacing in Lagrange’s equations (18.2.38), we get the equations

d ⎛ ∂R ⎞ ∂R − = 0, j = r + 1, r + 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.2.84)

which are of Lagrange type, corresponding to a mechanical system with s − r degrees of freedom; these s − r equations determine the generalized co-ordinates q j = q j (t ) ,

j = r + 1, r + 2,..., s . Replacing these co-ordinates in Routh’s function, it results

R = R (C 1 ,C 2 ,...,C r ; t ) and one can obtain the other generalized co-ordinates in the form qk − qk0 =

t

∫t

0

∂R d t = 0, k = 1, 2,..., r , ∂C k

(18.2.85)

the theorem being thus justified. Starting from (18.2.83), Hertz introduced a generalized kinetic energy of the form

T = T − ⎛⎜ V + ⎝

r

⎞

∑ C k qk ⎟⎠ ,

k =1

the sum being considered as being a potential of forces resulting from a unknown cyclic motion; as a matter of fact, Hertz tried to define the potential energy of a mechanical system by the cyclic motion of several “ignorable masses”, which are added to the palpable (visible) masses of the mechanical system S , which justifies the denomination given to the corresponding generalized co-ordinates. Let be the classical model of atom, called “rotator”, which has a motion of rotation about a fixed axis in the space. Because the kinetic energy T = (m / 2) (r 2 + r 2 θ2 + z 2 ) is thus that ∂T / ∂θ = 0 , the theorem of areas is = ∂L / ∂θ mr 2 θ = const ; taking into account r = const , it results θ = ω0t + θ0 , ω0 , θ0 = const . We notice that the function R of Routh plays the rôle of a kinetic potential for the new problem which remains to be solved. But we mention that in this catastatic mechanical system appear always terms linear in the generalized velocities too (even in case of the catastatic mechanical systems for which the kinetic energy is a positive definite quadratic form and which is acted upon by generalized forces which admit a simple quasi-potential).


81

18.2.3.8 Passing to an Equivalent System of Differential Equation of First Order In the absence of holonomic constraints, the space of configurations can be even the representative space E 3n . Indeed, the relation (18.2.1), written for a fixed t , leads to a metrics of the form ds 2 = g jk dq j dqk ,

(18.2.86)

where the coefficients of the Riemannian space R3n n

g jk =

∂r

∑ ∂qij

i =1

⋅

∂ri , j , k = 1, 2,..., s , ∂qk

(18.2.86')

can be obtained from the coefficients (18.2.15') taking masses equal to unity and maintaining the property det[ g jk ] ≠ 0 ; if the mechanical system is a free one, then s = 3n , hence an Euclidean space E 3 n , so that – by a linear transformation – we can obtain the normal form with constant coefficients

ds 2 =

∑ ⎡⎣⎢ ( dx1( i ) ) n

i =1

2

(

+ dx 2( i )

) + ( dx 3(i ) ) 2

2

⎤, ⎦⎥

(18.2.86'')

the square of the element of arc being expressed in the form of a sum of squares. If the time t is not fixed, ds 2 is an arbitrary polynomial of second degree in dq j (corresponding to the analogous form of the kinetic energy); but the character of the kinetic energy for a mechanical motion leads, by a convenient transformation, to an element of arc of the form (18.2.86'') too. But if there exist certain constraints, the space R3 n becomes a Riemannian space (of positive curvature), because – reducing the number of the configurative dimensions – the Euclidean character of the space is destroyed. In general, the state of a discrete mechanical system S is determined by its position and its motion in the neighbourhood of this position, hence by the set q j = q j (t ) , u j = q j = u j (t ), j = 1,2,..., s . We can thus conceive a representative space with 2s

dimensions, in which the representative point of co-ordinates q j , u j , j = 1, 2,..., s , does represent, at any moment, “the state” of the mechanical system S ; this space is called the state space or the phase space. In the case in which s = 3n , this space is 6n -dimensional and has not the character of a metric manifold; it is only an affine space and one cannot choose a common unity of measure for both types of co-ordinates, but it has a particular importance in the analytical representation of the motion. We have seen in Sect. 18.2.2.4 that one can replace Lagrange’s system of s equations of second order (18.2.45') by a system of 2s differential equations of first order (18.2.48), written in the normal form, the unknown functions being q j = q j (t ) and u j = u j (t ), j = 1, 2,..., s . This system of equations can be written also in the form


82

d q1 dq du1 du2 dqs dus dt = 2 = ... = = = = ... = = . 1 u1 u2 us ϕ1 ϕ2 ϕs

(18.2.87)

If we succeed to find 2s first integrals of the form fl (q1 , q2 ,..., qs , u1 , u2 ,..., us ; t ) = C l , l = 1,2,..., 2s , where C l are arbitrary constants, then we can determine the

unknown functions q j = q j (t ;C 1 ,C 2 ,...,C 2 s ) , u j = u j (t ;C 1 ,C 2 ,...,C 2 s ) , obviously, the relations dq j dt = u j , j = 1, 2,..., s , taking place too. The problem of motion of the mechanical system S is thus entirely solved. Let us suppose that the mechanical system S is catastatic, so that the time t does not appear explicitly in the expression of the kinetic energy; in this case,, the equations of motion take the form (18.2.47') (we assume to have only holonomic constraints, which – in this case – are scleronomic), where Christoffel’s symbols of second kind do not depend explicitly on time. As well, we suppose that neither the generalized forces Q j∗ do not depend explicitly on time ( ∂Q j∗ ∂t = 0, j − 1,2,.., s ). In this case, nor in the system of equations (18.2.87) the time does not intervene explicitly in the first 2s ratios; neglecting the ratio dt 1 , we will try to integrate the system of equations

dq1 dq du1 du2 dqs dus = 2 = ... = = = = ... = . u1 u2 us ϕ1 ϕ2 ϕs

(18.2.88)

If we succeed now to find the first integrals fl (q1 , q2 ,..., qs , u1 , u2 ,..., us ) = C l ,

l = 1,2,..., 2s − 1 , then we can determinate 2s − 1 unknown functions with respect to one the co-ordinates, e.g. q1 , in the form qk = qk (q1 ;C 1 ,C 2 ,...,C 2 s −1 ), k = 2, 3,..., s , u j = u j (q1 ;C 1 ,C 2 ,...,C 2 s −1 ), j = 1, 2,..., s .

Associating the equation dq1 u1 = dt 1 , we obtain t +τ =

dq

∫ u1 (q1 ;C1 ,C 2 ,...,Cs −1 ) ,

(18.2.88')

completing thus the integration constants by τ , the integration of the system of 2s linear equations (18.2.88) is thus reduced to the integration of a system of 2s − 1 linear equations and to a quadrature. Assuming that one can write a Jacobi first integral of the form (18.2.63) too, we can express then one of the generalized velocities, for instance q1 , in the form q1 = u1 (q1 , q2 ,..., qs , u2 , u3 ,..., us ; h ) , so that – instead of the system (18.2.88) – we

have to integrate a system of 2s − 2 linear differential equations


83

d q1 dq du2 du 3 d qs dus = 2 = ... = = = = ... = , u1 u2 us ϕ2 ϕ3 ϕs

(18.2.89)

with ϕk = ϕk (q1 , q2 ,..., qs , u1 , u2 ,..., us ), k = 2, 3,..., s . Thus, the complete system of integrals of the equations of motion (18.2.88) is formed by 2(s − 1) integrals of the system (18.2.89), by Jacobi’s first integral (18.2.63) and by the integral (18.2.88'). We put thus in evidence the trajectory of the representative point in the state space. Such a study has been made by Hamilton in the phase space and we will present it systematically in the next chapter.

18.3 Other Problems Concerning Lagrange’s Equations There are also other problems concerning Lagrange’s equations which present a particular interest, as the forms taken by these equations in some special cases or different systems of equations of motion equivalent to them. Besides these results, we will deal with some significant applications of the analytical methods introduced by Lagrange.

18.3.1 New Forms of Lagrange’s Equations In the following, we consider some new forms of Lagrange’s equations, as the central equation of Lagrange, the equations of Nielsen, the equations of Tsenov and the Mangeron–Deleanu equations. As well, we will put in evidence the form taken by these equations in case of continuous mechanical system, in case of a non-inertial frame of reference and in case of the phenomenon of collision. 18.3.1.1 Lagrange’s Central Equation Let be a particle Pi of position vector ri and of mass mi , which belongs to a discrete mechanical system S . We can write the obvious relation

(

d dri ⋅ δri dt dt

)

=

( ) ⎥⎤⎦ + ddrt ⋅ ⎡⎢⎣ ddt ( δr ) − δ ( ddrt ) ⎤⎥⎦ .

d2 ri ⎡ 1 dri ⋅ δri + δ ⎢ 2 dt ⎣ 2 dt

2

i

i

i

Taking into account the relations (18.2.1), we have T (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) =

( ),

dri 1 n mi 2 i∑ dt =1

2

wherefrom, using the relations (18.2.28) too, we obtain ∂T δq = ∂q j j

n

∑ mi

i =1

( )

dri ∂ dri ⋅ δq j = dt ∂q j dt

n

∑ mi

i =1

dri ∂ri ⋅ δq = dt ∂q j j

n

∑ mi

i =1

The principle of virtual work, written in the form (18.2.27), leads to

dri ⋅ δri . dt


84 n

∑ mi

i =1

d2 ri ⋅ δri = Q j δq j . dt 2

In this case, multiplying the above identity by mi and summing for all the indices i , it results d ⎛ ∂T ⎞ δq = Q j δq j + δΤ + dt ⎜⎝ ∂q j j ⎟⎠

( )

dri ⎡ d dri ⎤ ⋅ ( δri ) − δ . dt ⎢⎣ dt dt ⎥⎦

n

∑ mi

i =1

Observing that δΤ =

∂T ∂T δq j + δq , ∂q j ∂q j j

we can write this relation in the form ∂T ⎡ d ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q − Q ⎥ δq j = ∂q ⎢⎣ δq j − dt ( δq j ) ⎥⎦ j j ⎣ ⎝ j ⎠ ⎦ n dr dri d − ∑ mi i ⋅ ⎡ δ − ( δri ) ⎤ . ⎢ ⎥⎦ dt ⎣ dt dt i =1

( )

(18.3.1)

We get thus, after Hamel, Lagrange’s central equation (Hamel, G., 1927). In case of holonomic constraints, the operator relations (18.1.77) take place for the position vectors in E 3 ; as well, the operator relations (18.2.13) take always place for the generalized co-ordinates in Λs . In these conditions, the equation (18.3.1) leads to the fourth form of the fundamental equation (the equation (18.2.27')). 18.3.1.2 Mechanical Systems with Separable Variables. Systems of Liouville Type There are many problems, e.g. in celestial mechanical or in atomic physics, which lead to a Lagrangian for which the kinetic energy and the simple potential of the generalized forces are of the form T =

1 s Vj (q j )q2j , U = 2∑ j =1

s

∑U j (q j ) , j =1

(18.3.2)

where U j ,Vj , j = 1, 2,..., s , are functions of only one variable of class C 1 . In this case, Lagrange’s equations (18.2.29) read Vj (q j )qj +

dU j 1 dV j 2 q = , j = 1, 2,..., s . 2 dq j j dq j

Multiplying by q j and integrating with respect to time, we get

(18.3.3)


85

1 V (q )q 2 − U j (q j ) = C j , j = 1,2,..., s , 2 j j j

(18.3.3')

where C j , j = 1, 2,..., s , are integration constants. Summing with respect to the index j . one obtain the first integral of mechanical energy; but we have thus first integrals corresponding to s particular mechanical energies (eventually, to the mechanical energies of s particles). By s quadratures of the form t + τj =

Vj (q j ) 1 dq j , j = 1, 2,..., s , 2 ∫ U j (q j ) + C j

(18.3.4)

we introduce the integration constants τ j , j = 1, 2,..., s , too. Thus, one obtains the relations which link the time to each generalized co-ordinate, the problem being thus entirely solved. The respective mechanical systems are mechanical systems with separable variables. A problem similar is that of the discrete mechanical systems, called systems of Liouville type, for which we have s

T =

s 1 s uk (qk ) ∑Vj (q j ) q 2j , U = ∑ 2 k =1 k =1

∑U j (q j ) j =1 s

∑ uk (qk )

(18.3.5)

k =1

and where U j ,Vj and uk , j , k = 1, 2,..., s , are functions of only one variable. Obviously, the case considered above is a particular case of a system of Liouville type for which uk = 1 s = const . By the substitution Vj (q j ) =

dWj (q j ) W j = , j = 1, 2,..., s , q j dq j

we can write T =

u s 2 1 s Wj , U = ∑U j (Wj ), u = ∑ 2 k =1 u j =1

s

∑ uk (Wk ) .

k =1

(18.3.6)

Lagrange’s equations relative to the new generalized co-ordinate Wj take the form d ( uW k dt

1 ∂u

s

) − 2 ∂W ∑W j2

=

k j =1

∂U , k = 1, 2,..., s . ∂Wk

Multiplying by 2uW k , we get d 2 2 ∂u ( u Wk ) − uW k ∂W dt k

s

∑W j2 j =1

= 2uW k

∂U . ∂Wk


86

On the other hand, the first integral of mechanical energy is u s 2 Wj − U = h , h = const . 2∑ j =1

We are thus led to d 2 2 ( u Wk dt = 2W k

∂u ∂U + 2uW k ∂Wk ∂Wk

) = 2(h + U )Wk

∂ d [ ( h + U ) u ] = 2 [ huk (qk ) + U k (qk ) ] , k = 1, 2,..., s , ∂Wk dt

wherefrom one obtains the system of differential equations dW1 dW2 = = ... hu1 (q1 ) + U 1 (q1 ) + α1 hu2 (q2 ) + U 2 (q2 ) + α2 dWs dt = = 2 . u hus (qs ) + U s (qs ) + αs

(18.3.7)

Noting that u 2W k2 = 2(huk + U k ) + 2 αk , k = 1,2,..., s , summing with respect to all indices and taking into account the integral of mechanical energy, we find the condition s

∑ αk

k =1

= 0,

(18.3.7')

which must be verified by the constants αk , k = 1, 2,..., s , thus introduced. The equations (18.3.7) are with separate variables, the problem being reduced to quadratures. 18.3.1.3 Case of Continuous Mechanical Systems Let be a continuous mechanical system with a finite number of degrees of freedom, which occupies a domain V in E 3 and is acted upon by a volumic force F ( r ) (hence, by the force F dV on the element of volume); the torsor of these forces (resultant force and resultant moment) at the pole O is given by R =

∫V F (r )dV , MO

=

∫V r × F dV .

(18.3.8)

The kinetic energy is expressed in the form

T =

( ) dV ,

1 dr μ( r ) 2 ∫V dt

2

(18.3.9)

where μ ( r ) is the density, so that in the formulae (18.2.15), (18.2.15'), (18.2.15''), and (18.2.15''') we will have


87 g jk =

gj =

∂r

∫V μ(r ) ∂q j

∂r

∫V μ(r ) ∂q j

⋅

∂r dV , ∂qk

⋅ r dV , j , k = 1, 2,..., s , g 0 =

1 μ( r )r 2 dV . 2 ∫V

(18.3.9')

Analogously, the virtual work of the volumic forces will be given by δW =

∫V F ⋅ δr dV

= δq j

∂r

∫V F ⋅ ∂q j dV

= Q j δq j ,

so that the generalized forces will be Qj =

∂r

∫V F ⋅ ∂q j dV .

(18.3.10)

The principle of virtual work is expressed in the form ∂r ⎡ ⎤ ⎢ ∫V μ ( r )r ⋅ ∂q − Q j ⎥ δq j = 0 . j ⎣ ⎦

(18.3.11)

By a demonstration analogous to that in Sect. 18.2.2.1, where the sign plus is replaced by the sign integral, we find again the fourth form of the fundamental equation (18.2.27'). The results obtained are thus justified for rigid solids too.

18.3.1.4 Electromechanical Analogy We have seen in Sect. 8.2.2.10 that one can establish an interesting analogy between an R.L.C. circuit (an ohmic resistance R , a loading inductance L and a condenser of capacity C , connected in series with a generator having an electromotive force E (t ) ) and a particle P (of mass m → L , of damping coefficient k ′ → R and of coefficient of elasticity k → 1/C , acted upon by a given force F (t ) → E (t ) , where we have put in evidence the correspondence between various quantities), in a vibrating motion. Thus, we can introduce quantities analogue to those of mechanical nature in the form (we use well known notations)

1 ∂ ⎛ Rq 2 ⎞ 1 2 T = Wq 2 , Q = − ⎜ q , = − Rq , U = − 2 ∂q ⎝ 2 ⎟⎠ 2C

(18.3.12)

where (1/ 2)Rq 2 corresponds to Rayleigh’s function, q = q (t ) is the charge, while i = i (t ) = q (t ) is the intensity of the current. Lagrange’s equation d ⎛ ∂T ⎞ ∂T ∂U = +Q + F ⎜ ⎟− ∂q dt ⎝ ∂q ⎠ ∂q

leads to the equation (8.2.58).


88

The above considerations can be applied to complex R.L.C. circuits, corresponding to discrete mechanical systems with a finite number of degrees of freedom; using the mentioned electromechanical energy one obtains a system of differential equations which allows the determination of the electric charges q (t ) in each circuit.

18.3.1.5 Nielsen’s Equations We have established Lagrange’s equations by means of the d’Alembert–Lagrange principle (the principle of virtual work); analogously, Jourdain’s principle (18.1.57'), written in the form n

∑ mi ai

i =1

n

⋅ δvi = ∑ Fi ⋅ δvi ,

(18.3.13)

i =1

leads to an equivalent system of equations of motion. Starting from the relation (18.2.14) and taking into account (18.2.4'), we may write

∂ai ∂ 2 ri ∂r ∂v =2 qk + 2 i = 2 i , i = 1, 2,..., n , j = 1, 2,..., s , ∂q j ∂q j ∂qk ∂q j ∂q j

(18.3.14)

On the other hand, we notice that Jourdain’s principle can be written for a given position of the mechanical system, at a fixed moment, varying the velocities; thus, it results δvi =

∂vi ∂r δq j = i δq j , ∂q j ∂q j

(18.3.15)

where we took into account (18.2.28). In this case, Jourdain’s principle takes the form ⎡

n

∑ ⎢⎣ −mi ai i =1

⋅

∂vi ∂r ⎤ + Fi ⋅ i ⎥ δq j = 0 . ∂q j ∂q j ⎦

(18.3.13')

Using the expression of the kinetic energy, we get n

dT = dt

∑ mi vi

i =1

⋅ ai ,

∂T = ∂q j

n

∑ mi vi

i =1

⋅

∂vi , ∂q j

so that

( ) = ∑m v

∂ dT ∂q j dt

n

i =1

i i

⋅

∂ai + ∂q j

n

∑ mi ai

i =1

⋅

n ∂vi ∂v = 2 ∑ mi vi ⋅ i + qj ∂q j ∂ i =1

n

∑ mi ai

i =1

⋅

∂vi , ∂q j

where we have used the relation (18.3.14). Taking into account the expression (18.2.18) of the generalized forces, the form (18.3.13') of Jourdain’s principle leads to

( )

∂ dT ⎡ ∂T ⎤ ⎢ 2 ∂q − ∂q dt + Q j ⎥ δq j = 0 . j j ⎣ ⎦

(18.3.16)


89

Noting that the variations δq j of the generalized velocities are independent and arbitrary, in case of holonomic constraints, we can write the equations obtained in 1935 by J. Nielsen in the form (Dolapčev, Bl., 1966)

( )

∂ dT ∂T −2 = Q j , j = 1,2,..., s . ∂q j dt ∂q j

(18.3.17)

These equations are governing the motion of the representative point P in the space Λs . In case of the non-holonomic constraints (18.2.10), one must use the method of Lagrange’s multipliers. Taking into account the relation (we use the operator relation (18.2.13))

d ⎛ ∂vi ⎞ d ⎛ ∂ri ⎞ ∂vi , i = 1, 2,..., n , j = 1,2,..., s , = = ⎜ ⎟ dt ⎝ ∂q j ⎠ dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.18)

we can write the derivative ∂v d ⎛ ∂T ⎞ d n mi vi ⋅ i = = ⎜ ⎟ q j ∂ dt ⎝ ∂q j ⎠ dt i∑ =1

n

∑ mi ai

i =1

⋅

∂vi + ∂q j

n

∑ mi vi

i =1

⋅

d ⎛ ∂vi ⎞ dt ⎜⎝ ∂q j ⎟⎠

in the form

( )

d ⎛ ∂T ⎞ ∂ dT ∂T , j = 1, 2,..., s . = − dt ⎜⎝ ∂q j ⎟⎠ ∂q j dt ∂q j

We see thus that Nielsen’s equations are equivalent to Lagrange’s equations. We mention also that, in some cases, Nielsen’s equations allow a more simple calculation that Lagrange’s ones. We consider also Gauss’s principle (the principle of the least action) (18.1.60') or (18.1.66), written in the form n

∑ mi ai

i =1

⋅ δai =

n

∑ Fi

i =1

⋅ δai

(18.3.19)

for a given position and velocity of the mechanical system. at a fixed moment, varying only the accelerations; in this case, the relation (18.2.14) allows to write ∂ai ∂r = i , ∂qj ∂q j

(18.3.20)

∂ai ∂r δq = i δqj . ∂qj j ∂q j

(18.3.15')

so that δai =


90 The relation (18.3.19) takes the form n

⎡

∑ ⎢⎣ Fi

i =1

⋅

∂ri ∂r ⎤ − mi ai ⋅ i ⎥ δqj = 0 ∂q j ∂q j ⎦

(18.3.19')

and, taking into account the above results and (18.2.28), we obtain

( )

∂ dT ⎡ ∂T ⎤ ⎢ 2 ∂q − ∂q dt + Q j ⎥ δqj = 0 . j j ⎣ ⎦

(18.3.16')

Noting that, in case of holonomic constraints, the variations δqj of the generalized accelerations are independent and arbitrary, we find again the equations (18.2.17) of Nielsen. The principle of Gauss in finite form (18.2.60) can be written also in the form n

∑ mi ai

i =1

⋅ Δai − Q Δqj = 0 ,

(18.3.19'')

where we have put in evidence the finite variations of the accelerations ai and of the generalized accelerations qj ; this is the fifth form of the fundamental equation.

18.3.1.6 Tsenov’s Equations Starting from Gauss’s principle, written in the form (18.3.19) or in the form (18.3.19'), we can establish also other equations, equivalent to Lagrange’s equations. To do this, starting from the expression of the derivative dT / dt , let us calculate d2T = dt 2

n

n

i =1

i =1

∑ mi ai2 + ∑ mi vi

⋅ a(2) , i (2)

where we have introduced the accelerations of second order ai too). Using the equations (18.2.14), it results ) a(2 = i

(see Sect. 18.1.2.5

∂ri ∂2 ri ∂r d ⎛ ∂ri ⎞ + 2 qj + q qj qk + 2 i qj + ... j ⎜ ⎟ ∂q j ∂q j ∂qk ∂q j dt ⎝ ∂q j ⎠ 2 ∂r ∂r ⎞ ⎛ ∂ ri = i qj + 3 ⎜ qk + i ⎟ qj + ..., ∂q j ∂q j ⎠ ⎝ ∂q j ∂qk

wherefrom ) a(2 = i

∂ri ∂v qj + 3 i qj + ... , ∂q j ∂q j

(18.3.21)


91

so that ) ∂a(2 ∂r i = i , ∂qj ∂q j

(18.3.20')

) ∂a(2 ∂v i =3 i . ∂qj ∂q j

(18.3.14')

as well as

Taking into account these results, we obtain n ∂a ∂ ⎛ d2T ⎞ = 2 mi ai ⋅ i + ∑ ⎜ ⎟ ∂qj ⎝ dt 2 ⎠ ∂ qj i =1

n

∑ mi vi

n n ∂ai ∂r ∂v = 2 ∑ mi ai ⋅ i + 3 ∑ mi vi ⋅ i , ∂qj ∂ ∂ q qj j i =1 i =1 (2)

⋅

i =1

so that the relation (18.2.19') leads to

⎧ 1 ⎡ ∂T

n

∑ ⎨⎩ 2 ⎢⎣ 3 ∂q j

i =1

−

∂ ⎛ d2T ⎞ ⎤ ⎫ + Q j ⎬ δqj = 0 . ∂qj ⎜⎝ dt 2 ⎟⎠ ⎥⎦ ⎭

(18.3.22)

Because, in case of holonomic constraints, the variations δqj of the generalized accelerations are independent, we can write the equations of I. Tsenov in the form ∂ ⎛ d2T ⎞ ∂T −3 = 2Q j , j = 1, 2,..., s . ⎜ ⎟ 2 ∂qj ⎝ dt ⎠ ∂q j

(18.3.23)

Taking into account the previous relations, we find easily that

( )

1 ⎡ ∂ ⎛ d2T ⎞ ∂T ⎤ ∂ dT ∂T −3 = −2 , j = 1,2,..., s . 2 ⎢⎣ ∂qj ⎜⎝ dt 2 ⎟⎠ ∂q j ⎥⎦ ∂q j dt ∂q j

(18.3.24)

Hence, the equations obtained in 1962 by Tsenov are equivalent to Nielsen’s equations (hence, to Lagrange’s equations too), in the case of ideal holonomic constraints (Dolapčev, Bl., 1966).

18.3.1.7 The Mangeron–Deleanu Equations Starting from the expression of the derivative d2T / dt 2 , we observe that n d3T ) 3 mi ai ⋅ a(2 = + ∑ i dt 3 i =1 n d4T ( 3) = 4 ∑ mi ai ⋅ ai + 4 dt i =1

n

∑ mi vi

i =1

n

∑ mi vi

i =1

⋅ a(i 3 ) ,

⋅ ai

(4)

+ ...,


92

where, in the last expression, we have mentioned only terms which contain generalized accelerations till qj (and not generalized accelerations of higher order); by complete induction, one can show that n dmT ( m − 1) + m = m ∑ mi ai ⋅ ai dt i =1

n

∑ mi vi

i =1

⋅ a(i m ) + ... ,

(18.3.25)

where the terms which contain the generalized accelerations q (j m ) have been put in evidence. On the other hand, starting from (18.3.21) and using the operator relation (18.2.13), we obtain (3)

ai

=

∂ri ∂v q j + 4 i qj + ... ∂q j ∂q j

and, in general, by complete induction, =

(m )

ai

∂ri ( m + 1) ∂v ( m ) + (m + 1) i q j + ..., i = 1, 2,..., n , q ∂q j j ∂q j

(18.3.26)

wherefrom ∂a(i m ) ( m + 1)

∂q j

=

(m ) ∂ri ∂ai ∂v , ( m ) = (m + 1) i , i = 1, 2,..., n , j = 1, 2,..., s . ∂q j ∂q ∂q j j

(18.3.26')

We can thus calculate ∂ ∂q j

(m )

( m − 1) m n ⎛ d T ⎞ = m m a ⋅ ∂ai ⎜ m ⎟ ∑ i i ∂q ( m ) + ⎝ dt ⎠ i =1 j

n

= m ∑ mi ai ⋅ i =1

n

∑ mi vi

i =1

⋅

∂a(i m ) ∂q j

(m )

+ ...

n ∂ri ∂v + (m + 1) ∑ mi vi ⋅ i + ..., j = 1, 2,..., s . ∂q j ∂ qj i =1

(18.3.25')

The relation 1 ⎡ ∂ ⎛ dmT ⎞ ∂T ⎢ ⎜ ⎟ − (m + 1) m ⎢ ∂q ( m ) ⎝ dt m ⎠ ∂q j ⎣ j

( )

⎤ ∂ dT ∂T ⎥ = . −2 q d t qj ∂ ∂ j ⎥⎦

(18.3.24')

which generalizes the relation (18.3.24) is thus easily stated. Thus, we find the D. I. Mangeron–D. Deleanu equations (Lagrange’s generalized equations), obtained in 1962 in the form ∂ ∂q (j m )

m ⎛ d T ⎞ − (m + 1) ∂T = mQ , j = 1, 2,..., s , m ∈ ` . ⎜ m ⎟ j ∂q j ⎝ dt ⎠

(18.3.27)


93

These equations are equivalent to Lagrange’s equations, to Nielsen’s equations and to Tsenov’s equations, which result as particular cases for m = 1 and m = 2 , respectively. Let us consider a differential principle of the form n

∑ mi ai

( m − 1)

⋅ δai

i =1

=

n

∑ Fi

i =1

( m − 1)

⋅ δai

,m ∈`,

(18.3.28)

where we have put in evidence accelerations of higher order; assuming that this principle takes place for a given positions, a given velocity and given accelerations till the (m − 2) -th order inclusive, at a fixed moment, varying only the accelerations of the (m − 1) -th order, we can write δa(i m ) =

∂a(i m −1) (m ) ∂q j

δq (j m ) =

∂ri δq ( m ) , i = 1, 2,..., n , m ∈ ` . ∂q j j

(18.3.26'')

The differential principle (18.3.28) can be written in the form n

⎛

∑ ⎜⎝ Fi

i =1

⋅

∂ri ∂r ⎞ − mi ai ⋅ i ⎟ δqi( m ) = 0, m ∈ ` . ∂q j ∂q j ⎠

(18.3.28')

Starting from this principle we can obtain the Mangeron–Deleanu equations. Obviously, in this case, the ideal constraints are defined by the relation n

∑ Ri

i =1

⋅ δqi( m −1) = 0, m ∈ ` .

(18.3.29)

All the equations thus established can be used for holonomic mechanical systems, immaterial from the type of ideal constraints which are considered, which represents an interesting generalization from the mechanical point of view. In case of non-holonomic constraints, these equations are completed with constraint generalized forces, after Lagrange’s method; one uses the equations which correspond to different types of ideal constraints.

18.3.1.8 Case of a Non-inertial Frame of Reference If we report the discrete mechanical system S to a non-inertial frame of reference R (movable with respect to an inertial frame R ′ , considered fixed), then the

generalized co-ordinates q j , j = 1, 2,..., s , will specify its position with respect to this frame; as a matter of fact, the position of the system S is thus specified even with respect to the frame R ′ , because the motion of the frame R with respect to the frame R ′ is known. One uses Lagrange’s equations (18.2.29) in the form

d ⎛ ∂T ′ ⎞ ∂T ′ − = Q j , j = 1,2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.30)


94

where the kinetic energy with respect to the frame R ′ is given by T′ =

1 n mi ( vO′ + vi + ω × ri )2 , 2 i∑ =1

vi being the velocities relative to the frame

(18.3.30')

R , while Q j = Q j (ql , ql ; t ) are the

generalized forces; because ri = ri (ql ; t ) , one obtains T ′ = T ′(ql , ql ; t ) , the problem being then reduced to that corresponding to an inertial frame. Another method of calculation is based on the theory of relative motion (see Sect. 10.2.1); one introduces thus the generalized forces Q (j t ) = Q (j t ) (ql , ql ; t ) and (C )

Qj

= Qj

(C )

(ql , ql ; t ), j = 1, 2,..., s , corresponding to the transportation generalized

forces and to the Coriolis generalized forces, respectively. Lagrange’s equations become

d ⎛ ∂T ⎞ ∂T (t ) (C ) − = Q j′ , Q j′ = Q j + Q j + Q j , j = 1, 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.31)

where T = T (ql , ql ; t ) is the kinetic energy given by (calculated in the frame R )

T =

1 n mi vi2 . 2 i∑ =1

(18.3.31')

Gilbert elaborated in 1883 a mixed method, which is based – partially – on the theory of relative motion; one introduces thus a frame R with the origin at the pole O and with the axes parallel to the axes of the frame R ′ (which is not rotating about this frame). Hence, Lagrange’s equations read

d ⎛ ∂T ⎞ ∂T ∂U , j = 1,2,..., s , − = Qj + dt ⎜⎝ ∂q j ⎟⎠ ∂q j ∂q j

(18.3.32)

where U = U (ql , ql ; t ) is a potential given by

U =

n

∑ mi ri

i =1

⋅ aO′ = − M ρ ⋅ aO′ ,

(18.3.32')

corresponding to the centrifugal force. The kinetic energy T = T (ql , ql ; t ) with respect to the frame R is given by (see Sect. 11.2.2.2) T =

1 n 1 mi ( vi + ω × ri )2 = T + I Δ ω 2 + ω ⋅ KO , 2 i∑ 2 =1

(18.3.32'')


95

where KO and T are the moment of momentum with respect to the pole O and the kinetic energy, respectively, in the frame R , while I Δ is the moment of inertia of the mechanical system S with respect to the instantaneous axis of rotation Δ ; thus, the kinetic energy T with respect to the generalized co-ordinates and the generalized velocities is easily obtained. Using this method, Gilbert studies the influence of the rotation of the Earth on the motion of a heavy rigid solid of revolution, fixed at a fixed point O (different from C ), the principal axis of inertia Ox 3 being in rotation in a vertical plane linked to the Earth; the apparatus by which Gilbert verified experimentally the results theoretically obtained is called barogyroscope.

18.3.1.9 The Phenomenon of Collision In case of a phenomenon of collision, we start from the theorem of momentum, written in the form (13.1.8) for a particle Pi of the discrete mechanical system S subjected to constraints; a scalar product by ∂ri / ∂q j leads to mi vi′′ ⋅

∂ri ∂r ∂r ∂r − mi vi′ ⋅ i = Pi ⋅ i + PRi ⋅ i , ∂q j ∂q j ∂q j ∂q j

the position of the system S remaining the same before and after collision (we denoted by “second” the corresponding quantities after collision). Taking into account the relations (18.2.28) and summing for all the particles of the system S , we get the jump relations ⎛ ∂T ⎞ ⎛ ∂T Δ⎜ ≡⎜ ⎟ ⎝ ∂q j ⎠0 ⎝ ∂q j

⎞′′ ⎛ ∂T ⎟ − ⎜ ∂q ⎠ ⎝ j

⎞′ ⎟ = ⎠

Pj + PRj , j = 1, 2,..., s ,

(18.3.33)

where we have introduced the given generalized percussions and the constraint generalized percussions, respectively, in the form

Pj =

n

∑ Pi

i =1

⋅

∂ri ,P = ∂q j Rj

n

∑ PRi

i =1

⋅

∂ri , j = 1, 2,.., s , ∂q j

(18.3.33')

the index zero corresponding to an arbitrary moment of discontinuity t0 ∈ [ t ′, t ′′ ] , | t ′′ − t ′ |< ε, ε > 0 arbitrary.

Starting from the equations (18.2.32), integrating on the interval [t ′, t ′′ ] and passing to the limit in the sense of the theory of distributions, we can write

lim

∫

t ′′

t ′′ − t ′ → 0 + 0 t ′

t ′′ ∂T ⎛ ∂T ⎞ d⎜ dt = ⎟ − t ′′ −tlim ∫ ′ ∂q t ′ → + 0 0 ∂ q j ⎝ j ⎠

lim

∫

t ′′

t ′′ − t ′ → 0 + 0 t ′

(Q j + Rj )dt ;

noting that ∂T / ∂q j is a finite quantity and neglecting the given and constraint nonpercussive forces with respect to the corresponding percussive ones, we find again the jump relations (18.3.33), with


96 t ′′

Q j dt t ′′ − t ′ → 0 + 0 ∫t ′ lim

=

n

∂r

∑ ∂qij

i =1

⋅

=

t ′′

Fi t ′′ − t ′ → 0 + 0 ∫t ′

t ′′

lim

n

Fi dt = ∑ Pi t ′′ −t ′ → 0 + 0 ∫t ′ i =1

lim

⋅

⋅

∂ri dt ∂q j

∂ri = ∂q j

Pj ,

and similar relations for the constraint generalized percussions. Introducing the generalized momenta (18.2.80), considering Lagrange’s equations in the form (18.2.36) and observing that ∂U / ∂q j is a finite quantity, one can write the jump relations (18.3.33) in the form

( Δp j )0

= p j′′ − p j′ = Pj + PRj , j = 1, 2,..., s ,

(18.3.33'')

We state thus Theorem 18.3.1 (theorem of generalized momentum). The jump of the generalized momentum of a discrete mechanical system subjected to constraints, corresponding to a generalized co-ordinate at a moment of discontinuity, is equal to the sum of the given and constraint generalized percussions, which correspond to the same generalized co-ordinate and which act upon the same system at that moment. This theorem corresponds to the Theorem 13.1.2 stated in the space E 3 . Let us consider, in general, the case of a mechanical system S subjected to holonomic constraints, the position of which at the moment t ′ is definite by the generalized co-ordinates q1′ , q2′ ,..., qs′ (after eliminating the holonomic constraints); at the moment t0 ∈ [ t ′, t ′′ ] , h new holonomic constraints are suddenly introduced, so that the motion of the system S is suddenly perturbed. while at the moment t ′′ the generalized velocities have finite variations, passing from q j′ to q j′′, j = 1, 2,..., s , the position of the considered system remaining practically the same. The preceding constraints of the system S are permanent, while the new introduced constraints can become permanent or can disappear after the interval of percussion. The new introduced constraints are expressed by relations of the form ϕl (q1 , q2 ,..., qs ) = 0, l = 1, 2,..., h . Making a change of variable by which one replaces the co-ordinates ql by the co-ordinates rl = ϕl , the new constraints imposed to the system S will be expressed in the form rl = 0, l = s − h + 1, s − h + 2,..., s ; we assume to choose the generalized co-ordinates just in this form, having qs − h + 1 = 0, qs − h + 2 = 0,..., qs = 0 . If the new constraints introduced are temporary, then these co-ordinates will no more be equal to zero after the moment t ′′ ; otherwise, they remain equal to zero. The arbitrary virtual generalized displacements δq1 , δq2 ,..., δqs − h take place, while the virtual generalized displacements δqs − h + 1 , δqs − h + 2 ,..., δqs must vanish. We remain with Lagrange’s equations

d ⎛ ∂T ⎞ ∂T − = Q j , j = 1, 2,..., s − h , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(18.3.34)


97

for the interval of percussion, the constraints generalized forces new introduced disappearing from the calculation, because the corresponding virtual work vanishes Rs − h + 1 δqs − h + 1 + Rs − h + 2 δqs − h + 2 + ... + Rs δqs = 0 .

(18.3.34')

Noting that the variations of the generalized velocities are due only to the constraint generalized forces, the intensity of which has increased very much but which have disappeared from the calculation in the considered system of generalized co-ordinates, it results that ∂T / ∂q j and Q j are finite quantities; by integration on the interval of percussion and by passing to the limit, we find the jump relations ⎛ ∂T ⎞ ⎛ ∂T Δ⎜ ≡⎜ ⎟ ⎝ ∂q j ⎠0 ⎝ ∂q j

⎞′′ ⎛ ∂T ⎟ − ⎜ ∂q ⎠ ⎝ j

⎞′ ⎟ = 0, j = 1, 2,..., s − h , ⎠

(18.3.34'')

which are linear and homogenous with respect to the differences q j′′ − q j′ , j = 1, 2,..., s . One can thus state that the generalized momenta corresponding to the generalized co-ordinates, which do not vanish at the moment of collision, have the same values before and after this phenomenon (have not a jump). The generalized co-ordinates q1 , q2 ,..., qs have such values, in the equations (18.3.34''), that qs − h + 1 , qs − h + 2 ,..., qs do vanish at the moment of collision. In exchange, the generalized velocities qs − h + 1 , qs − h + 2 ,..., qs must not necessarily vanish, neither before, nor after collision; they equate to zero after collision if the new introduced constraints are permanent. In this last case, the relations (18.2.34'') determine entirely the generalized velocities q1′′, q2′′,..., qs′′− h , hence the state of generalized velocities after collision. Beside this particular case, we have at our disposal only s − h equations for s unknown generalized velocities, so that we must make supplementary hypotheses for the mathematical modelling of the mechanical phenomenon.

Fig. 18.8 A circular disc rolling without sliding on an axis in a vertical plane

As Beghin and Rousseau have shown in 1903, in case of a non-holonomic mechanical system, the constraint generalized forces are finite, so that they disappear from calculation, as well as the given generalized forces; hence, in this case, the jump relations (18.3.34'') can be applied too. Let be, e.g., a homogenous circular disc, of mass M and radius R , which moves in the vertical plane Ox1x 2 ; the disc strikes the Ox1 -axis at the moment t0 and continues to roll


98

without sliding on this axis. Before the collision, the position of the disc depends on three parameters: the co-ordinates ρ1 and ρ2 of the centre C of the disc and the angle of rotation θ , taken counterclockwise (Fig. 18.8). As an effect of the collision, two new constraints are introduced: the contact with the Ox1 -axis (hence ρ2 = R ) and the rolling of the disc on this axis ( ρ1 = Rθ ). Choosing conveniently the origin O , one introduces the generalized co-ordinates q1 = ρ1 , q2 = ρ2 − R , q 3 = ρ1 − Rθ , so that the constraint relations are expressed in the form q2 = 0, q 3 = 0 . Denoting by IO = MiO2 the moment of inertia of the disc with respect to the pole O , where iO is the corresponding radius of gyration, we can write the kinetic energy in the form T =

1 M ( iO2 θ2 + ρ12 + ρ22 2

1

( ) ( q

⎡ iO ⎣ R

) = 2M ⎢

2

1

⎤ − q3 )2 + q12 + q22 ⎥ . ⎦

We have at our disposal only a relation of the form (18.3.34''), corresponding to the generalized co-ordinate q1 , which does not vanish at the collision moment, that is

( ) [q ′′ − q ′ − ( q ′′ − q ′ ) ] + q ′′ − q ′ = 0 . iO R

2

1

1

3

3

1

But q 3 = 0 after collision, hence we have q3′′ = 0 too. Returning to the old variables, we can write

( iR ) ( ρ ′′ − Rθ′ ) + ρ ′′ − ρ ′ = 0 , O

2

1

1

1

the velocity of the centre of the disc in a rolling motion on the Ox1 -axis being given by ρ1′′ =

R ( Rρ1′ + iO2 θ ′ ) R2 + iO2

.

(18.3.35)

If Rρ1′ + iO2 θ ′ = 0 , then the disc stops.

18.3.2 Applications In what follows we present several applications, interesting by themselves, to illustrate the results previously obtained: the motion of a particle in various cases of loading, the problem of two particles and the problem of two centres; as well, we consider some cases of motion of the rigid solid: the plane motion of a bar, the double pendulum, the sympathetic pendulums etc.


99

18.3.2.1 Motion of a Particle Using the expression T = (m / 2)( x12 + x22 + x 32 ) of the kinetic energy, in case of a single particle P , Lagrange’s equations (18.2.29) take the form ( q j = x j , j = 1, 2, 3 ) d ( mx j dt

∂r

) = F ⋅ ∂x

j

+F⋅

∂x k ik = F ⋅ ik δkj = F ⋅ i j = Fj = Q j , k = 1, 2, 3 , ∂x j

so that we find again the equations of motion (6.1.22'). In the case of cylindrical co-ordinates q1 = r , q2 = θ , q 3 = z , the kinetic energy (see the formula (5.1.13'')) has the form T = (m / 2)(r 2 + r 2 θ2 + z 2 ) , resulting Lagrange’s equations ( r = r cos θi1 + r cos θi2 + z i 3 ) d ∂r ( mr ) − mr θ2 = F ⋅ = F ⋅ (cos θi1 + sin θi2 ) = F ⋅ ir = Fr = Qr , dt ∂r d ∂r ( mr 2 θ ) = F ⋅ ∂θ = F ⋅ [ r ( − sin θi1 + cos θi2 ) ] = F ⋅ ( r iθ ) = rFθ = Qθ , dt d ∂r ( mz ) = F ⋅ = F ⋅ i 3 = Fz = Qz , dt ∂z

being thus led to the equations of motion (6.1.26), (6.1.26'). In particular, we obtain the plane motion in polar co-ordinates. As well, in the case of spherical co-ordinates q1 = r , q2 = θ , q 3 = ϕ (see the formula (5.1.12'')), we can write T = (m / 2)(r 2 + r 2 θ2 + r 2 sin2 θϕ 2 ) , while Lagrange’s equations read ( r = r sin θ cos ϕi1 + r sin θ sin ϕi2 + r cos θi 3 ) d ∂r ( mr ) − mr ( θ2 + sin2 θϕ 2 ) = F ⋅ dt ∂r = F ⋅ (sin θ cos ϕi1 + sin θ sin ϕi2 + cos θi 3 ) = F ⋅ ir = Fr = Qr , d ∂r mr 2 θ ) − mr 2 sin θ cos θϕ 2 = F ⋅ ( dt ∂θ = F ⋅ [ r ( cos θ cos ϕi1 + cos θ sin ϕi2 − sin θ i 3 ) ] = F ⋅ ( r i θ ) = rFθ = Qθ , d ∂r ( mr 2 sin2 θϕ ) = F ⋅ ∂ϕ dt = F ⋅ [ r ( sin θ sin ϕi1 + sin θ cos ϕi2 ) ] = F ⋅ ( r iϕ ) = rFϕ = Qϕ ,

wherefrom we get the equations of motion (6.1.25), (6.1.25'). If the support of the force F pierces the Ox 3 -axis, then Qϕ = 0 , so that we can write the first integral

mr 2 sin2 θϕ = const ,

(18.3.36)

according to which the projection of the particle P on the plane Ox1x 2 moves after the law of areas.


100

Analogously, we find again the equations (6.1.24''') in arbitrary curvilinear coordinates. Let be, e.g., the quadric Γ x12 x 22 x 32 + + −1 = 0 α1 α2 α3

(18.3.37)

and the quadrics homofocal to this one x12 x 22 x 32 + + −1 = 0. α1 − λ α2 − λ α3 − λ

(18.3.37')

If, to fix the ideas, α1 > α2 > α3 , then the equation (18.3.37') represents an ellipsoid for λ < α3 , an one-sheet hyperboloid if α3 < λ < α2 or a two-sheet hyperboloid if α2 < λ < α1 . Through each point P (x1 , x 2 , x 3 ) pass three homofocal quadrics Γ i , i = 1, 2, 3 . Supposing that the co-ordinates x1 , x 2 , x 3 are given, the equation (18.3.37') of third degree in λ has three real roots q 3 < α3 < q2 < α2 < q1 < α1 , as one can easily verify; the root q1 leads to a two-sheet hyperboloid, the root q2 leads to an one-sheet hyperboloid, while the root q 3 correspond to an ellipsoid. These three quadrics are orthogonal two by two. We obtain thus the elliptical co-ordinates (introduced by Jacobi) q1 , q2 , q 3 of the point P . Obviously, we have x12 x 22 x 32 ( λ − q1 ) ( λ − q2 ) ( λ − q 3 ) + + −1 = , − − − f (λ ) α1 λ α2 λ α3 λ f (λ ) = ( α1 − λ ) ( α2 − λ ) ( α2 − λ ) ;

multiplying by αi = λ and making then λ = αi , i = 1, 2, 3 , we can express the Cartesian co-ordinates as functions of the elliptic co-ordinates in the form x i2 =

( αi − q1 ) ( αi − q2 ) ( αi − q 3 ) , i ≠ j ≠ k ≠ i , i , j , k = 1, 2, 3 . ( αj − αi ) ( αk − αi )

(18.3.38)

Calculating the logarithmic derivatives, we obtain the element of arc ds 2 =

1 ( A dq 2 + A2 dq22 + A3 dq 32 ) , 4 1 1

(18.3.39)

where Ai = =

( qi

x12

( qi − a1 )

2

+

x 22

( qi − a 2 )

2

+

x 32

( q 3 − a1 )2

− q j ) ( qi − qk ) ( ! ) , i ≠ j ≠ k ≠ i , i , j , k = 1, 2, 3. f (qi )

(18.3.39')


101

If the intersections of the surfaces q j = const and qk = const taken two by two are the curves C i , i ≠ j ≠ ≠ k ≠ i , i , j , k = 1, 2, 3 (Fig. 18.9), then the element of arc on these curves is given by dsi =

1 Ai dqi ( ! ) , i = 1, 2, 3 , 2

(18.3.40)

so that ds 2 = dsi dsi .

(18.3.39'')

Fig. 18.9 Elliptic co-ordinates in E 3

The kinetic energy is calculated in the form T =

( )

m ds 2 dt

2

=

m ( A q2 + A2q22 + A3q32 ) , 8 1 1

(18.3.41)

allowing thus to obtain Lagrange’s equations. We decompose the given force F along the tangents to the curves C 1 ,C 2 ,C 3 , hence let be F = F1 + F2 + F3 ; the corresponding virtual work will be F ⋅ δr = Qi δqi = Fi δsi . Taking into account (18.3.40), it results Qi =

1 F 2 i

Ai ( ! ) , i = 1, 2, 3 .

(18.3.42)

In case of a plane motion, we can use elliptic co-ordinates in the plane Ox1x 2 , the corresponding formulae being obtained by particularization (we make x 3 = 0, q 3 = 0 ) from the above ones. The homofocal conics will be an ellipse and a hyperbola (Fig. 18.10). If the given forces are conservative or quasi-conservative, then we can use the results in Sect. 18.1.1.3 concerning natural systems. Lagrange’s equations can be written also in the vector form


102

d ⎛ ∂T ′ ⎞ ∂T ′ = F, ⎜ ⎟− ∂r dt ⎝ ∂v ⎠

(18.3.43)

with respect to a non-inertial frame of reference R (the generalized co-ordinates are taken with respect to this frame); in this case, the kinetic energy is given by T ′ = (m / 2)( vO′ + v + ω + r )2 . We notice that vO′2 is a known function of time and can be expressed as the total derivative with respect to time of another function; hence, this term can be omitted from calculation (Lagrange’s function – inclusive the kinetic energy – is determined excepting the total derivative with respect to time of an arbitrary function). On the other hand, we notice that m vO′ ⋅ ( v + ω × r ) = m vO′ ⋅

dr ⎡ d ( vO′ ⋅ r ) ⎤ = m⎢ − aO′ ⋅ r ⎥ ; t dt d ⎣ ⎦

Fig. 18.10 Elliptic co-ordinates in E 2

the total derivative with respect to time can be, as well, neglected. Thus, we can use the conventional kinetic energy 1 1 T = m v2 + m v ⋅ ( ω × r ) − m aO′ ⋅ r . 2 2

(18.3.44)

We can calculate dT = m ( v + ω × r ) ⋅ dv − m [ aO′ + ω × v + ω × ( ω × r ) ] ⋅ dr ,

wherefrom ∂T ∂T = m ( v + ω × r ), = − m [ aO′ + ω × v + ω × ( ω × r ) ] . ∂v ∂r

(18.3.44')

The equation (18.3.43) leads thus to the equation of motion m

dv = F − m [ aO′ + ω × r + ω × ( ω × r ) ] − 2m ω × v , dt

(18.3.45)


103

written with respect to the non-inertial frame of reference R ; the transportation force and the Coriolis force are thus put in evidence (see Sect. 10.2.1.1 too)

18.3.2.2 Case of a Particular Conservative Force We consider the motion of a particle P acted upon by a conservative force which derives from the simple potential

U ( r , θ , ϕ ) = f (r ) +

g (θ ) h (ϕ ) + 2 , r2 r sin2 θ

(18.3.46)

expressed in spherical co-ordinates. Noting that Qr = ∂U / ∂r , Qθ = ∂U / ∂θ and Qϕ = ∂U / ∂ϕ , we obtain Lagrange’s equations in the form h (ϕ ) ⎤ ⎡ ⎢⎣ g ( θ ) + r 2 sin2 θ ⎥⎦ , d 2 ⎡ sin 2 θ ⎤ , 2 ( r 2 θ ) − r 2 sin 2 θϕ 2 = g ′( θ ) − h (ϕ ) 2 dt mr 2 ⎣⎢ r sin 4 θ ⎦⎥ h ′(ϕ ) d 2 ( r sin2 θϕ ) = 2 2 , dt mr sin θ

r − r ( θ2 + sin2 θϕ 2

1

) = m f ′(r ) −

2 mr 3

(18.3.47)

where we have denoted by “prime” the derivative of the functions f , g and h with respect to the corresponding variables. The third equation (18.3.47) can be written also in the form mr 2 sin2 θϕ ( r 2 sin2 θϕ ) =

m d ( r 4 sin 4 θϕ 2 2

) = h ′(ϕ )ϕ dt

= h ′(ϕ )dϕ = dh (ϕ ),

so that we obtain the first integral r 4 sin 4 θϕ 2 =

2 [ h (ϕ ) + C 1 ] , m

(18.3.48)

where C 1 is an integration constant; as well, the second equation (18.3.47) reads 2C 1 sin 2 θ d 1 sin 2 θ ⎡ 4 2 2 ⎡ ( r 2 θ ) − g ′( θ ) ⎤ = 2 r sin 4 θϕ 2 − h (ϕ ) ⎤ = , m ⎦⎥ mr 2 sin 4 θ ⎣⎢ dt ⎦⎥ r sin 4 θ ⎣⎢ mr 2

where we took into account the first integral previously found. Multiplying by r 2 θ , one obtains 2r 2 θ d ( r 2 θ ) = d ( r 4 θ2 =

) = m ⎡⎢⎣ g ′( θ )θ dt + 2C 1 2

(

)

2 ⎡ 1 ⎤, g (θ ) − C1 d m ⎢⎣ sin2 θ ⎦⎥

sin θ cos θ ⎤ θ dt ⎥⎦ sin 4 θ


104 resulting the first integral r 4 θ2 =

2 m

⎡ g (θ ) + C1 + C 2 ⎢⎣ sin2 θ

⎤, ⎥⎦

(18.3.48')

where C 2 is a new integration constant. Eliminating the terms r θ2 and r sin2 θϕ 2 between the first equation (18.3.47) and the first integrals (18.3.48), (18.3.48'), we obtain r − 2

C2 1 = f ′(r ) . 3 m mr

Multiplying by rdt and integrating, we get the third first integral r 2 + 2

C2 1 = [ 2 f (r ) + C 3 ] , m mr 2

(18.3.48'')

where C 3 is a new integration constant. As a matter of fact, one can obtain this result starting from the first integral of the mechanical energy and taking into account the first integrals previously obtained. The first integral (18.3.48'') contains only one space variable, so that C dr 1 ⎡ 2 f (r ) − 2 22 + C 3 ⎤ , = ± f1 (r ), f1 (r ) = dt m ⎣⎢ ⎦⎥ r

(18.3.49)

wherefrom, by a quadrature, one gets t as a functions of r and then r = r (t ) , introducing the fourth integral constant C 4 . Analogously, the first integral (18.3.48') leads to

r

2

dr dθ 2 = ± , g1 ( θ ) = m f1 (r ) g1 ( θ )

⎡ g (θ ) − C1 + C ⎤ . 2 ⎥ ⎢⎣ ⎦ sin2 θ

(18.3.49')

By two quadratures, we obtain θ = θ (r ) and then θ = θ (t ) , a new integration constant C 5 being introduced. Finally, the first integral (18.3.48) allows to write dϕ dθ 2 = ± 2 , h1 (ϕ ) = [ h (ϕ ) + C 1 ] , m h1 (ϕ ) sin θ g1 ( θ )

(18.3.49'')

where we have used the previous result; hence, one obtains ϕ = ϕ ( θ ) and then one gets ϕ = ϕ (t ) , appearing the integration constant C 6 . The integration constants C k , k = 1, 2,..., 6 , are then determined by means of the initial conditions. The case of loading of a particle P , considered above, is useful in many particular cases of great interest.


105

18.3.2.3 Problem of Two Particles Let be two particles P1 and P2 of masses m1 and m2 , m1 ≥ m2 , respectively; the mass centre O is on the segment of a line P1P2 , so that m1r1 = m2 r2 , r1 = OP 1 , r2 = OP 2 , r1 ≤ r2 (Fig. 18.11). In spherical co-ordinates, we have P1 (r1 , θ , ϕ ) , P2 (r2 , π − θ , π + ϕ ) , and the kinetic energy can be expressed in the form

Fig. 18.11 Problem of two particles

T =

m1 2 m r1 + r12 θ2 + r12 sin2 θϕ 2 ) + 2 ( r22 + r22 θ2 + r22 sin2 θϕ 2 ) . ( 2 2

We introduce the notations m1r12 + m2 r22 = mr 2 , m1r12 + m2 r22 = mr 2 , with

r1 + r2 = r ,

(18.3.50)

where r = P1P2 ; we get r1 =

m2 m1 r , r2 = r, m1 + m2 m1 + m2

so that m1r12 + m2 r22 =

m1m2 2 m1m2 2 r , m1r12 + m2 r22 = r . m1 + m2 m1 + m2

Comparing with the notations which have been introduced, we must have 1 1 1 = + . m m1 m2

(18.3.50')

Thus, the motion of relation about the mass centre O is specified by the generalized coordinates q1 = r , q2 = θ , q 3 = ϕ , corresponding to three degrees of freedom. The kinetic energy will be thus given by


106 T =

1 m ( r 2 + r 2 θ2 + r 2 sin2 θϕ 2 ) . 2

(18.3.51)

Assuming that the two particles are acted upon only by forces of Newtonian attraction, the potential function being U = f

m1m2 , r

(18.3.51')

we obtain the case considered in the preceding section, with f (r ) = fm1m2 / r and g ( θ ) = h (ϕ ) = 0 ; the problem is thus reduced to quadratures.

18.3.2.4 Problem of Two Centres Let be a particle subjected to the attraction of two fixed centres C 1 and C 2 , in an inverse proportion to the squares of the distances rk = PC k , k = 1, 2 , to these centres; the potential of the corresponding forces is U =

k1 k2 + , r1 r2

(18.3.52)

where k1 , k2 > 0 are positive constants. We can study this problem in spherical coordinates too, by means of the results obtained in Sect. 18.3.2.2. But we can study the problem also with the aid of the elliptical co-ordinates introduced in Sect. 18.3.2.1. If, in this order of ideas, we denote q1 = λ =

1 1 ( r1 + r2 ) , q2 = μ = ( r1 − r2 ) , 2 2

(18.3.53)

then the curves λ = const represent ellipses, the semi-major axes of which are equal to λ , and the curves μ = const are hyperbolae the semi-major axes of which are equal to μ ; these curves are homofocal (they have common foci C 1 and C 2 ) and orthogonal one to the other. Observing that min(r1 + r2 ) = 2c , max r1 − r2 = 2c , 2c = C 1C 2 , it results the relation of condition λ ≥ c ≥ μ ; in this case, the equations of the families of ellipses and hyperbolae will be

x12 x2 x2 x2 + 2 2 2 = 1, 12 + 2 2 2 = 1 . 2 λ λ −c μ c −μ

(18.3.54)

By logarithmic differentiation, we can calculate the kinetic energy in the form

T =

m 2 ( x + x22 2 1

)=

μ 2 m 2 ⎛ λ 2 λ − μ2 ) ⎜ 2 + ( 2 2 c 2 − μ2 ⎝λ −c

Taking into account (18.3.53), we obtain the potential

⎞ ⎟. ⎠

(18.3.55)


U =

107

1 [ ( k1 + k2 ) λ − ( k1 − k2 ) μ ] , λ 2 − μ2

(18.3.55')

observing thus that we have a system of Liouville type (see Sect. 18.3.1.2). Because u1 (q1 ) = λ2 , u2 (q2 ) = − μ2 , U 1 (q1 ) = (k1 + k2 )λ , U 2 (q2 ) = (k1 − k2 ) μ, m m , V2 (q2 ) = − 2 , 2 λ −c μ − c2 dμ dλ dW1 = m , dW2 = m , 2 2 2 c − μ2 λ −c u = λ2 − μ2 , α1 = α, α2 = − α, V1 (q1 ) =

2

the system of equations (18.3.7) leads to

∫

dμ = ψ(μ)

2 β , m 1

dλ − μ2 ϕ (λ ) ∫

dμ = ψ(μ )

2 (t + β2 ), m

dλ − ϕ (λ )

∫ ∫λ

2

(18.3.56)

with the notations ϕ (λ ) = (λ 2 − c 2 )[ hλ 2 + (k1 + k2 )λ + α ], ψ ( μ ) = ( μ2 − c 2 )[ h μ2 + (k1 − k2 )μ + α ],

(18.3.56')

where h is the constant of mechanical energy, while α, β1 and β2 are three other integration constants. We denote by λ1 , λ2 and μ1 , μ2 pairs of two simple roots, differing from ±c , of the equations ϕ (λ ) = 0 and ψ ( μ ) = 0 , respectively, and let be the integrals ω11 =

λ2

∫λ

1

ω21 =

λ2

∫λ

1

λ

2

dλ , ω12 = ϕ (λ ) dλ , ω22 = ϕ (λ )

dμ , ψ(μ) μ2 2 dμ ∫μ1 μ ψ ( μ ) . μ2

∫μ

1

(18.3.57)

We introduce also the quantities ν1 and ν2 by relation ω11 ν1 + ω12 ν2 = πβ1 , ω21ν1 + ω22 ν2 = π (t + β2 ) .

(18.3.58)

The relations (18.3.56) define thus λ and μ as double periodical (elliptic) functions of β1 and t + β2 or (with the relations (18.3.58)) of ν1 and ν2 . If a relation of the form


108

n1 ω11 + n2 ω12 = 0, n1 , n2 ∈ ` , takes place, hence if ω11 and ω12

are commensurable

quantities, then the motion is strictly periodic with the period 2T = 2(n1 ω21 + n2 ω22 ) . If the roots λ1 , λ2 and μ1 , μ2 , respectively, are double, then we have so called motions of libration (called periodical conditioned motions too), important in the classical quantum theory.

18.3.2.5 Motion of an Electrized Particle in an Electromagnetic Field Let be an electrized particle P of electric charge q (eventually, an electron), which is in motion with the velocity v in an electromagnetic field { E, B } , where E is the intensity of the electric field and B is the magnetic induction (see Sect. 18.1.1.4). Lorentz’s generalized force (18.1.36) derives from the generalized quasi-potential (18.2.37'). We obtain thus Lagrange’s kinetic potential 1 2

L = m v2 + q ( A ⋅ v − A0 ) ,

(18.3.59)

where A = A ( x1 , x 2 , x 3 ; t ) is the vector magnetic quasi-potential, while the scalar one is A0 = A0 (x1 , x 2 , x 3 ; t ) . If one observes that dL / dvi = mvi + qAi , i = 1, 2, 3 , then Lagrange’s equations (18.3.38) read d ( mvi + qAi ) = q ( v j Aj ,i − A0,i dt

) = 0, i

= 1, 2, 3 .

We assume that the velocities do not depend on the point but only on the time. Calculating the total derivative dAi / dt = ( ∂Ai / ∂x j )v j + Ai , Lagrange’s equations become m

dvi = q ⎡⎣ ( Aj ,i − Ai , j ) v j − Ai − A0,i ⎤⎦ , i = 1, 2, 3 . dt

(18.3.60)

The relations (18.1.36') and (18.1.36'') can be scalarly written in the form Bi =∈ijk Ak , j , Ei = −( Ai + A0,i ), i = 1, 2, 3 .

It is seen that the equations (18.3.60) represent, in fact, the projections on the three axes of co-ordinates of Newton’s equation of motion m

dv = q(E + v × B) , dt

(18.3.60')

where we have Lorentz’s generalized force (18.1.36); for this, one takes into account the above relations and the relation ( v × B )i =∈ijk v j Bk =∈ijk v j ∈klm Am ,l

= ( δil δjm − δim δjl ) v j Am ,l = ( Aj ,i − Ai , j ) v j .


109

To take into account the relativistic aspect of the problem, one must introduce the mass m = m0 1 − (v / c )2 , where m0 is the mass at rest, while c is the velocity of propagation of light in vacuum.

18.3.2.6 Motion of the Rigid Solid A free rigid solid S has six degrees of freedom and we can use, e.g., the ′ , i = 1, 2, 3 , q 4 = ψ , q 5 = θ , q 6 = ϕ , where xOi ′ generalized co-ordinates qi = xOi are the co-ordinates of a point O of it with respect to an inertial frame of reference R ′ , while ψ , θ and ϕ are Euler’s angles, which specify the rotation about the point O . If a certain point P0 of the solid is compelled to slide (without friction) on a given surface, then it will have only two degrees of freedom, appearing thus a holonomic constraint relation; analogously, if the point P0 is compelled to stay on a given curve, then there result two holonomic constraint relations. Especially, if to the point P0 is imposed a given motion, then the rigid solid is subjected to three holonomic constraints; in particular, the point P0 (taken – in this case – as a pole O of a non-inertial frame R ) can be fixed. In each of these cases, one searches convenient generalized coordinates, by eliminating the respective constraints; thus, in case of a rigid solid with a fixed point, we can choose as generalized co-ordinates Euler’s angles, the kinetic energy being of the form (15.1.13), where I 1 , I 2 , I 3 are the principal moments of inertia with respect to the fixed point. We notice that ψ is a hidden co-ordinate, so that d ⎛ ∂T ′ ⎞ = Qψ . dt ⎜⎝ ∂ψ ⎟⎠

(18.3.61)

For instance, in case of the motion of a heavy homogenous rigid solid of revolution of mass M , which slides frictionless on a fixed horizontal plane (see Sect. 17.1.2.4), the kinetic energy reads T′ =

1 {M ( ρ1′2 + ρ 2′2 ) + [ Mf ′2 ( θ ) + J ] θ2 + J ψ 2 sin2 θ + I 3 (ϕ + ψ cos θ )}, 2 (18.3.62)

where ρ1′ , ρ2′ and ρ3′ = f ( θ ) are the co-ordinates of the mass centre C with respect to the inertial frame of reference R ′ and where J = I 1 = I 2 ; the potential of its weight is U = − Mgf ( θ ) .

(18.3.62')

We can use thus the generalized co-ordinates q1 = ρ1′ , q2 = ρ2′ , q 3 = ϕ, q 4 = ψ, q 5 = θ . The first four equations of Lagrange are written in the form ( ρ1′ , ρ2′ , ψ and ϕ are hidden and ignorable co-ordinates) d ⎛ ∂T ′ ⎞ d ⎛ ∂T ′ ⎞ d ∂T ′ ⎞ d ⎛ ∂T ′ ⎞ = 0, = 0, ⎜⎛ = 0, ⎜ = 0, ⎟ ⎜ ⎟ ⎜ ⎟ dt ⎝ ∂ρ1′ ⎠ dt ⎝ ∂ρ 2′ ⎠ dt ⎝ ∂ϕ ⎠ dt ⎝ ∂ψ ⎟⎠


110 being thus led to the first integrals

ρ1′ = νC′01 , ρ 2′ = νC′02 , ϕ + ψ cos θ = ω30 ,

J ψ sin2 θ + I 3 (ϕ + ψ cos θ ) cos θ = KC′ 3 ,

(18.3.63)

where we have used the notations in Sect. 17.1.2.4. Instead of the fifth equation of Lagrange, we can use the first integral of mechanical energy, the problem being thus entirely formulated. An important case is that of the rigid solid for which three non-collinear points have a given motion in a fixed (or mobile) plane; the problem is thus reduced to the motion of a mechanical system in a plane (or of a plane solid rigid), which has only three degrees of freedom, corresponding to the translation of a point (two degrees of freedom) and to the rotation about this point (one degrees of freedom). In the case of a system of rigid solids, one must keep in mind the connections between them. E.g., a hinge between two rigid solids involves the introduction of three holonomic constraints (we can imagine a free rigid solid, the second one having a point with a given motion).

18.3.2.7 Plane Motion of a Rigid Straight Bar Let be a rigid straight bar AB which moves frictionless in the plane O ′x1′x 2′ , being acted upon by forces in direct proportion to the mass and to the distance to the O ′x1′ -axis (we have chosen a system of particular forces, to fix the ideas). The position of the bar is specified by the co-ordinates ρ1′ and ρ2′ of the mass centre C and by the angle θ made with the fixed axis O ′x1′ (Fig. 18.12); we can thus choose the generalized co-ordinates q1 = ρ1′ , q2 = ρ2′ , q 3 = θ .

Fig. 18.12 Plane motion of a straight rigid bar

Using Koenig’s theorem, we can write the kinetic energy in the form T′ =

1 ⎡ M ( ρ1′2 + ρ 2′2 ) + I θ2 ⎦⎤ , 2⎣

(18.3.64)

where M is the mass of the bar and I is its moment of inertia with respect to the centre of mass C . Let us consider a bar element dx along the Cx -axis, which is attracted by the O ′x1′ -axis by the force − k μ ( ρ2′ + x sin θ )dx , k = const , where μ is the linear


111

density of the bar; this force −(k μ / 2)( ρ2′ + sin θ )2 dx . Noting that B

∫ A μ dx

= M,

B

∫A μx dx

derives

= 0,

B

∫A μx

2

from

the

simple

potential

dx = I ,

we get U = −

2 k B k ′ + ( sin ) ρ x θ μ dx = − ( M ρ2′2 + I sin2 θ ) . 2 2 ∫A 2

(18.3.64')

One obtains, easily, k Q1 = 0, Q2 = − kM ρ2′ , Q3 = − I sin 2 θ . 2

(18.3.64'')

We are thus led to Lagrange’s equations

ρ1′ = 0, ρ2′ = − k ρ2′ , 2 θ = − k sin 2 θ .

(18.3.65)

The mass centre C describes a sinusoid along the O ′x1′ -axis, its projection on this axis having a uniform motion; the bar oscillates about the O ′x1′ -axis with a frequency equal (in case of small oscillations with sin θ ≅ θ , hence sin 2 θ ≅ 2 θ ) to the frequency of the oscillations of the projection of the centre C on the O ′x 2′ -axis.

18.3.2.8 Double Pendulum We return to the double pendulum, considered in Sect. 17.1.1.2. This mechanical system has two degrees of freedom and we can choose as generalized co-ordinates the angles made by O1C 1 and O2C 2 , respectively, with the descendent vertical ( q1 = θ1 , q2 = θ2 ); we have denoted by O1 the suspension pole of the solid S1 with the mass centre C 1 and by O2 the hinge between this solid and the solid S2 , with the mass centre C 2 (see Fig. 17.1, a too). The kinetic energy of the mechanical system S ≡ {S1 , S2 } is given by T′ =

{

}

2 2 1 2 1 1 I θ + M ⎡ ρ ′(2 ) ⎤ + ⎡⎣ ρ2′(2 ) ⎤⎦ + I 2 θ22 , 2 1 1 2 2 ⎣ 1 ⎦ 2

(18.3.66)

where I 1 is the moment of inertia of the rigid solid S1 with respect to the O ′x1′ -axis, while I 2 is the moment of inertia of the rigid solid S2 , of mass M 2 , with respect to (2 ) (2 ) the axis which passes through the centre C 2 of co-ordinates ρ1′ , ρ2′ and is parallel

to the axis O1′x 3′ . We notice that ρ1′

(2 )

we obtain thus

= l cos θ1 + l2 cos θ2 , ρ2′

(2 )

= l sin θ1 + l2 sin θ2 ;


112 T′ =

1 1 ( I + M 2l 2 ) θ12 + 2 ( I 2 + M 2l22 ) θ22 + M 2ll2 θ1θ2 cos( θ2 − θ1 ) . 2 1

(18.3.66')

As well, the potential of the weight forces is U = M1gl1 cos θ1 + M 2 g ( l cos θ1 + l2 cos θ2 ) .

(18.3.67)

Lagrange’s equations read

( I1

+ M 2l 2 ) θ1 + M 2ll2 ⎡⎣ cos ( θ2 − θ1 ) θ2 − sin ( θ2 − θ1 ) θ22 ⎤⎦ = − ( M 1l1 + M 2l ) g sin θ1 ,

( I2

+ M 2l22 ) θ2 + M 2ll2 ⎡⎣ cos ( θ2 − θ1 ) θ1 + sin ( θ2 − θ1 ) θ12 ⎤⎦

(18.3.68)

= − M 2l2 g sin θ2 .

We find thus again (17.1.4), which have been studied and integrated by Bradistilov and by Anca Zlătescu. The small oscillations of the double pendulum have been considered in Sect. 17.1.1.2.

18.3.2.9 Sympathetic Pendulums The sympathetic pendulum has been considered directly with the aid of Newton’s equations in Sect. 17.1.1.3. We take again the problem in case of the two physical pendulums of masses M 1 and M 2 , the gravity forces being applied at the mass centres C 1 and C 2 , respectively; we denote O1C 1 = l1 , O2C 2 = l2 , where O1 and O2 are the

poles through which pass the axes of suspension. The two pendulums are connected by an elastic spring Q1Q2 , so that the point Q1 is on the straight line O1C 1 , while the point Q2 is on the straight line O2C 2 , with O1Q1 = O2Q2 = a (Fig. 18.13). The spring Q1Q2 is characterized by the elastic constant k , so that the magnitude of the force

which arises in the spring is given by k | O1O2 − Q1Q2 | , where k = const (unlike the case considered in Sect. 17.1.1.3, where the elastic constant depends on the mass of the physical pendulum upon which acts the spring, the position of the points Q1 and Q2 playing no one rôle). The position of this mechanical system is specified by the generalized co-ordinates q1 = θ1 and q2 = θ2 , these ones being the angles made by O1C 1 and O2C 2 , respectively, with the vertical line. The kinetic energy is given by T =

1 ( I θ2 + I 2 θ22 ) , 2 1 1

(18.3.69)

where I 1 and I 2 are the moments of inertia of each pendulum with respect to the corresponding axis of suspension. The potential U is expressed in the form U = g (M 1l1 cos θ1 + M 2l2 cos θ2 ) −

k (O O − Q1Q2 2 1 2

)2 .


113

But Q1Q22 = [O1O2 + a ( sin θ2 − sin θ1 ) ] + a 2 (cos θ2 − cos θ1 )2 2

= O1O22 + 2aO1O2 ( sin θ2 − sin θ1 ) + 2a 2 [ 1 − cos ( θ2 − θ1 ) ] .

In the case of small oscillations, we have sin θ1 ≅ θ1 , sin θ2 ≅ θ2 , cos θ1 ≅ 1 − θ12 / 2 , cos θ2 ≅ 1 − θ22 / 2 , so that

Q1Q22 + 2aO1O2 ( θ2 − θ1 ) + a 2 ( θ2 − θ1 )2 = [O1O2 + a ( θ2 − θ1 ) ] . 2

Hence, in case of small oscillations, it results U = g ( M1l1 + M 2l2 ) − −

1 ( M1gl1 + ka 2 ) θ12 2

1 ( M 2 gl2 + ka 2 ) θ22 + ka 2 θ1θ2 . 2

(18.3.69')

Fig. 18.13 Sympathetic pendulums

Lagrange’s equations read ( L = T + U ) k ∂ ( O O − Q1Q2 )2 = 0, 2 ∂θ1 1 2 k ∂ 2 I 2 θ2 + M 2 gl2 sin θ2 + O1O2 − Q1Q2 ) = 0 . ( 2 ∂θ2 I 1 θ1 + M 1gl1 sin θ1 +

(18.3.70)

In case of small oscillations, we get I 1 θ1 + M 1gl1 θ1 + ka 2 ( θ1 − θ2 ) = 0, I 2 θ2 + M 2 gl2 θ2 + ka 2 ( θ1 − θ2 ) = 0 .

(18.3.70')

In the particular case of two identical physical pendulums ( M 1 = M 2 = M , l1 = l2 = l , I 1 = I 2 = I ), we may write


114

∂2 ∂2 2 θ θ ω θ θ 0, + + + = ( ) ( ) ( θ1 − θ2 ) + ω 2 ( θ1 − θ2 ) = 0 , 0 1 2 1 2 ∂t 2 ∂t 2

where we have introduced the pulsations ( l ′ is the length of the synchronous mechanical pendulum) ω02 =

Mgl g 1 = , ω 2 = ( Mgl + 2ka 2 I l′ 2

) = ω02

+

2ka 2 . I

(18.3.71)

Hence, we obtain 1 [ ( θ 0 + θ20 ) cos ω0t + ( θ10 − θ20 ) cos ωt ] 2 1 1 1 0 1 θ1 + θ20 ) sin ω0t + ( θ10 − θ20 ) sin ωt ⎤⎥ . + ⎡⎢ ( 2 ⎣ ω0 ω ⎦ 1 θ2 (t ) = [ ( θ10 + θ20 ) cos ω0t − ( θ10 − θ20 ) cos ωt ] 2 1 1 0 1 − ⎡⎢ ( θ + θ20 ) sin ω0t − ω ( θ10 − θ20 ) sin ωt ⎤⎥ , 2 ⎣ ω0 1 ⎦ θ1 (t ) =

(18.3.72)

where θk0 = θk (0) , θk0 = θk (0), k = 1, 2 , corresponding to the initial moment t = 0 . If θ10 = θ0 , θ20 = 0, θ10 = θ20 = 0 , then we get θ0 (ω ( cos ω0t + cos ωt ) = θ0 cos 0 2 θ (ω θ2 (t ) = 0 ( cos ω0t − cos ωt ) = θ0 sin 0 2

θ1 (t ) =

− ω )t (ω cos 0 2 − ω )t (ω sin 0 2

+ ω )t , 2 + ω )t , 2

(18.3.72')

These results correspond to those in Sect. 17.1.1.3, excepting to the different definition of the elastic constant k .

Chapter 19 Hamiltonian Mechanics The motion of the representative point P ∈ Λs is governed by a system of s differential equations of second order (Lagrange’s equations) in Lagrangian mechanics. In 1834, W. R. Hamilton had the idea to use a representative space with 2s dimensions, the motion of a representative point in this space being specified by a system of 2s linear differential equations (Hamilton’s equations); The new formulation (in the frame of Hamiltonian mechanics) is equivalent to the Lagrangian formulation for discrete mechanical systems with holonomic, ideal constraints, being put a restrictive condition: these systems must be natural (as a matter of fact, the considered systems must admit a Lagrangian). These equations have remarkable analytical properties, leading to a rigorous and elegant mathematical formulation of the quantic model of mechanics. Especially, the Hamilton– Jacobi partial differential equation allows to pass from the matric quantum mechanics to the undulatory mechanics. In this order of ideas, after a study of Hamilton’s equations, we will consider in detail the Hamilton–Jacobi method (Arnold, V.I., 1976; Dobronravov, V.V., 1976; Goldstein, H., 1956; Lurie, A.I., 2002; Routh, E.J., 1892, 1898; Ter Haar, D., 1964).

19.1 Hamilton’s Equations After establishing some results with a general character, including Hamilton’s equations, we make a study of several expressions which play a special rôle: Lagrange’s brackets and Poisson’s brackets; these results will be illustrated by some applications to particular mechanical systems.

19.1.1 General Results To pass from the space of configurations to the phase space, one introduces the canonical co-ordinates and one obtains Hamilton’s canonical equations; after putting in evidence some properties of the latter ones, one considers Routh’s equations, as well as other equivalent equations. 19.1.1.1 Canonical Co-ordinates. Phase Space. Associate Expressions To study a natural discrete mechanical system S of n particles, subjected to holonomic ideal constraints, we consider Lagrange’s equations (18.2.38) (where we have introduced the kinetic potential L = T + U ), which describe the motion of the representative point P (q1 , q2 ,..., qs ) in the space of considerations Λs . To pass from this P.P. Teodorescu, Mechanical Systems, Classical Models, © Springer Science+Business Media B.V. 2009

115


116

system of equations of second order to a system of equations of first order, one has an infinity of possibilities to proceed (e.g., as in Sect. 18.2.3.8). Thus, in Sect. 18.2.3.6 there have been introduced the generalized momenta by means of relations (18.2.80). Taking into account (18.2.81) and (18.2.15), (18.2.15'), and (18.2.15''), we can write p j = g jk qk + g j + U j , j = 1, 2,..., s .

(19.1.1)

Introducing the normalized algebraic complement g jk of the element g jk of the determinant g = det[ g jk ] ≠ 0 , which verify the relations (18.2.44), we obtain, easily, q j = g jk pk − g jk (gk + U k ), j = 1, 2,..., s .

(19.1.1')

Let Λ2′s be the space of the points P (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) ; let us introduce also the representative space Γ 2s of the representative points P (q1 , q2 ,..., qs , p1 , p2 ,..., ps ) . We notice that the formulae (19.1.1), (19.1.1') establish a one-to-one correspondence between the two spaces. A state of the considered discrete system S is represented, as we have shown in the preceding chapter, by a representative point in the space Λs , hence by a point ′ , and this one is represented, in a one-to-one mode, by a representative in the space Λ2s point in the space Γ 2s . The representative space Γ 2s is called the phase space (or the Gibbs’s space). The co-ordinates q1 , q2 ,..., qs , p1 , p2 ,..., ps of the corresponding representative point P (the set of generalized co-ordinates and the set of generalized momenta) are called canonical co-ordinates (Hamiltonian co-ordinates); the generalized momenta p j are coordinates conjugate to the generalized co-ordinates q j . We replace thus the study of the motion of the mechanical system S in E 3 by the study of the motion of the representative point P in the space Γ 2s . Having to do only with holonomic constraints, which are eliminated by passing to generalized co-ordinates, the representative point P ∈ Γ 2s is a free point. To study the motion of the point P , we will express the quantities of energetical nature used in the space Λs by means of the canonical co-ordinates in the space Γ 2s ; by passing to canonical co-ordinates, a function F (q j , q j ; t ) becomes a new function F (q j , p j ; t ) , called the associate expression of the function F , so that F (q j , q j ; t ) = F (q j , p j ; t ) . The quantities (18.2.15'), (18.2.15'') read 1 1 1 g jk q j qk = g jk p j pk − g jk (g j + U j ) pk + g jk (g j + U j )(gk + U k ), 2 2 2 g j q j = g jk g j pk − g jk g j ( gk + U k ),

where we took into account the relations (18.2.44). The kinetic energy will take the form

Hamiltonian Mechanics

117 T = T2 + T1 + T0 ,

(19.1.2)

where we have put in evidence a quadratic form, a linear form and a constant with respect to the generalized momenta, given by (we notice that Tk ≠ Tk , k = 0,1, 2 ) T2 =

1 jk 1 g p j pk , T1 = − g jkU j pk , T0 = − g jk (g j gk − U jU k ) + g 0 . 2 2

(19.1.2')

We get thus the associate expression of the kinetic energy by means of the canonical coordinates. If the generalized forces derive from a simple quasi-potential, then we have U j = 0, j = 1, 2,..., s , so that T1 = 0 ; as well, if the mechanical system is scleronomic, then it results g j = 0, j = 1,2,..., s , and g 0 = 0 , so that T0 = (1/ 2)g jkU jU k . If both conditions are fulfilled (the mechanical system is scleronomic and the generalized forces derive from a simple quasi-potential), then we obtain T1 = T0 = 0 , so that T = T2 . In this case, the relations (19.1.1) read p j = g jk qk ,

(19.1.1'')

the generalized momenta vanishing together with the generalized velocities. This property does no more take place if the mentioned conditions are not fulfilled. After a partial differentiation of (18.2.15) with respect to q j , we multiply by q j and sum for all the values of the index j ; it results q j

∂T2 ∂T1 ∂T0 ∂T = q j + q j + q j = 2T2 + T1 , ∂q j ∂q j ∂q j ∂q j

where we used Euler’s theorem concerning the homogeneous functions. Adding T1 + 2T0 to both members of this relation and using the relation (18.2.81) and the notations (18.2.15''), (18.2.15'''), we may write T =

1 1 q p + (g − U j )q j + g 0 , 2 j j 2 j

(19.1.3)

obtaining thus a new remarkable expression for the kinetic energy. By an analogous calculation, the associate expression of the generalized quasipotential (18.2.22) will be of the form

U = U 1 + U 0 , U 1 = g jk pk , U 0 = − g jkU j (gk + U k ) + U 0 ,

(19.1.4)

hence a sum of a linear form and a constant with respect to the generalized momenta. In case of a simple quasi-potential (U j = 0 ), we have U = U 0 = U 0 . Observing that T1 + U 1 = 0 , the associate expression of the kinetic potential (18.2.34) is given by


118

1 L = L2 + L0 , L2 = T2 , L0 = − g jk (g j + U j )(gk + U k ) + g 0 + U 0 , 2

(19.1.5)

hence it will be a sum a quadratic form and a constant with respect to the generalized momenta. In case of a scleronomic mechanical system and of forces which derive from a simple quasi-potential we have L0 = U 0 , while L2 is a positive definite form in the generalized momenta. We notice that one can make the transformation (18.2.80) in case of a non-natural mechanical system too, if the condition (18.2.34''') is fulfilled. Indeed, in this case the Jacobian of the functions ∂L ∂q j is the Hessian of the kinetic potential; thus, the theorem of implicit functions allows to calculate q j = q j (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ,

j = 1,2,..., s . Obviously, these relations are linear with respect to the generalized momenta, so that the most important results previously obtained remain valid. 19.1.1.2 Donkin’s Theorem. Legendre’s Transformation Let be a function X = X (x1 , x 2 ,..., xs ) of class C 2 , for which the Hessian is non-zero

⎡ ∂2 X ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂x j ∂x k ⎦

(19.1.6)

Let us consider a transformation of variables generated by the function X in the form yj =

∂X , j = 1, 2,..., s . ∂x j

(19.1.7)

We can state Theorem 19.1.1 (Donkin). A transformation (19.1.7) for which the condition (19.1.6) is fulfilled being given, there exists a transformation, inverse to this one, generated by a function Y = Y (y1 , y2 ,..., ys ) of class C 1 , by means of the relations xj =

∂Y , j = 1, 2,..., s . ∂y j

(19.1.7')

The two functions are linked by the relation (we assume that the variables x i are expressed with the aid of the variables y j ; as well, the function Y too) X + Y = x jyj .

(19.1.8)

If X = X ( x1 , x 2 ,..., xs ; α1 , α2 ,..., αm ) , where αk , k = 1, 2,..., m , are given parameters, then we have Y = Y (y1 , y2 ,..., ys ; α1 , α2 ,..., αm ) too, so that


119 ∂Y ∂X =− , k = 1,2,..., m . ∂αk ∂αk

(19.1.9)

Indeed, the Hessian of the function X coincides with the Jacobian of the functions in the second member of the relations (19.1.7); on the basis of the condition (19.1.6), the theorem of implicit functions shows that, using the mentioned relations, we can obtain x j = x j (y1 , y2 ,..., ys ), j = 1, 2,..., s . Calculating the function Y from the relation (19.1.8) and expressing it by means of the variables y j , j = 1, 2,..., s , we obtain ∂x ∂Y ∂ ∂X ∂x k . = ( x y − X ) = k yk + x j − ∂y j ∂y j k k ∂y j ∂x k ∂y j

Taking into account the relations (19.1.7), we get the relations (19.2.7'), which are thus justified. If in the function X intervene also the parameters αj , j = 1, 2,..., m , then these ones intervene in the direct transformation (19.1.7) too; hence, we have to do with x j = x j (y1 , y2 ,..., ys ; α1 , α2 ,..., αm ) , j = 1, 2,..., s , as well as with Y = Y (y1 , y2 ,..., ys ; α1 , α2 ,..., αm ) . We can calculate ∂x ∂Y ∂ ∂X ∂x k ∂X ∂X , = − = − ( x y − X ) = k yk − ∂α j ∂α j k k ∂α j ∂x k ∂α j ∂α j ∂α j

where we took into account (19.1.7). The theorem stated by Donkin in 1854 is thus completely proved. The passing from the variables x j to the variables y j , considered above, is known as Legendre’s transformation. For instance, in the mathematical modelling of thermodynamics, where X = H is the enthalpy, −Y = G is Gibbs’s function of state, s = 1 , x1 = S is the entropy, while y1 = T is the absolute temperature, takes place the relation H − G = ST ,

(19.1.10)

the product ST being the bound energy; the relations ∂H ∂G = T, = −S ∂S ∂T

(19.1.10')

take place. 19.1.1.3 Canonical Equations We will use Donkin’s theorem and Legendre’s transformation to establish the equations of motion of the representative point P ∈ Γ 2s . We take thus X = L , x j = q j , y j = p j , α j = q j , j = 1, 2,..., s , as well as αm = t , m = s + 1 ; because of


120

the condition (18.2.34'''), the condition (19.1.6) is fulfilled. We introduce the function Y = H , called Hamilton’s function (Hamiltonian), in the form

H = p j q j − L .

(19.1.11)

If H = p j q j − L , then it results H = H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) . We may write pj =

∂L ∂H , , q = ∂q j j ∂p j

∂L ∂H ∂L ∂H , =− , = ∂q j ∂q j ∂t ∂t

(19.1.12) (19.1.12')

for j = 1,2,..., s . If in Lagrange’s equations (18.2.38) we take into account the first relations (19.1.12), then we can replace these relations by the equivalent system pj =

∂L ∂L . , p j = − ∂q j ∂q j

(19.1.13)

Using the first relations (19.1.12') and associating the last relations (19.1.12), there result the equations of motion (given by Hamilton in 1834) of the representative point P in the space Γ 2s in the normal form q j =

∂H ∂H , p = − , j = 1, 2,..., s . ∂p j j ∂q j

(19.1.14)

These equations are the canonical equations of Hamiltonian mechanics (Hamilton’s equations) (Hamilton, W.R., 1890). Using the relation of definition (19.1.11), we get ∂H ∂L ⎞ ∂qk ⎛ ∂L ⎞ ⎛ = − p j + ⎜ pk − ⎟ ∂q + ⎜ p j − ∂q ⎟ , ∂p j ∂ q ⎝ j j ⎠ k ⎠ ⎝ ∂ q ∂H ∂ L ⎛ ⎞ k . = q j + ⎜ pk − ∂p j ∂qk ⎟⎠ ∂p j ⎝

If the system (19.1.13) is verified, then one obtains Hamilton’s equations. On the other hand, if the canonical equations are verified, then the second relations obtained above lead to the first equations (19.1.13); indeed, one obtains a homogeneous system of linear algebraic equations for which the determinant of the coefficients det[ ∂qk / ∂p j ] is the inverse of the determinant det[ ∂p j / ∂qk ] , which is thus the Hessian (18,2,34'''), hence being non-zero (if L = T + U , then this determinant is equal to 1/ g ). Then, the first above relations lead to the last equations (19.1.13). We can thus state that Hamilton’s equations are equivalent to the equations (19.1.13), which – at their turn – are equivalent to Lagrange’s equations.


121

We can calculate the first variation of the Lagrangian in the form δL =

∂L ∂L δq + δq = p j δq j + p j δq j , ∂q j j ∂q j j

(19.1.15)

which represents the sixth form of the basic equation; taking into account the operator relation (18.1.77), this relation takes the remarkable form δL =

d ( p δq ) , dt j j

(19.1.15')

in case of holonomic constraints. We can also write δ ( p j q j − L

) = q j δp j

− p j δq j .

For Hamilton’s function, we obtain, analogically, δΗ =

∂H ∂H δq + δp . ∂q j j ∂p j j

Using the relation of definition of Hamilton’s function and comparing these results, we find again the canonical equations (19.1.14). Hamilton’s equations form a system of 2s differential equations of first order in the unknown functions q j = q j (t ), p j = p j (t ), j = 1, 2,..., s . Taking into account the one-to-one link between the generalized velocities and the generalized momenta, we can put the initial conditions in the form

q j (t0 ) = q j0 , p j (t0 ) = p j0 .

(19.1.14')

Hence, we must know the position P0 of the representative point P at the initial moment. The number of the unknowns and of the equations became double, but their order diminished by a unity, which represents a great advantage from the point of view of computation; we remark also some properties of symmetry of the system of equations (19.1.14). We can write Hamilton’s equations in the matric form R = ZHR

(19.1.14'')

too, where R = [ q1 , q2 ,..., qs , p1 , p2 ,..., ps ]T , ∂H ∂H ∂H ∂H ∂H ∂H ⎤ T , ,..., , , ,..., HR = ⎡⎢ ∂qs ∂p1 ∂p2 ∂ps ⎥⎦ ⎣ ∂q1 ∂q 2

are column matrices, respectively, and Z is the matrix

(19.1.14''')


122

⎡ 0 I⎤ Z = ⎢ ⎥. ⎣ −I 0 ⎦

(19.1.14IV)

The matrices 0 and I are the null and the unit matrices, respectively, with s rows and s columns. If, in Donkin’s theorem, we take X = T , x j = q j , y j = p j , j = 1,2,..., s , so that the generalized momenta are given by p j = ∂T / ∂q j , j = 1,2,..., s , while the function Y = Φ is specified by Φ = p j q j − T ,

(19.1.16)

then we obtain Hamilton’s equations in the form q j =

∂Φ ∂Φ , p = − + Q j + Rj , j = 1,2,..., s , ∂p j j ∂q j

(19.1.16')

corresponding to a non-natural and non-holonomic mechanical system, hence to Lagrange’s equations (18.2.32). If Q j , j = 1, 2,..., s , is the non-conservative part of the generalized force, then Hamilton’s equations read q j =

∂H ∂H , p j = − + Q j + Rj , j = 1,2,..., s . ∂p j ∂q j

(19.1.16'')

In case of holonomic constraints ( Rj , j = 1,2,..., s ) and of quasi-conservative forces which derive from a simple potential ( Q j = 0, j = 1, 2,..., s ) we find again the canonical equations; in case of a generalized potential appears a supplementary term due to the modality in which the generalized momenta have bean introduced. The advantage to introduce these quantities by the relations (18.2.80) is thus put in evidence; as a matter of fact, we can thus consider the case of an arbitrary Lagrangian too. 19.1.1.4 General Theorems. First Integrals For the initial value problem (19.1.14), (19.1.14') one can state Theorem 19.1.2 (of existence and uniqueness; Cauchy–Lipschitz). If Hamilton’s function H is of class C 1 with respect to the space variables and of class C 0 with respect to time on the ( 2s + 1 )-dimensional interval D , specified by q j0 − Q j0 ≤ q j ≤ q j0 + Q j0 , p j0 − Pj0 ≤ p j ≤ p j0 + Pj0 , t0 − T0 ≤ t ≤ t0 + T0 , Q j0 , Pj0 ,T0 = const, j = 1,2,..., s ,

and defined in the space Cartesian product of the phase space (of canonical co-ordinates q1 , q2 ,..., qs , p1 , p2 ,..., ps ) by the time space (of co-ordinate t ), and if Lipschitz’s conditions


123

∂ ∂ H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) − H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∂q j ∂q j μj s ⎡ μ 1 ⎤ ≤ ⎢⎣ τ λk qk − qk + λk pk − pk ⎥⎦ , T k∑ =1 ∂ ∂ H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) − H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∂p j ∂p j ≤

1 νjT

∑ ⎡⎢⎣ λk s

k =1

qk − qk +

τ ⎤ p − pk ⎥ , μλk k ⎦

hold for j = 1,2,..., s , where T > 0 is a time constant independent on q j , p j and t , the time constant τ and the constant μ of the nature of a mass are equal to unity, while λk , μj , ν j are constants equal to unity (dimensionless if the corresponding generalized co-ordinate is a length and having the dimension of a length if the corresponding generalized co-ordinate is non-dimensional), then there exists a unique solution q j = q j (t ), p j = p j (t ) of the system (19.1.14), which satisfies the initial conditions (19.1.14') and is definite an the interval t0 − T ≤ t ≤ t0 + T , where q j0 τ p j0 ⎡ T ≤ min ⎢T0 , λ j , ,T V μλj V ⎣

⎤ ∂H ⎡ τ ∂H ⎤ ⎥ , V = max ⎢ μλ ∂q , λj ∂p in D ⎥ , j j ⎣ j ⎦ ⎦

without summation with respect to the index j . The continuity of the derivatives of first order of the function H with respect to the canonical co-ordinates on the interval D ensures the existence of the solution, according to Peano’s theorem. Lipschitz’s conditions must be also fulfilled for the uniqueness of the solution; but the latter conditions can be replaced (as in the case of other theorems of uniqueness) by other ones less restrictive, according to which the partial derivative of second order of the function H with respect to the canonical coordinates must exist and must be bounded in absolute value on the interval D . As a matter of fact, the conditions in the Theorem 19.1.2 are sufficient conditions of existence and uniqueness, which are not necessary too. As in the case of other analogous theorems (e.g., the Theorem 18.2.1) one can obtain a prolongation of the solution for t ∈ [t1 , t2 ] , corresponding to a certain interval of time in which the considered phenomena takes place, or even for t ∈ [t0 , ∞ ) or t ∈ ( ∞, t1 ] or t ∈ ( −∞, ∞ ) .

We can state also theorems on the continuous or analytical dependence of the solution on a parameter, analogue to the Theorems 6.1.3 and 6.1.4; as well, we state Theorem 19.1.3 (on the differentiability of the solution). If, in the neighbourhood of a point P (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∈ D , the functions ∂H / ∂q j , ∂H / ∂p j , j = 1,2,..., s , H = H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) , are of class C k , then the solution


124

q j (t ) and p j (t ) of the system (19.1.14) which satisfy the initial conditions (19.1.14')

are of class C k +1 in a neighbourhood of the point P . We call first integral of the system of canonical equations (19.1.14) a function f = f (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) of class C 1 , which is reduced to a constant along the integral curves (trajectories in the phase space) q j = q j (t ; c1 , c2 ,..., c2 s ), p j = p j (t ; c1 , c2 ,..., c2 s ), j = 1, 2,..., s ,

(19.1.17)

of this system ( ck , k = 1, 2,..., 2s , are, obviously, integration constant); hence df =

∂f ∂f dq j + dp j + f dt = 0 ∂q j ∂p j

(19.1.18)

or df ∂f ∂f q + p + f = 0 = dt ∂q j j ∂p j j

(19.1.18')

along the integral curves. The canonical equations (19.1.14) can be written in the form d q1 dq dp1 dp2 dqs dps dt = 2 = ... = = = = ... = = ∂H ∂H ∂H ∂H ∂H ∂H 1 − − − ∂p1 ∂p2 ∂ps ∂q1 ∂q2 ∂qs

(19.1.19)

too, for integration being necessary 2s independent first integrals. If we determine l ≤ 2s first integrals fk (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) = ck , ck = const, k = 1, 2,..., l ,

(19.1.20)

the matrix ∂ ( f1 , f2 ,..., fl ) ⎡ ⎤ M = ⎢ ⎣ ∂ (q1 , q2 ,..., qs , p1 , p2 ,..., ps ) ⎥⎦

being of rank l , then all these integrals are functionally independent (independent first integrals) and we can express l unknown functions of the system (19.1.19) with respect to the other ones; replacing in this system of equations, the problem is reduced to the integration of a system of equations with 2s − l unknowns (hence, with a smaller number of unknowns). If l = 2s , then all the first integrals are independent, so that the system (19.1.20) determines all the unknown functions. We notice that, if l > 2s , then the first integrals (19.1.20) are no more independent; hence, one can set up at the most 2s independent first integrals.


125

Solving the system (19.1.20) for l = 2s , we obtain (the matrix M is a square one of order 2s , for which det M ≠ 0 ) the solution (19.1.17), hence a general integral of the system of equations (19.1.14). One puts thus in evidence 2s integration constants. Imposing the initial conditions (19.1.14'), it results q j (t0 ; c1 , c2 ,..., c2 s ) = q j0 , p j (t0 ; c1 , c2 ,..., c2 s ) = p j0 , j = 1,2,..., s ; the conditions (19.1.14') being independent,

we may write ⎡ ∂ (q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ) ⎤ det ⎢ ⎥ ≠0 ∂ (c1 , c2 ,..., c2 s ) ⎣ ⎦

and we get ck = ck (t0 ;q10 , q 20 ,..., qs0 , p10 , p20 ,..., ps0 ) , k = 1, 2,..., 2s , using the theorem of implicit functions. Thus, it results, finally (for j = 1,2,..., s ), q j = q j (t ; t0 , q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ), p j = p j (t ; t0 , q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ).

(19.1.21)

Hence, in the frame of the conditions of the theorem of existence and uniqueness, the canonical equations and the corresponding initial conditions determine, univocally, the motion of the discrete mechanical system S , subjected to holonomic ideal constraints, in a finite interval of time; by prolongation, the statement can be valid for any t . The deterministic aspect of mechanics is thus put in evidence in the Hamiltonian presentation too. If f1 = c1 and f2 = c2 are two distinct first integrals, then f ( f1 , f2 ) = 0 is a first integral too; any other first integral is expressed, in general, as a function of the 2s distinct first integrals of Hamilton’s equations. 19.1.1.5 Hamilton’s Function Starting from the relation of definition (19.1.11), passing to canonical co-ordinates, taking into account (18.2.80), (18.2.34') and using Euler’s theorem concerning homogeneous functions, we can write Hamilton’s function in the form

∂L H = p j q j − L = q − L = 2L2 + L1 − L = L2 − L0 . ∂q j j If we take into account also the relations (18.2.34''), it results H =

E , E = T2 − T0 + V ,

(19.1.22)

where E is the generalized mechanical energy, given by (18.2.68) and expressed in canonical co-ordinates (V = −U in case of a simple quasi-potential or V = −U 0 in case of a generalized quasi-potential). In case of a scleronomic mechanical system


126

( T2 = 0, T0 = 0 ) we have E = T2 + V = T + V = E , obtaining the mechanical energy of the considered mechanical system. Observing, on the basis of the formulae (18.2.15') and (19.1.1'), that 2T2 − g jk g jl [ pl − (gl + U l )]g km [ pm − (gm + U m )] = g jk [ p j − ( g j + U j )][ pk − ( gk + U k )],

we can represent Hamilton’s functions in the form H = H 2 + H1 + H 0 ,

(19.1.23)

where H 2 = T2 = H0 =

1 jk g p j pk , H 1 = − g jk (g j + U j ) pk , 2

1 jk g (g j + U j )( gk + U k ) − g 0 + V , 2

(19.1.23')

hence a sum of a quadratic form, of a linear form and of a constant with respect to the generalized momenta. In case of scleronomic constraints we have g 0 = 0 , g j = 0, j = 1,2,..., s , so that H 1 = − g jkU j pk , H 0 =

1 jk g U jU k + V , 2

(19.1.23'')

and in case of generalized forces which derive from a simple potential it result U j = 0 , j = 1,2,..., s , so that H 1 = − g jk g j pk , H 0 =

1 jk g g j gk − g 0 + V ; 2

(19.1.23''')

if both conditions are fulfilled, then H 1 = 0, H 0 = V . If g jk = δ jk , then we have g jk = δ jk , where δ jk is Kronecker’s tensor; we can write T2 =

1 1 1 1 δ jk q j qk = q j q j , H 2 = T2 = δ jk p j pk = p j p j , 2 2 2 2

(19.1.24)

hence in the form of a sum of squares. In this case, the generalized co-ordinates are orthogonal (they are called normal co-ordinates), the generalized momenta having the same property. We can calculate ∂H ∂H dH = q j + p + H . dt ∂q j ∂p j j


127

Taking into account the equations (19.1.14), we can state that the relation ∂H dH = = H dt ∂t

(19.1.25)

takes place along the integral curves of Hamilton’s system of equations. If Hamilton’s function does not depend explicitly on time ( H = 0 , hence E = 0 ), then the generalized mechanical energy H (q1 , q 2 ,..., qs , p1 , p2 ,..., ps ) =

E (q1 , q 2 ,..., qs , p1 , p2 ,..., ps ) = h , h = const, (19.1.26)

is a first integral of the canonical system (19.1.14). This first integral corresponds to Jacobi’s first integral or to a first integral of Jacobi type of Lagrange’s system of equations, which can be written if L = 0 (see Sect. 18.2.3.4); as a matter of fact, H = 0 leads to L = 0 and reciprocally, on the basis of the last relation (19.1.12'). The mechanical system is, in this case, a generalized conservative system (as in the case in which L = 0 ). If the mechanical system is scleronomic, then we find again the first integral of the mechanical energy expressed in canonical co-ordinates ( H = T + V = E = h ), as a particular case; we have to do with a simple or generalized potential (U = 0 ) in this case, so that ∂ (T2 − T0 )/ ∂t + ∂V / ∂t = 0 . We notice that to have H = 0 it is not necessary that the mechanical system be scleronomic. Indeed, from (19.1.23') we see that it is necessary and sufficient to have g jk = 0, g j + U j = 0, g0 = V ;

(19.1.27)

the first relations above lead also to g jk = 0 for j , k = 1, 2,..., s , Taking into account (18.2.15'), (18.2.15''), we can state that these relations can take place only and only if the derivative ∂ri / ∂q j , i , j = 1, 2,..., s , do not depend explicitly on time, i.e. if ri = ri′ + ri′′ , ri′ = ri′ (q1 , q2 ,..., qs ) , ri′′ = ri′′(t ), i = 1, 2,..., s . This statement remains

valid for Jacobi’s first integral or for the first integrals of Jacobi type too; in particular, for ri′′ = 0, i = 1, 2,..., s , when the mechanical system is scleronomic, then one obtains the integral of mechanical energy. 19.1.1.6 Reduction of the Number of First Integrals As we have seen above, the function H is a first integral of the canonical system of equations if H = 0 ; hence, it is necessary to determine other 2s − 1 first integrals. If we write Hamilton’s equations in the form (19.1.19), then we notice that – in this case – we can neglect the last ratio in this system; indeed, to integrate the remaining system of equations, there are sufficient 2s − 1 first integrals, which do not depend explicitly on time. Thus we can determine 2s − 1 canonical co-ordinates as functions of one of them. To fix the ideas, we assume that ∂H / ∂p1 ≠ 0 ; we find thus the generalized co-


128

ordinates q j = q j (q1 ; c1 , c2 ,..., c2 s −1 ), j = 2, 3,..., s , and the generalized momenta pi = pi (q1 ; c1 , c2 ,..., c2 s −1 ), i = 1, 2,..., s . In this case, it results the function of

Hamilton H = H (q1 ; c1 , c2 ,..., c2 s −1 ) , where we have put in evidence the integration constants. From the first and the last ratio (19.1.19) we get −1

⎛ ∂H ⎞ t = ∫⎜ ⎟ dq1 + c2 s = F (q1 ; c1 , c2 ,..., c2 s ) , ⎝ ∂p1 ⎠

(19.1.28)

the function F containing he last integration constant c2 s too. Observing that the derivative ∂F / ∂q1 = ( ∂H / ∂p1 )−1 ≠ 0 , the theorem of implicit functions allows to calculate q1 = q1 (t ; c1 , c2 ,..., c2 s ) . Introducing then in the above results, we obtain the general integrals (19.1.17). The system of equations (19.1.19) is of the form (15.1.24), so that one can use the theory of Jacobi’s multiplier (see Sect. 15.1.1.4), as well as the theory of the last multiplier (see Sect. 15.1.1.5). In this case, x j = q j , xs + j = p j , X j = ∂H / ∂p j , Xs + j = −∂H / ∂q j , j = 1, 2,..., s , and x 2 s + 1 = 1 , n = 2s + 1 . We notice that the

divergence of the vector which has as components the denominators in (19.1.19) vanishes ∂ ∂q1

∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ⎛ ∂H ⎞ ⎜ ∂p ⎟ + ∂q ⎜ ∂p ⎟ + ... + ∂q ⎜ ∂p ⎟ s ⎝ s ⎠ ⎝ 1⎠ 2 ⎝ 2 ⎠ ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ∂ + − + − + ... + ⎜− ⎟ + ( I ) = 0, ∂p1 ⎜⎝ ∂q1 ⎟⎠ ∂p2 ⎜⎝ ∂q2 ⎟⎠ ∂ps ⎝ ∂qs ⎠ ∂t

hence the condition (15.1.30) is fulfilled; a constant is thus a multiplier of the sequence of equations (19.1.19), while – according to the Theorem 15.1.4'' – any non-constant multiplier is a first integral of this system of differential equations. Hence, there are sufficient 2s − 1 first integrals to determine the general solution of the respective system. If H = 0 , then there remain sufficient 2s − 2 first integrals for a complete study of the canonical equations. For instance, in case of the problem of two particles – at a first view – there are necessary 2 ⋅ 6 = 12 first integrals; according to what has been shown above, in case of scleronomic mechanical systems and of conservative forces there are sufficient 2 ⋅ (6 − 1) = 10 first integrals. If the two particles are subjected to no one constraint, being acted upon only by forces of Newtonian attraction, we are just in the mentioned case; the conservation theorem of momentum leads to six first integrals, the conservation theorem of the moment of momentum allows to write three first integrals and the conservation theorem of mechanical energy given also a first integral, so that the integration of the corresponding system of canonical equations is reduced to quadratures. Let be a scleronomic conservative Hamiltonian system with two degrees of freedom. It is possible, in this case, to write the integral of the mechanical energy, i.e. H = (q1 , q2 , p1 , p2 ) = h , h = const ; one obtains the differential system


129 d q1 dq 2 dp1 dp2 dt = = = = . ∂H ∂H ∂H ∂H 1 − − ∂p1 ∂p2 ∂q1 ∂q2

Let us suppose that another first integral f (q1 , q2 ; p1 , p2 ) = c , c = const , which does not contain explicitly the time, is known. The two first integrals being independent, we can calculate p j = p j (q1 , q2 ; h , c ), j = 1, 2 ; replacing in Hamilton’s equations, this system is reduced to dq1 dq = 2 = dt , ∂H ∂H ∂p1 ∂p2

with the unknown functions qk = qk (t ), k = 1, 2 , for which the last multiplier M = det[ ∂ ( p1 , p2 )/ ∂ (h , c )] satisfies the equation (see Sect. 15.1.1.5 too)

∂M ⎞ ⎛ ∂M ⎞ ∂ ⎛⎜ M ⎟ ∂ ⎜ M ∂p ⎟ ∂ p ⎝ ⎝ 1 ⎠ 2 ⎠ + = 0. ∂q1 ∂q2

Hence, ⎛ ∂H

∂H

⎞

∫ M ⎜⎝ ∂p2 dq1 − ∂p1 dq2 ⎟⎠ = const is a first integral of the last differential system. We notice that H and f depend on h and c by means of the generalized momenta p1 and p2 . Because dH / dh = 1 and dH / dc = 0 , it results

∂H ∂p1 ∂H ∂p2 ∂H ∂p1 ∂H ∂p2 + = 1, + = 0. ∂p1 ∂h ∂p2 ∂h ∂p1 ∂c ∂p2 ∂c One obtains analogous results from df / dh = 0 , df / dc = 1 . The determinant M being non-zero, we can write ∂p ∂H = 2,M ∂p1 ∂c ∂p ∂f = 2,M M ∂p1 ∂h

M

∂p ∂H = − 1, ∂p2 ∂c ∂p ∂f = − 1. ∂p2 ∂h

The last first integral becomes ∂χ = const, χ(q1 , q2 ; h , c ) = ∫ ( p1 dq1 + p2 dq2 ) . ∂c


130

Then, taking into account M det[ ∂ (H , f )/ ∂ ( p1 , p2 )] = 1 , the reduced system of differential equations allows to write ∂f ∂f dt = M ⎛⎜ dq 2 − dq1 ⎞⎟ , p p ∂ ∂ ⎝ 1 ⎠ 2

wherefrom t =

∂χ + const . ∂h

The integration of the differential system is thus reduced to a quadrature, which determines the function χ . 19.1.1.7 Routh’s Equations It is convenient, in some cases, to use as parameters which describe the motion of the mechanical system S a part of the Hamiltonian co-ordinates and a part of the Lagrangian ones; one obtains thus a “mixed method” to approach the problem. We assume that the motion is specified by Hamilton’s variables (denoted by Greek indices) q α , pα , α = 1, 2,..., r and Lagrange’s variables (denoted by Roman indices) q j , q j , j = r + 1, r + 2,..., s , the set of which form Routh’s variables. We effect a partial Legendre transformation of the form pα =

∂L , α = 1, 2,..., r , ∂qα

(19.1.29)

which allows to determine qα = qα (q β , pβ , q j , q j ; t ) = qα (q1 , q2 ,..., qs , p1 , p2 ,..., pr , qr + 1 , qr + 2 ,..., qs ; t ), α = 1, 2,..., r .

Applying a variation δ to Lagrange’s function L = L (q α , qα , q j , q j ; t ) , we obtain δL =

r

∂L

∑ ⎝⎛⎜ ∂qα δqα

α =1

+

∂L δqα ⎞⎟ + ∂qα ⎠

∂L ⎛ ∂L ⎜ ∂q δq j + ∂q δq j j j j =r +1 ⎝ s

∑

⎞ ⎟. ⎠

Taking into account (19.1.29), we notice that r

δ ∑ pα qα = α =1

r

∂L

∑ ∂qα δqα

α =1

+

r

∑ qα δpα .

α =1

wherefrom r ∂L ⎛ r ⎞ δ ⎜ ∑ pα qα − L ⎟ = − ∑ ⎛⎜ δq α − qα δpα ⎞⎟ − ∂ q α ⎝ ⎠ ⎝ α =1 ⎠ α =1

∂L ⎛ ∂L ⎞ ⎜ ∂q δq j + ∂q δq j ⎟ . j j ⎠ j =r +1 ⎝ s

∑


131

We introduce Routh’s function

R =

∑ pαqα − L r

,

(19.1.30)

α =1

where “the cap” indicates the expression of the generalized velocities qα , α = 1, 2,...,r , by means of Routh’s variables, so that R = R (q α , pα , q j , q j ; t ) ; it results δR =

r

∂R

∑ ⎛⎜⎝ ∂qα

α =1

δq α +

∂R δpα ⎞⎟ + ∂pα ⎠

∂R ⎛ ∂R ⎞ ⎜ ∂q δq j + ∂q δq j ⎟ . j j ⎠ j =r +1 ⎝ s

∑

Identifying the above relations, where we take into account the notation (19.1.30), we can write ∂L ∂R ∂R , qα = , α = 1, 2,..., r , = ∂q α ∂q α ∂pα ∂L ∂R ∂L ∂R ,− , j = r + 1, r + 2,..., s . − = = ∂q j ∂q j ∂q j ∂q j −

Lagrange’s equations (18.2.38) led thus Routh, in 1876, to the equations qα =

∂R ∂R , p α = − , α = 1, 2,..., r , ∂pα ∂q α

d ⎛ ∂R ⎞ ∂R − = 0, j = r + 1, r + 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(19.1.31) (19.1.31')

called Routh’s equations (see Sect. 18.2.3.7 too); we remark that a part of these equations are of Hamilton type (Routh’s function plays the rôle of Hamilton’s function) and another part of Lagrange type (Routh’s function plays the rôle of Lagrange’s kinetic potential), which justifies the name of “mixed method” given above. In the case in which the generalized co-ordinates q α , α = 1,2,..., r , are ignorable we find again the Routh–Helmholtz theorem (see Sect. 18.2.3.7 too). If the holonomic constraints to which is subjected the mechanical system S are also scleronomic, then the kinetic energy is expressed in the form T =

1 s s ∑ g jk (qr +1 , qr + 2 ,..., qs )qj qk . 2∑ j = 1 k =1

(19.1.31'')

We also assume that the generalized forces are quasi-conservative, deriving from a simple quasi-potential U = U (q1 , q2 ,..., qs ; t ) . In this case, pα =

∂Τ = ∂qα

r

∑ gαβ qβ

β =1

+

s

∑

j =r +1

g α j q j , α = 1, 2,..., r .

(19.1.32)


132

Because [ g αβ ] is a square matrix of rank r and det[ g αβ ] ≠ 0 , corresponding to the condition (18.2.34'''), we get r

∑ g αβ pβ

qα =

β =1

−

s

∑

j =r +1

hα j q j , α = 1, 2,..., r ,

(19.1.32')

where g αβ is the normalized algebraic complement of the element g αβ in det[ g αβ ] , while r

∑ g αβ g β j ,

hα j =

β =1

α = 1, 2,..., r , j = r + 1, r + 2,..., s .

(19.1.32'')

Obviously, the coefficients g αβ and hα j are functions only on the palpable coordinates q j , j = r + 1, r + 2,..., s . Observing that g jk =

r ∂2T ∂ ⎛ ∂T ∂T ∂qα = + ∑ ⎜ ∂q j ∂qk ∂q j ⎝ ∂qk α =1 ∂qα ∂qk ∂ ⎛ ∂T ⎞ ∂ r = qα − hαk pα , ⎜ ⎟ ∂q j ⎝ ∂qk ⎠ ∂q j α∑ =1

⎞ ⎟ qα ⎠

there results r

∑ gα j hαk

g jk = g jk −

α =1

= g jk −

r

r

∑ ∑ g αβ gα j g βk , j , k

α =1 β =1

= r + 1, r + 2,..., s . (19.1.33)

Analogically, ∂2T ∂ = = ∂pα ∂pβ ∂pα

g αβ

=

r

∑ g γβ δγα

γ =1

g jα ∂ = ∂q j

r

∑ g βα pβ

β =1

∂T ∂qγ ∂ = ∂ p ∂ p α β γ =1 r

∑ ∂qγ

r

∑ g γβ pγ

γ =1

= g αβ , α, β = 1,2,..., r ,

∂2T ∂ = = ∂q j ∂pα ∂q j

r

∂T ∂qβ ∂pα

∑ ∂qβ

β =1

= 0, α = 1, 2,..., r , j = r + 1, r + 2,..., s ,

so that we can write s 1 r r 1 s T = ∑ ∑ g αβ pα pβ + g q q . ∑ ∑ 2 α =1 β =1 2 j = r + 1 k = r + 1 jk j k

We denote

(19.1.34)


133 T∗ =

s 1 s g q q . ∑ ∑ 2 j = r + 1 k = r + 1 jk j k

(19.1.35)

Let us introduce Routh’s potential U∗ = U −

1 r r αβ ∑ g pα p β 2 α∑ =1 β =1

(19.1.36)

and the generalized potential Π = U∗ +

s

r

∑ ∑ hα j pαqj .

j =r + 1 α =1

(19.1.36')

Taking into account the relation of definition (19.1.30) and the generalized velocities (19.1.32'), we obtain Routh’s function in the form

R = −(T ∗ + Π ) .

(19.1.37)

From (19.1.34), (19.1.35), and (19.1.36), it results T −U = T∗ −U∗.

(19.1.38)

Lagrange’s equations (19.1.31') can be thus considered as being the equations of motion of a reduced system, which admits −R (given by (19.2.37)) as kinetic potential and which has the same mechanical energy as the initial system. If, in particular, the kinetic energy of the initial mechanical system does not contain products of the generalized velocities qα of the ignorable co-ordinates, hence if g α j = 0, α = 1, 2,..., r , j = r + 1, r + 2,..., s , then the kinetic energy T is obtained as a sum of two quadratic forms: a form in the generalized velocities qα and a form in the generalized velocities q j ; such a mechanical system is a gyroscopic uncoupled one. In this case, hα j = 0, α = 1,2,..., r , j = r + 1, r + 2,..., s , while Π = U ∗ , as well, T∗ =

s 1 s g q q . ∑ ∑ 2 j = r + 1 k = r + 1 jk j k

(19.1.35')

As it can be easily seen, if the initial system is conservative (U = 0 ), then the reduced system has the same property (U ∗ = 0 ). If one has ∂U / ∂q α = 0 too, then it results ∂L / ∂q α = 0, α = 1, 2,..., r , so that the ignorable co-ordinates are hidden ones too. The motions in which vary only the hidden co-ordinates are called hidden motions; such type of motion occurs, e.g., in case of mechanical systems which contain gyroscopes. H. Hertz dealt in 1894 with these problems, considering the potential energy of a conservative system as being the kinetic energy of some hidden motions.


134 19.1.1.8 Cyclic Co-ordinates

A generalized co-ordinate q α which does not intervene explicitly in Hamilton’s function ( ∂H / ∂q α = 0 ) is called, after Helmholtz, cyclic co-ordinate; let us admit thus that the first r co-ordinates are cyclic ( α = 1,2,...,r ). Taking into account the first relation (19.1.12'), we can state that an ignorable co-ordinate (for which ∂L / ∂q α = 0 ) is a cyclic one too. We are thus led to r first integrals pα = bα , bα = const , α = 1,2,...,r ; taking into account the results in the preceding subsection, we can state that the cyclic co-ordinates do not intervene explicitly neither in Routh’s function, so that the equations (19.1.31) lead to the same results as the Routh–Helmholtz theorem (see Sect. 18.2.3.7). The Lagrange type equations (19.1.31) remain to be integrated. If we deal with the system of canonical equations, after the determination of the mentioned r first integrals, we remain with the system of equations (19.1.14) for j = r + 1, r + 2,..., s (a system of 2(s − r ) partial differential equations of first order with 2(s − r ) unknown functions). Hamilton’s function will be of the form H = H (q j , p j , bα ; t ) ≡ H (qr + 1 , qr + 2 ,..., qs , pr + 1 , pr + 2 ,..., ps , b1 , b2 ,..., br ; t ) ;

thus, by integration of the reduced canonical system, we get q j = q j (t ; ak , bk , bα ) ≡ q j (t ; ar + 1 , ar + 2 ,..., as , b1 , b2 ,..., bs ) , p j = p j (t ; ak , bk , bα ) ≡ p j (t ; ar + 1 , ar + 2 ,..., as , b1 , b2 ,..., bs ), ak , bk = const, j , k = r + 1, r + 2,..., s .

Replacing in Hamilton’s function, we obtain H = H (t ; ak , a j , bk , bα ) ≡ H (t ; ar + 1 , ar + 2 ,..., as , b1 , b2 ,..., bs ) .

Introducing in the first r equations of the first group of equation (19.1.14), we find also the cyclic co-ordinates qα =

∂H

∫ ∂bα

dt + aα , α = 1, 2,..., r .

(19.1.39)

Hence, q α = q α (t ; aα , bα , bj ) ≡ qα (t ; a1 , a2 ,..., as , b1 , b2 ,..., bs ) , only the cyclic coordinates depending on all the integration constant. If, in particular, all the generalized co-ordinates are cyclic ( r = s ), then all the generalized momenta are constant, while H = H (t ;b1 , b2 ,..., bs ) ; the cyclic co-ordinates are obtained by s quadratures, intervening the other s integration constants. If the mechanical system is also scleronomic, then it results ∂H / ∂bj = ω j , ω j = const , j = 1, 2,..., s , the generalized co-ordinates being of the form q j = ω j t + a j , j = 1, 2,..., s ,

(19.1.39')


135

which justifies the denomination of cyclic co-ordinates. The problem of integration of Hamilton’s equations (19.1.14) is thus reduced to the determination of a transformation of generalized co-ordinates to maintain the form of the canonical equations, the new generalized co-ordinates being cyclic. As we will see, the intervention of the one or more cyclic co-ordinates may lead to a simplification of many methods of calculation. 19.1.1.9 The Pfaff Form of Hamilton. The Bilinear Covariant. Whittaker’s Equations Let be the differential form ω =

n

∑ Xi (x1 , x 2 ,..., xn )dxi ,

(19.1.40)

i =1

where the functions Xi , i = 1, 2,..., n , of class C 1 are definite on a certain domain in \n ; we assume that ω is not a total differential. An expression of this form is called Pfaffian (a Pfaff form). Let us suppose that x j = x j (u1 , u2 ), j = 1, 2,..., n , depending on two parameters. Let be the differential quantities dx j =

∂x j ∂x j du1 , δx j = du . ∂u1 ∂u2 2

Noting that d ( δx j

)=

∂2 x j d u d u , δ ( dx j ∂u1 ∂u2 2 1

)=

∂2 x j du d u ∂u2 ∂u1 1 2

and assuming that x j are functions of class C 2 in the two parameters, we have d( δx j ) = δ(dx j ) , according to Schwartz’s theorem. If we denote

ωd =

n

n

i =1

i =1

∑ Xi dxi , ωδ = ∑ Xi δxi

(19.1.41)

and take into account the permutation relation of the differential operators, then we can write ω ′ = ω ( d, δ ) = δωd − dωδ =

n

∑ ω j δx j , j =1

⎛ ∂Xi ∂X j ⎞ dx , j = 1, 2,..., n , ωj = ∑ ⎜ − ∂x i ⎟⎠ i i = 1 ⎝ ∂x j n

(19.1.41')

obtaining thus the bilinear covariant of the form ω . Let us make the change of variables

x j = x j (y1 , y2 ,..., yn ), x j ∈ C 1 , j = 1, 2,..., n ,

(19.1.42)


136

so that det[ ∂x j / ∂yk ] ≠ 0 on a domain in \n . Denoting ∂x

n

∑ Xi ∂yki , k

Yk =

i =1

= 1, 2,..., n ,

(19.1.42')

we obtain

ω = ω′ =

n

∑ ωk δyk , ωk

=

k =1

n

∑Yk dyk ,

k =1

∂Y

∑ ⎛⎜⎝ ∂ykl n

l =1

−

(19.1.43)

∂Yk ⎞ dy , k = 1, 2,..., n . ∂yl ⎟⎠ l

(19.1.43')

The Pfaff form ω and its bilinear covariant ω ′ maintain their form. The necessary and sufficient conditions to have ω ′ ≡ 0 for any δx j , considered arbitrary, are ω j = 0, j = 1, 2,..., n , being thus led to the differential system associate to the Pfaff form n

⎛ ∂X

∑ ⎜⎝ ∂x ji

i =1

−

∂X j ⎞ dx = 0, j = 1, 2,..., n . ∂x i ⎟⎠ i

(19.1.44)

The transform of this system, by the change of variables (19.1.42), will – obviously – be ∂Y

∑ ⎛⎜⎝ ∂ykl n

l =1

−

∂Yk ⎞ dy = 0, k = 1, 2,..., n . ∂yl ⎟⎠ l

(19.1.44')

As it can be easily seen, if ω is a total differential, then ω ′ = 0 ; consequently, we can state that the Pfaff forms ω and

c ω + dF = c ω +

n

∂F

∑ ∂xi

i =1

dx i ,

where c = const , and F = F ( x1 , x 2 ,..., x n ) is a function of class C 2 , have the same bilinear covariant. By definition, the Pfaff form of Hamilton is ω = p j dq j − H dt ,

(19.1.45)

where we have introduced the canonical variables q j , p j and the time t (hence, n = 2s + 1 ); the bilinear covariant of the form is

ω ′ = δωd − dωδ = δp j dq j − δHdt − ( dp j δq j − dH δt )

(

)

∂H ∂H ∂H ⎛ ⎞ ⎛ ⎞ dt ⎟ δq j + ⎜ dq j − dt ⎟ δp j + dH − dt δt . = − ⎜ dp j + ∂ ∂ ∂t q p j j ⎝ ⎠ ⎝ ⎠


137

The differential system of Hamilton associated to the Pfaff form is formed by the canonical equations (19.1.14) and the differential outcome (19.1.25) of them. By introducing the notion of exterior differential in the sense of Elie Cartan, the Pfaff form of Hamilton can lead to a new differential principle of mechanics; choosing conveniently the Hamiltonian, one can build up new mathematical models (e.g., the invariantive model of O. Onicescu). In case of a generalized conservative system ( H = 0 ) one obtains the first integral (19.1.26) of the generalized mechanical energy. Assuming that ∂H / ∂p1 ≠ 0 (at least one the components of the generalized momentum must be non-zero, otherwise the motion of the mechanical system could not take place), the theorem of implicit functions leads to p1 = − K (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; h ) . The corresponding Pfaff form will be ω =

s

∑ p j dq j

j =2

− Kdql − hdt ,

(19.1.46)

where K plays the rôle of Hamilton’s function and q1 the rôle of time; indeed, according to a previous notice, the form ω and the form ω + hdt ( hdt is an exact differential) have the same bilinear covariant. In this case, the associate differential system will be q j′ =

∂K ∂K dK ∂K , , p′ = − , j = 1, 2,..., s , = ∂p j j ∂q j d q1 ∂q1

(19.1.47)

where q j′ = dq j / dq1 , p j′ = dp j / dq1 , j = 1,2, 3,..., s ; these 2(s − 1) equations of Hamilton type are known as Whittaker’s equations. One can thus determine the canonical co-ordinates q j = q j (q1 ; a2 , a 3 ,..., as , b2 , b3 ,..., bs , h ), p j = p j (q1 ; a2 , a 3 ,..., as , b2 , b3 ,..., bs , h ), j = 1, 2,..., s ;

(19.1.47')

replacing then in the function K , we obtain K = K (q1 ; a2 , a 3 ,..., as , b2 , b3 ,..., bs , h ) , hence the generalized momentum p1 too. Finally, Hamilton’s equation q1 = ∂H / ∂p1 allows to put in evidence the dependence on time of the canonical co-ordinates, introducing the last integration constant too. Besides, these considerations can be put in connection to those made in Sect. 19.1.1.6. Denoting

L =

s

∑ p j q j′

j =2

−K,

(19.1.48)

where L ′ = L ′(q1 , q2 ,..., qs , q2′ , q 3′ ,..., qs′ ) , and effecting a Legendre transformation, we can replace Whittaker’s equations (of Hamilton type) by Jacobi’s equations (of Lagrange type) (see Sect. 18.2.3.5).


138

19.1.2 Lagrange’s Brackets. Poisson’s Brackets In what follows one introduces Lagrange’s and Poisson’s brackets, in connection to the system of canonical equations; these brackets allow to determine first integrals of the mentioned equations and put in evidence interesting properties of the integration constant. 19.1.2.1 Lagrange’s Brackets. Lagrange’s Theorem Let be the function x i = x i ( ξ1 , ξ2 ,..., ξn , η1 , η2 ,..., ηn ), yi = yi ( ξ1 , ξ2 ,..., ξn , η1 , η2 ,..., ηn ), i = 1, 2,..., n ,

(19.1.49)

of class C 1 . We call Lagrange’s brackets (denoted by square brackets) the sums ⎡ ∂ (x , y ) ⎤

∂x ∂y ⎞ ⎛ ∂x ∂yi − i i ⎟, ∂ ∂ ξ ξk ∂ξ j ⎠ k ⎦ k ⎣ i =1 i =1 ⎝ n n ∂x ∂y ⎞ ⎡ ∂ (x , y ) ⎤ ⎛ ∂x ∂y [ η j , ηk ] = ∑ det ⎢ ∂ ( η i , ηi ) ⎥ = ∑ ⎜ ∂ηi ∂η i − ∂η i ∂ηi ⎟, j j j ⎠ k ⎦ k k ⎣ i =1 i =1 ⎝ n n ∂x i ∂yi ⎞ ⎛ ∂x i ∂yi ⎡ ∂ ( x i , yi ) ⎤ [ ξj , ηk ] = ∑ det ⎢ ∂ ( ξ , η ) ⎥ = ∑ ⎜ ∂ξ ∂η − ∂η ∂ξ ⎟, k k ⎦ k j ⎠ j j ⎣ i =1 ⎝ i =1 n

n

[ ξj , ξk ] = ∑ det ⎢ ∂ ( ξ i , ξi ) ⎥ = ∑ ⎜ ∂ξ i j j

(19.1.49')

for j , k = 1, 2,..., n ; the denomination of brackets is given because these quantities are sums of brackets. Lagrange’s brackets are, obviously, anticommutative (skewsymmetric with respect to the indices j and k )

[ ξj , ξk ] = − [ ξk , ξj ], [ η j , ηk ] = − [ ηk , η j ], [ ξj , ηk ] = − [ ηk , ξj ] .

(19.1.49'')

In quantum mechanics it is used to denote Lagrange’s brackets in the form { ξ j , ξk } ,

{ η j , ηk } , { ξj , ηk } . Let be q j = q j (t ; a , b ) , p j = p j (t ; a , b ), j = 1,2,..., s , the integrals of the canonical system (19.2.14), where we have put in evidence the integration constants a and b ; the corresponding Lagrange’s bracket is written in the form (the two constants play the rôle of parameters) ∂q j ∂a [ a ,b ] = ∂q j ∂b

Let us calculate

∂p j ∂q j ∂p j ∂q j ∂p j ∂a = − . ∂p j ∂a ∂b ∂b ∂a ∂b

(19.1.50)


139

∂q j ∂p j ∂q j ∂p j ∂q j ∂p j ∂q j ∂p j d [ a ,b ] = + − − , dt ∂a ∂b ∂a ∂b ∂b ∂a ∂b ∂a

where we assume that the two parameters are independent on t , so that (d / dt )( ∂ / ∂c ) = ( ∂ / ∂c )(d / dt ) , where c can be a or b ; taking into account Hamilton’s equations, we may write ∂q j ∂2 H ∂qk ∂2 H ∂pk = + , ∂c ∂qk ∂p j ∂c ∂pk ∂p j ∂c ∂p j ∂2 H ∂qk ∂2 H ∂pk = − − , ∂c ∂qk ∂q j ∂c ∂pk ∂q j ∂c

where, analogously, we make – successively – c equal to a or b . We get ∂q ∂q j ⎞ ∂p ∂p j ⎞ ∂2 H ⎛ ∂qk ∂q j ∂2 H ⎛ ∂pk ∂p j d [ a ,b ] = − k + − k ⎜ ⎟ dt ∂qk ∂q j ⎝ ∂a ∂b ∂b ∂a ⎠ ∂pk ∂p j ⎜⎝ ∂a ∂b ∂b ∂a ⎟⎠ ∂q ∂p j ⎞ ∂q ∂q j ⎞ ∂2 H ⎛ ∂qk ∂p j ∂2 H ⎛ ∂pk ∂q j . + − k + − k ⎜ ⎟ ⎜ ∂qk ∂p j ⎝ ∂a ∂b ∂b ∂a ⎠ ∂pk ∂q j ⎝ ∂a ∂b ∂b ∂a ⎟⎠

Observing that the mixed derivatives of Hamilton’s function do not depend on the order of differentiation (we assume that H ∈ C 2 ), it results that the first two sums vanish (the derivatives of the function H are symmetric with respect to the indices k and j , while the brackets are skew-symmetric with respect to the same indices); if in the last sums we replace the dummy indices k and j by j and k , respectively, then we remark that the last two sums vanish too. Hence, d[a , b ]/ dt = 0 , so that

[ a , b ] = const ;

(19.1.51)

we can state Theorem 19.1.4 (Lagrange). Lagrange’s bracket corresponding to two integration constants is constant along the trajectory of the representative point P in the phase space Γ 2s . 19.1.2.2 Poisson’s Bracket. Properties Let be the functions ξi = ξi (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ), ηi = ηi (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ), i = 1, 2,..., n ,

of class C 1 . We call Poisson’s brackets (denoted by usual brackets) the sums

(19.1.52)


140 ⎡ ∂( ξ , ξ ) ⎤

⎛ ∂ξ j ∂ξk ∂ξ j ∂ξk ⎞ , − ∂yi ∂yi ∂x i ⎟⎠ i i ⎦ ⎣ i =1 i =1 n n ⎡ ∂ ( η j , ηk ) ⎤ ⎛ ∂η j ∂ηk ∂η j ∂ηk ⎞ , = ∑ det ⎢ = ∑⎜ − ⎥ ∂yi ∂x i ⎟⎠ ⎣ ∂ ( x i , yi ) ⎦ i =1 ⎝ ∂x i ∂yi i =1 n n ⎡ ∂ ( ξ j , ηk ) ⎤ ⎛ ∂ξ j ∂ηk ∂ξ j ∂ηk ⎞ = ∑⎜ − , = ∑ det ⎢ ⎥ ∂yi ∂x i ⎟⎠ ⎣ ∂ (x i , yi ) ⎦ i =1 ⎝ ∂x i ∂yi i =1 n

( ξj , ξk ) = ∑ det ⎢ ∂ (xj , yk ) ⎥ ( η j , ηk ) ( ξj , ηk )

=

n

∑ ⎜⎝ ∂xi

(19.1.52')

for j , k = 1, 2,..., n ; these brackets are anticommutative too (skew-symmetric with respect to the indices j and k )

( ξj , ξk ) = − ( ξk , ξj ) , ( η j , ηk ) = − ( ηk , η j ) , ( ξ j , ηk ) = − ( ηk , ξj ) .

(19.1.52'')

In quantum mechanics, Poisson’s brackets are usually denoted in the form [ ξ j , ξk ] , [ η j , ηk ] , [ ξ j , ηk ] .

Let us put in evidence the connection between Lagrange’s and Poisson ’s brackets in a unitary form; we will consider 2n functions f j = f j (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ) , j = 1, 2,..., 2n , for which ∂ ( f1 , f2 ,..., f2 n ) ⎡ ⎤ det ⎢ ≠ 0; ⎣ ∂ (x1 , x 2 ,..., x n , y1 , y2 ,..., yn ) ⎥⎦

so that, in this case, we can calculate x i = x i ( f1 , f2 ,..., f2 n ) , yi = yi ( f1 , f2 ,..., f2 n ) , i = 1, 2,..., n . Observing that 2n

2 n ∂f ∂y ∂f ∂x k k = δ jk , ∑ i = 0, ∂fi i = 1 ∂x j ∂fi

∑ ∂xij

i =1 2n

2 n ∂f ∂y ∂f ∂x k k = 0, ∑ i = δ jk , ∂fi ∂ y ∂ f j i i =1

∑ ∂yij

i =1

∂fp ∂xl ∂fp ∂yl ⎞ + = δpq , ∂fq ∂yl ∂fq ⎟⎠

∑ ⎛⎜⎝ ∂xl n

l =1

for j , k = 1, 2,..., n and p , q = 1, 2,..., 2n , we obtain, easily, 2n

∑ [ fi , fj ] ( fi , fk ) = δ jk , j , k

i =1

In detail, we can write

= 1, 2,..., 2n .

(19.1.53)


141

2n

∑ {[ ξi , ξj ] ( ξi , ξk ) + [ ηi , ξj ] ( ηi , ξk )} = δ jk

,

∑ {[ ξi , ξj ] ( ξi , ηk ) + [ ηi , ξj ] ( ηi , ηk )} = δ jk

,

i =1 n

i =1 n

∑ {[ ξi , η j ] ( ξi , ξk ) + [ ηi , η j ] ( ηi , ξk )} = δ jk ,

.

(19.1.53')

i =1 2n

∑ {[ ξi , η j ] ( ξi , ηk ) + [ ηi , η j ] ( ηi , ηk )} = δ jk

,

i =1

for j , k = 1, 2,..., n . In case of a Hamiltonian system, let be the functions

ϕ = ϕ (q j , p j ; t ) ,

ψ = ψ (q j , p j ; t ) of class C ; the corresponding Poisson’s bracket is 1

[ ϕ, ψ ] =

∂ϕ ∂q j

∂ψ ∂q j

∂ϕ ∂p j

∂ψ ∂p j

=

∂ϕ ∂ψ ∂ϕ ∂ψ − . ∂q j ∂p j ∂p j ∂q j

(19.1.54)

We mention the obvious properties (ϕ , ϕ ) = 0, (ϕ ,C ) = 0, (C ϕ, ψ ) = C (ϕ , ψ ), C = const, (ϕ , ψ ) = −( ψ , ϕ ), ( − ϕ , ψ ) = −(ϕ , ψ ) ,

(19.1.55)

as well as the properties ( f ,q j ) = −

∂f ∂f , ( f , pj ) = − , j = 1, 2,..., s , ∂p j ∂q j

(q j , qk ) = 0, ( p j , pk ) = 0, (q j , pk ) = −( p j , qk ) = δ jk , j , k = 1, 2,..., s .

(19.1.55') (19.1.55'')

Starting from the relation of definition (19.1.54), it results, as well, (ϕ1 + ϕ2 , ψ ) = (ϕ1 , ψ ) + (ϕ2 , ψ ), (ϕ1ϕ2 , ψ ) = ϕ1 (ϕ2 , ψ ) + ϕ2 (ϕ1 , ψ ),

(19.1.55''')

where we have used the formulae of differentiation of the sum and of the product of two functions. The formula of differentiation of a function of function leads to (ϕ , ψ ) = (ϕ , ψ ) =

m

∂ϕ

∑ (Pj , ψ ) ∂Pj

,

∂ϕ ∂ψ ⎛ ∂ϕ ∂ψ − P Pk ∂Q j ∂ ∂ k j =1 k =1 m

m

∑ ∑ ⎜⎝ ∂Qj

(19.1.56)

j =1

⎞ ⎟(Q j , Pk ) , ⎠

(19.1.56')


142

where ϕ = ϕ (Q j , Pj ) , ψ = ψ (Q j , Pj ) , with Q j = Q j (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ), Pj = Pj (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ), j = 1, 2,..., m .

We can calculate ∂2 ϕ ∂ψ ∂ϕ ∂2 ψ ∂2 ϕ ∂ψ ∂ϕ ∂2 ψ ∂ ( ϕ, ψ ) = + − − ∂t ∂t ∂q j ∂p j ∂q j ∂t ∂p j ∂t ∂p j ∂q j ∂p j ∂t ∂q j ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ϕ ∂ψ ∂ϕ ∂ψ = − + − , ∂q j ∂p j ∂p j ∂q j ∂q j ∂p j ∂p j ∂q j

where we assume the continuity of the derivatives of second order and where the point indicates the differentiation with respect to time; there results ∂ (ϕ , ψ ) = (ϕ , ψ ) + (ϕ, ψ ) . ∂t

(19.1.57)

Introducing the differential operator Dϕ =

∂ϕ ∂ ∂ϕ ∂ , − ∂q j ∂p j ∂p j ∂q j

(19.1.58)

where ϕ ∈ C 1 is a given function, we can express Poisson’s bracket in the form ( ϕ, ψ ) = Dϕ ψ .

(19.1.58')

Let be χ = χ (q j , p j ; t ) ; we assume that the functions ϕ , ψ and χ are of class C 2 . Taking into account the expression of Poisson’s brackets in the form (19.2.58'), we can write ( ϕ, ( ψ , χ ) ) − ( ψ , ( ϕ , χ ) ) = ( Dϕ Dψ − Dψ Dϕ ) χ = Dχ ∂ϕ ∂ψ ∂2 ∂ϕ ∂2 ϕ ∂ ∂ϕ ∂ψ ∂2 ⎡ ∂ϕ ∂2 ψ ∂ = ⎢ + − − ∂q j ∂qk ∂p j ∂pk ∂q j ∂p j ∂pk ∂qk ∂q j ∂pk ∂p j ∂qk ⎣ ∂q j ∂p j ∂qk ∂pk 2 2 2 ∂ϕ ∂ ψ ∂ ∂ϕ ∂ψ ∂ ∂ϕ ∂ ψ ∂ ∂ϕ ∂ψ ∂2 − − + + ∂p j ∂q j ∂qk ∂pk ∂p j ∂qk ∂q j ∂pk ∂p j ∂q j ∂pk ∂qk ∂p j ∂pk ∂q j ∂qk ∂ψ ∂ϕ ∂2 ∂ψ ∂2 ϕ ∂ ∂ψ ∂ ∂2 ⎛ ∂ψ ∂2 ϕ ∂ −⎜ + − − ∂q j ∂qk ∂p j ∂pk ∂q j ∂p j ∂pk ∂qk ∂q j ∂pk ∂p j ∂qk ⎝ ∂q j ∂p j ∂qk ∂pk ∂ψ ∂2 ϕ ∂ ∂ψ ∂ϕ ∂2 ∂ψ ∂2 ϕ ∂ ∂ψ ∂ϕ ∂2 ⎞ ⎤ − − + + χ. ∂p j ∂q j ∂qk ∂pk ∂p j ∂qk ∂q j ∂pk ∂p j ∂q j ∂pk ∂qk ∂p j ∂pk ∂q j ∂qk ⎟⎠ ⎦⎥


143

Observing that the operators of second order are of the form

∂ψ ∂ϕ ⎞ ∂2 ⎛ ∂ϕ ∂ψ ⎜ ∂q ∂q − ∂q ∂q ⎟ ∂p ∂p = 0 j j k ⎠ k ⎝ j k or of the form

−

∂ϕ ∂ψ ∂2 ∂ψ ∂ϕ ∂2 + = 0, ∂q j ∂pk ∂p j ∂qk ∂p j ∂qk ∂q j ∂pk

the commutator D will be also an operator of first order; we remain with ∂χ ⎛ ∂ϕ ∂2 ψ ∂ϕ ∂2 ψ ∂ψ ∂2 ϕ ∂ψ ∂2 ϕ ⎞ + − − ⎜ ∂qk ⎝ ∂q j ∂q j ∂pk ∂q j ∂p j ∂pk ∂q j ∂p j ∂pk ∂p j ∂q j ∂pk ⎟⎠ ∂χ ⎛ ∂ϕ ∂2 ψ ∂ϕ ∂2 ψ ∂ψ ∂2 ϕ ∂ψ ∂2 ϕ ⎞ − + − − ∂pk ⎜⎝ ∂p j ∂q j ∂qk ∂q j ∂p j ∂qk ∂q j ∂p j ∂qk ∂p j ∂q j ∂qk ⎟⎠ = Dχ ( ψ, ϕ ) = − Dχ (ϕ , ψ ) = − ( χ,(ϕ , ψ ) ) .

Finally, we get the Poisson–Jacobi identity ( ϕ , ( ψ , χ ) ) + ( ψ , ( χ , ϕ ) ) + ( χ, ( ϕ , ψ ) ) = 0 .

(19.1.59)

As a matter of fact, in the left member of this identity one observes that each term will have as factor a derivative of second order of one of the functions ϕ , ψ or χ . But all the derivatives of second order of the function χ vanish, as it has been shown; the expression in the above identity is obtained by circular permutations of the first Poisson bracket, so that the derivatives of second order of the functions ϕ and ψ will be equal to zero too, the identity being thus proved. If we take ϕ = q j or ϕ = p j and ψ = H , we obtain (corresponding to the formulae (19.1.55'))

(qj ,H ) =

∂H , ( pj , H ∂p j

)=

−

∂H , j = 1, 2,..., s . ∂q j

(19.1.60)

Hamilton’s equations (19.1.14) take the form q j = ( q j , H ) , p j = ( p j , H

)j

used in the quantic modelling of mechanics.

= 1, 2,..., s ,

(19.1.60')


144

19.1.2.3 First Integrals of the Canonical System Let be f = f (q j , p j ; t ) a first integral of Hamilton’s canonical system, which is constant along the trajectory of the representative point P ∈ Γ 2s ; thus, we will have ∂f ∂f df = q + p + f = 0 . ∂q j j ∂p j j dt

Taking into account the equations (19.1.14), this necessary condition becomes

( f , H ) + f = 0 ,

(19.1.61)

where we use Poisson’s bracket. As well, starting from the partial differential equation of first order ∂f ∂H ∂f ∂H − + f = 0 , ∂q j ∂p j ∂p j ∂q j

we can write the associate characteristic system (the Lagrange–Charpit sequence) in the form (19.1.19), hence in the form of Hamilton’s equations. Hence, we can state Theorem 19.1.5 (Poisson). The function f = f (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ∈ C 1 is a first integral of the canonical system (19.1.14) only and only if the relation (19.1.61) (which is thus a necessary and sufficient condition) holds. If f = 0 , hence if the function f does not depend explicitly on time ( f = f (q j , p j ) ), then the condition (19.1.61) becomes (f ,H ) = 0 .

(19.1.61')

In the particular case f = H we find again the first integral of the generalized mechanical energy. Performing the partial differentiation of the relation (19.1.61) with respect to time and taking into account (19.1.57), we can write ( f , H ) + ( f , H ) + f = 0 ; if H = 0 , corresponding to a generalized conservative mechanical system; then we remain with the relation

( f , H ) +

∂f =0 ∂t

and we can state Theorem 19.1.6 (Poisson). If f = f (q j , p j ; t ) is a first integral for Hamilton’s equations of a generalized conservative mechanical system, then f , f,... are, as well, first integrals for this system. If we make χ = H in the Poisson–Jacobi identity, that one takes the form (ϕ ,( ψ, H )) + ( ψ,( H , ϕ )) + (H ,(ϕ , ψ )) = 0 ; assuming that


145

(ϕ , H ) + ϕ = 0, ( ψ, H ) + ψ = 0

(19.1.62)

and taking into account the properties (19.1.55), (19.1.57), we can write ((ϕ , ψ ), H ) +

∂ (ϕ , ψ ) = 0 . ∂t

(19.1.62')

We state thus Theorem 19.1.7 (Jacobi–Poisson). If we suppose that the functions ϕ = ϕ (q j , p j ; t ), ψ = ψ (q j , p j ; t ) ∈ C 2 are first integrals of the canonical system (19.1.14), then Poisson’s bracket (ϕ , ψ ) is a first integral of the system. Starting from the first integrals H , ϕ , ψ , one can build up new first integrals of the form ϕ , ϕ,..., ψ , ψ,...,( H , ϕ ),( H , ψ ),(ϕ,(ϕ, ψ )),(ϕ ,( H , ψ )) etc., by means of the last two theorems. Thus, one can obtain, at a given moment, a constant or a first integral which is not independent on the previous first integrals; indeed, the canonical system admits at the most 2s distinct first integrals. If, by setting up a Poisson bracket of the first integrals ϕ and ψ , we get (ϕ , ψ ) = C , C = const

(19.1.63)

(ϕ , ψ ) = C (ϕ, ψ ), C = const ,

(19.1.63')

or

which is – practically – the same thing, then we say that the two first integrals are canonically conjugate. Moreover, if, in particular, (ϕ , ψ ) = 0 ,

then we say that the first integrals More general, let us consider α = 1, 2,...,r ; after S. Lie, the ( fα , fβ ) = fαβ ( f1 , f2 ,..., fr ), α, β

(19.1.64)

ϕ and ψ are in involution.

the functions fα = fα (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) , set of these functions forms a functional group if = 1, 2,..., r , hence if Poisson’s bracket of two

functions of this set is a function of the functions which belong to the set. These notions have been used by S. Lie in the study of Hamilton’s system of equations. 19.1.2.4 Theorem of Lie. Theorem of Liouville Let us suppose that r first integrals f j = f j (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) of the canonical system are known, so that

( f j , H ) + fj = 0, j = 1, 2,..., r . Assuming that these first integrals are in involution, we can write

(19.1.65)


146

( f j , fk ) ≡ 0, j , k = 1, 2,..., r .

(19.1.65')

Thus, the system of functions { f j , j = 1, 2,..., r } will be a system in involution. Let us introduce the conjugate variables p0 and q 0 = t , so that ∂H / ∂p0 = 0 , ∂f j / ∂p0 = 0, j = 1, 2,..., r . Let be, in general, the functions ϕ and ψ of variables q 0 , q1 , q2 ,..., qs , p0 , p1 , p2 ,..., ps . We can set up Poisson’s bracket ∂ϕ ∂ψ ∂ϕ ∂ψ − + (ϕ , ψ ) , ∂t ∂p0 ∂p0 ∂t

(ϕ , ψ )0 =

(19.1.66)

where we put the index zero so to specify the introduction of the variables q 0 and p0 ; if ∂ϕ / ∂p0 = ∂ψ / ∂p0 = 0 , then we have (ϕ , ψ )0 = (ϕ, ψ ) . Denoting f0 = H + p0 and observing that fj + ( f j , H ) = ( f j , f0 )0 = 0 , we can express the relations (19.1.65), (19.1.65') in the unitary form ( f j , fk )0 = 0, j , k = 1, 2,..., r .

(19.1.67)

We assume that the equations f j = c j , c j = const, j = 1, 2,..., r , and H + p0 = 0 lead to pk = ϕk (q1 , q2 ,..., qs , pr +1 , pr + 2 ,..., ps , c1 , c2 ,..., c2 s ; t ), p0 = H

= − H = ϕ0 (q1 , q2 ,..., qs , pr + 1 , pr + 2 ,..., ps , c1 , c2 ,..., c2 s ; t ) ,

pk = ϕk

(19.1.68)

for k = 1, 2,..., r . If u is one of the variables q 0 , q1 , q2 ,..., qs , p0 , pr +1 , pr + 2 ,..., ps , then we can write ∂f j + ∂u

r

∂f j ∂ϕα = 0, j = 1, 2,..., r , ∂u

∑ ∂pα

α =0

or ∂f j = ∂u

r

∂f j ∂ ( pα − ϕα ) , j = 1, 2,..., r . ∂u

∑ ∂pα

α =0

We notice that these equations become identities if u is one of the variable p0 , p1 , p2 ,..., pr . We can thus write ∂f j ∂fk = ∂qi ∂pi

r

r

∂f j ∂f ∂ ( pα − ϕα ) ∂ ( pβ − ϕβ ) , i = 0,1, 2,..., s . ∂qi ∂pi

∑ ∑ ∂pα ∂pkβ

α =0 β =0

Taking the double of the skew-symmetric part with respect to the indices j and k , it results

( fj , fk )0

=

r

r

∂f j ∂f

∑ ∑ ∂pα ∂pkβ ( pα

α =0 β =0

− ϕα , pβ − ϕβ )0 , j , k = 0,1, 2,..., m .

(19.1.69)


147

The conditions (19.1.67) read r

∂f j

∑ ∂pα z α( k )

∂f

r

∑ ∂pkβ ( pα

= 0, z α( k ) =

α =0

β =0

− ϕα , pβ − ϕβ ) ,

α, j , k = 0,1, 2,..., r .

(19.1.67')

Fixing k , one obtains a system of r + 1 linear algebraic equations with det[ ∂f j / ∂pα ] ≠ 0 , by the previous hypothesis; we get z ( k ) = 0 , hence also a system of r + 1 linear algebraic equations with the same non-zero determinant of the coefficients. Finally, ( pα − ϕα , pβ − ϕβ ) = 0 or ( p j − ϕ j , pk − ϕk )0 = 0, j , k = 0,1, 2,..., r .

(19.1.70)

In an expanded form, we can write ∂ϕ j ∂ϕk ⎞ ⎛ ∂ϕ j ∂ϕk ⎜ ∂q ∂p − ∂p ∂p ⎟ σ σ σ σ ⎠ σ =r +1 ⎝ ∂ϕ j ∂ϕ = − k , j , k = 0,1, 2,..., r . ∂qk ∂q j

( ϕj , ϕk )sσ =r +1

=

s

∑

(19.1.70')

These conditions show that the functions f1 , f2 ,..., fr are first integrals of the canonical system, being two by two in involution. Replacing p j by ϕ j , j = 1, 2,..., r , −H by ϕ0 and dt by dq 0 in the Pfaff form of Hamilton (19.1.45), we obtain the form ω =

r

∑ ϕ j dq j

+

j =0

s

∑

pσ dq σ ,

(19.1.71)

σ =r +1

the bilinear covariant of which is given by ∂ϕ j ⎞ δq ρ ⎟ dq j ∂ p ρ j =0 ⎝ k =0 ρ =r +1 ⎠ s s r s s ∂ϕ j ∂ϕ j ⎛ ⎞ ∂pσ + ∑ ∑ δpρ dq σ − ∑ ⎜ ∑ dqk + ∑ dpρ ⎟ δq j ∂ ∂ ∂ p q p ρ ρ k j =0 ⎝ k =0 σ =r +1 ρ =r +1 ρ =r +1 ⎠ ω ′ = δωd − dωδ =

r

⎛

s

∂ϕ j

∑ ⎜ ∑ ∂qk

δqk +

s

∑

⎡ ⎛ r ∂ϕ j ∂ϕ ⎞ ∑ ⎢ ⎜ ∑ ∂qk − ∂q kj ⎟ dq j k =0 ⎣ ⎝ j =0 ⎠ s s s r ∂ϕ s ⎛ ∂ϕk ∂ϕk ∂pσ j ⎤ ⎞ dpρ ⎥ δqk + ∑ ⎜ ∑ dq j − ∑ dpρ ⎟ δq σ − ∑ dq ρ − ∑ ∂ ∂ ∂ p q p ∂ q ρ σ ρ ρ ⎦ ⎠ ρ =r +1 ρ =r +1 σ =r +1 ⎝ j = 0 ρ =r +1 −

∂pσ ∑ ∑ ∂pρ dpρ δqσ = σ =r +1 ρ =r +1 s

s

+

s

⎛

r

∂ϕ j

∑ ⎜ ∑ ∂pρ dq j

ρ =r +1 ⎝ j = 0

+

r

∂pσ ⎞ dq σ ⎟ δpρ . ∂ p ρ ⎠ σ =r +1 s

∑


148

The associate differential system, obtained by equating to zero the brackets which multiply the differential variations δq σ and δpσ , is written in the form r ∂ϕ ∂ϕ j j dq j , dpσ = − ∑ dq j , σ = r + 1, r + 2,..., s . ∂ ∂ p q σ σ j =0 j =0 r

dq σ = − ∑

(19.1.72)

The equations obtained by annulling the square brackets which multiply the differential variations δqk are identically verified if one takes into account the conditions (19.1.70') and the equations (19.1.72). The functions ϕ j , j = 0,1, 2,..., r , being thus determined, the system (19.1.72) becomes a system of 2(s − r ) equations with 2(s − r ) unknown functions. Corresponding to Morera’s method of integration, we introduce an arbitrary parameter λ , so that q j = q j (λ ), j = 0,1, 2,..., r . Denoting r

K (qr + 1 , qr + 2 ,..., qs , pr + 1 , pr + 2 ,..., ps , c1 , c2 ,..., c2 s ; λ ) = − ∑ ϕ j j =0

dq j , dλ

(19.1.73)

the system of equation (19.1.71) reads q σ′ =

dq σ ∂K dpσ ∂K = , pσ′ = = − , σ = r + 1, r + 2,..., s . dλ ∂pσ dλ ∂q σ

(19.1.72')

As a matter of fact, starting from the Pfaff form ω =

s

∑

σ =r +1

pσ dq σ − Kdλ ,

(19.1.71')

one obtains the same result. We can thus state Theorem 19.1.8 (S. Lie). If the system of 2s canonical equations (19.1.14) admits r , r ≤ s , first integrals in involution, resolvable with respect to r , of the generalized momenta, then the integration of this system is reduced to the integration of a system of 2(s − r ) canonical equations in s − r pairs of conjugate variables and to quadratures. The generalized co-ordinates q1 , q2 ,..., qr are then determined, using the relation ϕ k = −

∂H , k = 1, 2,..., r . ∂qk

(19.1.74)

In the particular case r = s , we notice that ∂ϕ j / ∂pk = 0, j , k = 0,1, 2,..., s , and from (19.1.70') it results ∂ϕ j ∂ϕk = , j , k = 0,1, 2,..., s . ∂qk ∂q j

(19.1.75)


149

Thus, we can state that the existence (necessary and sufficient conditions) of a function S = S (q 0 , q1 , q2 ,..., qs ; a1 , a2 ,..., as ) is ensured, so that ∂S ∂S = ϕj , = S = ϕ0 = − H ∂q j ∂q 0

p j = ϕj

, j = 1, 2,..., s .

(19.1.75')

Hence, dS =

s

∂S

∑ ∂q j

j =0

dq j =

s

∑ ϕ j dq j ,

j =0

(19.1.75'')

and we can state Theorem 19.1.9 (Liouville). If the system of 2s canonical equations (19.1.14) admits s first integrals in involution, resolvable with respect to the generalized momenta, then the integration of this system is reduced to quadratures. We notice that the function S satisfies the partial differential equation of first order ∂S ∂S ∂S ⎞ , ,..., S + H ⎛⎜ q 0 , q1 ,..., qs , = 0, ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.1.76)

called the Hamilton–Jacobi equation, and the condition

⎡ ∂2S ⎤ ⎡ ∂ϕ j ⎤ det ⎢ = det ⎢ ⎥ ⎥ ≠ 0, ⎣ ∂ak ⎦ ⎣ ∂q j ∂ak ⎦ assuming that the function ϕ j

(19.1.76')

are resolvable with respect to the constant

ak , j , k = 1, 2,..., s ; in this case, S is the complete integral of the Hamilton–Jacobi

equation (see Sect. 19.2.1.1 too). If H = 0, fj = 0, j = 1,2,..., s , hence if the Hamiltonian and the first integrals do not contain explicitly the time, then the functions ϕ j have the same property, so that S (q1 , q2 ,..., qs , a1 , a2 ,..., as ; t ) = S (q1 , q2 ,..., qs , a1 , a2 ,..., as ) + T (t ; a1 , a2 ,..., as ) .

From (19.1.75'), it results T = ϕ0 ; but (19.1.75) shows that ϕ0 = − h , h = const , obtaining S = − ht + S . Let be a generalized conservative mechanical system ( H = 0 ), the canonical system (19.1.14) of which admits s − 1 independent first integrals f j , j = 1, 2,..., s − 1 , in involution, which do not contain explicitly the time ( fj = 0 ). The s th first integral is H and the other first integrals must verify the conditions ( f j , H ) = 0, j = 1, 2,..., s − 1 ; hence, H is also in involution with these first integrals and we can apply Liouville’s theorem.


150

If, in case of the motion of a heavy rigid solid about a fixed point, beside the first integral of the moment of momentum and the first integral of the kinetic energy we can set up a third integral, in involution with the first two ones, then the integration of the equations of motion is reduced to quadratures, according to Liouville’s theorem. 19.1.2.5 Other Formulations in the Spaces Λs and Γ 2s We have built up the Lagrangian formalism in the space Λs to study the motion of the representative point P , while in the space Γ 2s we have used the Hamiltonian formalism to determine the motion of the corresponding representative point; but these formalisms are not the only ones which can be used. For instance, starting from the relation (19.2.15'), we notice that one can write d δ ⎡ ( pj q j ⎣⎢ dt

)−L

⎤ = q δp + q δp j j j j ⎦⎥

too. Denoting ϕ =

d (p q dt j j

)−L

(19.1.77)

,

we obtain qj =

∂ϕ ∂ϕ , q = , j = 1, 2,..., s , ∂p j j ∂p j

(19.1.77')

hence, relations of the form (19.1.13); one obtain thus, easily, a system of differential equations of second order

d ⎛ ∂ϕ ⎞ ∂ϕ − = 0, j = 1,2,..., s , dt ⎜⎝ ∂p j ⎟⎠ ∂p j

(19.1.77'')

analogue to Lagrange’s equations (the place of the generalized co-ordinates is taken by the generalized momenta). But, to can set up the function ϕ , we must solve the system of equations (19.1.13) with respect to q j and q j , function of pk and pk , which is difficult – in general – this system being non-linear in the generalized co-ordinates. We can start also from the relation δ ( q j p j − L

) = q j δp j

− p j δq j .

We denote

ψ = q j p j − L

(19.1.78)

and obtain the system of equations qj =

∂ψj ∂ψj , pj = − . ∂p j ∂q j

(19.1.78')


151

In this case too, the difficulty consists in expressing the generalized co-ordinates as functions of the derivatives with respect to time of the generalized momenta, using the second subsystem (19.1.13).

19.1.3 Applications We consider, firstly, the motion of a single particle; we pass then to a Hamiltonian treatment of a system of two or more particles, as well as to the case of a rigid solid or of a system of rigid solids. 19.1.3.1 Motion of a Particle In general, the generalized momentum is defined by the relation (18.2.80). In case of a generalized quasi-potential intervenes the vector part in the form (18.2.81); to simplify the presentation, we assume, in what follows, that the generalized forces derive from a simple quasi-potential, the generalized momenta being thus given by the relations (18.2.80'). We use the results obtained in Sect. 18.3.2.1. In case of a single particle P (x1 , x 2 , x 3 ) , the representative point in Γ 6 is expressed by P (q1 , q2 , q 3 , p1 , p2 , p3 ), q j = x j , j = 1, 2, 3 , so that the kinetic energy is given by

T = (m / 2)( x12 + x22 + x 32 ) , while the generalized momenta (equal to the usual components of the momentum) are p j = mx j , j = 1, 2, 3 .

(19.1.79)

Hamilton’s function becomes (we use the formula of definition (19.1.11) with (18.2.3.4)) 1 p p − U (x1 , x 2 , x 3 ; t ) , 2m j j

H =

(19.1.79')

while the canonical equations (19.1.14) read x j =

1 ∂U p , p = , j = 1, 2, 3 . m j j ∂x j

(19.1.79'')

In cylindrical co-ordinates ( q1 = r , q2 = θ , q 3 = z ), we obtain

pr = mr , pθ = mr 2 θ , pz = mz .

(19.1.80)

Hamilton’s function being H =

(

)

1 1 pr2 + 2 pθ2 + pz2 − U (r , θ , z ; t ) , 2m r

There result the canonical equations

(19.1.80')


152

1 1 1 pr , θ = pθ , z = pz , m m mr 2 ∂U ∂U 1 2 ∂U pr = p + , p θ = , p z = . 3 θ ∂ r ∂ θ ∂z mr r =

(19.1.80'')

If ∂U / ∂θ = ∂U / ∂z = 0 , then θ and z are cyclic co-ordinates. As well, in spherical co-ordinates ( q1 = r , q2 = θ , q 3 = ϕ ) we can write

pr = mr , pθ = mr 2 θ, pϕ = mr 2 sin2 θϕ ,

(19.1.81)

so that H =

(

)

1 1 1 pr2 + 2 pθ2 + 2 pϕ2 − U (r , θ , ϕ ; t ) . 2m r r sin2 θ

(19.1.81')

We get Hamilton’s equations 1 1 1 pr , θ = p , ϕ = pϕ , 2 θ 2 m mr mr sin2 θ ∂U 1 1 pr = pθ2 + pϕ2 + , ∂r mr 3 sin2 θ ∂U ∂U cot θ p θ = pϕ2 + , pϕ = . 2 2 ∂ϕ ∂ϕ mr sin θ

r =

(

)

(19.1.81'')

If ∂U / ∂ϕ = 0 , then ϕ is a cyclic co-ordinate. We notice that, in case of a simple potential of the form (18.3.46), we find a system of canonical equations equivalent to Lagrange’s equations (18.3.47). In case of a central force of Newtonian attraction we have U = k / r , k > 0 , so that ∂U / ∂r = − k / r 2 , ∂U / ∂θ = ∂U / ∂ϕ = 0 .

As a matter of fact, in case of a central force the trajectory is a plane curve; if the intensity of this force is function only on the distance to the fixed pole, Hamilton’s function will be given by (in polar co-ordinates, in the plane of the motion, the fixed pole being the origin of the co-ordinate axes) H =

(

)

m 2 1 1 ( r + r 2 θ2 ) − U (r ) = 2m pr2 + 2 pθ2 − U (r ) . 2 r

(19.1.82)

There result Hamilton’s equations in Γ 4 r =

1 1 1 2 ∂U pr , θ = p , pr = pθ + , p θ = 0 , 2 θ m ∂r mr mr 3

(19.1.82')

θ being a cyclic co-ordinate; we obtain pθ = mC , C = const , and then r 2 θ = c ,

hence the first integral of areas. Eliminating the generalized momentum pr and taking into account the first integral previously obtain, we can write


153 r =

C2 1 ∂U + . 3 m ∂r r

(19.1.82'')

We find thus the equation of motion of the particle along the vector radius (see Sect. 8.1.1.1, formula (8.1.3)); the first integral H = h corresponds to the first integral (8.1.6'). In elliptical co-ordinates (see Sect. 18.3.2.1), the formula (18.3.41) leads to p1 =

m m m A q , p = A2q2 , p3 = A3q3 ; 4 1 1 2 4 4

Hamilton’s function will be given by H =

2 ⎛ p12 p22 p32 ⎞ + + −U , m ⎜⎝ A1 A2 A3 ⎟⎠

(19.1.83)

with U = U (q1 , q2 , q 3 ; t ) , allowing thus to write the canonical equations in the form q1 = p j =

4 p1 4 p2 4 p3 , q = , q = , m A1 2 m A2 3 m A3

p32 ∂A3 ⎞ 2 ⎛ p12 ∂A1 p22 ∂A2 + + ⎟ , j = 1, 2, 3 . m ⎜⎝ A12 ∂q j A22 ∂q j A32 ∂q j ⎠

(19.1.83')

The generalized momenta p j , given by (19.1.79), are the components of the momentum H = p ; the components of the moment of momentum KO = r × p will be KOi = ∈ijk x j pk = m ∈ijk x j xk , j = 1, 2, 3 .

(19.1.84)

We can thus form Poisson’s brackets ∂p j ∂pk ∂p j ∂pk , − ∂x i ∂pi ∂pi ∂x i ∂ (∈jlm xl pm ) ∂ (∈knr x n pr ) ∂ (∈jlm xl pr ) ∂ (∈knr x n pm ) (KOj , KOk ) = − ∂x i ∂pi ∂pi ∂x i ( p j , pk ) =

=∈jlm ∈kni x n pm − ∈jli ∈kir xl pr = ( ∈rji ∈kli + ∈jli ∈kri ) xl pr = ( δrk δ jl − δ jr δlk ) xl pr =∈ijk ∈ilr xl pr , ∂p j ∂ (∈klm xl pm ) ∂p j ∂ (∈klm xl pm ) − ∂x i ∂pi ∂pi ∂x i = −δij δil ∈klm pm = − ∈kjm pm ,

( p j , KOk ) =


154 so that

( p j , pk ) = 0,(KOi , KOk ) =∈ijk KOi , ( p j , KOk ) =∈ijk pi , j , k = 1, 2, 3 .

(19.1.85)

If some of the above quantities are conserved (are first integrals), then also the other quantities have the same property (the Jacobi–Poisson theorem leads to new first integrals). For instance, if p1 , KO 2 and KO 3 are first integrals, then p2 , p3 and KO 1 are first integrals too; other first integrals cannot be obtained, the cycle being closed. If a particle P is acted upon by an elastic central force of intensity in direct proportion with the distance to the pole O , then the potential U is given by U = −(k / 2)(x12 + x 22 + x 32 ), k = const ; the second group of canonical equations (19.1.79'') reads p j = − kx j , j = 1, 2, 3 .

(19.1.79''')

As it can be easily verified, (19.1.84) are three first integrals of the canonical equations. The first equation of the first subsystem of equations (19.2.79'') and the first equation (19.1.79''') (for j = 1 ) give the first integral f1 = mx12 +

1 2 p . k 1

(19.1.86)

As well, Poisson’s bracket (KO 3 , f1 ) leads to the first integral f2 =

1 p p + mx1x 2 . k 1 2

(19.1.86')

One observes easily that one cannot obtain new first integrals starting from the first integrals (19.1.84), (19.1.86), and (19.1.86') and using the Jacobi–Poisson theorem; indeed, if one could obtain such a first integral, then the functions q j , p j , j = 1, 2, 3 , could be expressed by means of the six integration constants, the generalized momenta being thus constant (which is not possible, because the motion would be annulled). A first integral which (unlike the previous ones) contains explicitly the time is f3 = m ωx1 cos ωt − p1 sin ωt , ω =

k . m

A partial differentiation with respect to time leads to a new first integral f4 = f3 = − m ω 2 x1 sin ωt − p1 ω cos ωt ,

which is not distinct from the previous ones, because f 2 f32 + ⎛⎜ 4 ⎞⎟ = kf1 . ⎝ω⎠

(19.1.86'')


155

As a matter of fact, the maximum number of first integrals in case of a free particle is six. 19.1.3.2 Problem of Two Particles Be means of the results in Sect. 18.3.2.3 and of the formulae (18.3.51), (18.3.51') for the kinetic energy T and for the simple potential U , respectively, we notice that we can use a formula of the form (19.1.81'); there results Hamilton’s function H =

(

)

fm1m2 1 1 1 pr2 + 2 pθ2 + 2 pϕ2 − . r 2m r r sin2 θ

(19.1.87)

Hence, the canonical equations read 1 1 1 pr , θ = pθ , ϕ = pϕ , m mr 2 mr 2 sin θ fm1m2 1 cot θ , p θ = pθ2 + pϕ2 − pϕ2 , pϕ = 0 . 2 r sin θ mr 2 sin θ r =

pr =

1 mr 3

(

)

(19.1.87')

We see that ϕ is a cyclic co-ordinate, so that pϕ = const is a first integral. We remain only with four canonical equations, determining thus the functions r = r (t ), θ = θ (t ), pr = pr (t ), pθ = pθ (t ) ; we obtain then the generalized angular velocity ϕ and, by a quadrature, we can calculate ϕ = ϕ (t ) too. 19.1.3.3 Motion of an Electrized Particle in an Electromagnetic Field Starting from the results obtained in Sect. 18.3.2.5, we find the generalized momenta pi = mvi + qAi , i = 1, 2, 3 ; the relation of definition (19.1.11) leads to H =

1 1 1 pi ( pi − qAi ) − ( pi − qAi ) ( pi − qAi ) − qAi ( pi − qAi ) + qA0 , 2m m m

where we took account (18.3.59). Finally, we can write Hamilton’s function in the form H =

1 1 ( p − qAi ) ( pi − qAi ) + qA0 = ( p p − 2qpi Ai + q 2 Ai Ai 2m i 2m i i

) + qA0

.

(19.1.88) One obtains thus the canonical equations νi = xi =

1 q ( p − qAi ) , pi = ( p j − qAj ) Aj ,i + qA0 , i = 1, 2, 3 . m i m

(19.1.88')

19.1.3.4 Motion of a Discrete System of Particles In case of a free discrete system of particles Pi of masses mi , i = 1, 2,..., n , the space of configurations is just E 3 n (we have s = 3n ). The kinetic energy is given by (we use the co-ordinates x (j i ) , j = 1, 2, 3, i = 1, 2,..., n )


156 T =

(

1 n (i ) mi ⎡⎢ x1 ⎣ 2 i∑ =1

) + ( x2( i ) ) + ( x3( i ) ) 2

2

2

⎤. ⎥⎦

(19.1.89)

Assuming, for the sake of simplicity, that the given forces derive from a simple quasi-potential

(i ) U = U (x j ; t ) , we

obtain

Hamilton’s

function H = T − U .

Noting that p(j i ) = mi x (j i ) , j = 1, 2, 3, i = 1, 2,..., n , it results

H =

(

1 n 1 ⎡ (i ) ⎢ p1 2 i∑ = 1 mi ⎣

) + ( p2( i ) ) + ( p3( i ) ) 2

2

2

⎤. ⎥⎦

(19.1.89')

The canonical equations are written in the form (i )

x j

=

1 (i ) (i ) ∂U p , p j = ( i ) , j = 1, 2, 3, i = 1, 2,..., n . mi j ∂x j

(19.1.90)

Introducing the components of the momentum of the mechanical system Hl =

n

∑ pl( i ) , l

i =1

= 1, 2, 3 ,

(19.1.91)

and the components of the moment of momentum of the same system KOl =

n

∑ ∈jkl

i =1

x j pk , l = 1, 2, 3 , (i ) (i )

(19.1.91')

where we use the summation convention of the dummy indices for j , k = 1, 2, 3 , we can calculate Poisson’s brackets as in Sect. 19.1.3.1; we obtain thus the relations

( H j , H k ) = 0, ( KOj , KOk ) =∈jkl KOl , ( H j , KOk ) =∈jkl Hl , j , k = 1, 2, 3 ,

(19.1.92)

analogues to the relations (19.1.85). Obviously, we are led to analogous conclusions in what concerns eventual first integrals too. In particular, one can obtain all six first integrals previously considered in the case in which the mechanical system is isolated, hence it is subjected only to the action of internal forces Fij , i ≠ j , i , j = 1, 2,..., n ; the torsor of these forces is, obviously, equal to zero R = MO =

n

n

∑ ∑ Fij

i =1 j =1 n

n

∑ ∑ rij

i = 1 j =1

= 0,

(19.1.93)

× Fij = 0 .

(19.1.93')


157

Mayer considered, in 1878, internal forces with a more general character that the Newtonian ones (definite in Sects. 1.1.1.11; see. 2.2.2.8 too), hence forces which verify only the global conditions (19.1.93), (19.1.93'), hence the conditions of equilibrium of a non-deformable mechanical system (as the mechanical system would be a rigid one at a given moment). In this context, we can state (we denote rij = rj − ri , i , j = 1, 2,..., n ) Theorem 19.1.10 (Mayer). The most general quasi-conservative internal forces which derive from a generalized quasi-potential U and verify the condition (19.1.93) are those for which

U = U ( r12 , r13 ,..., rn −1,n , r12 , r13 ,..., rn −1,n ; t ) .

(19.1.94)

Theorem 19.1.10' (Mayer). The most general quasi-conservative internal forces which derive from a generalized quasi-potential U and verify the condition (19.1.93) are those for which U = U ( r1 , r2 ,..., rn , r1 , r2 ,..., rn , r12 , r13 ,..., rn −1,n ; t ) .

(19.1.94')

From both theorems taken together, we get Theorem 19.1.11 (Mayer). The most general quasi-conservative internal forces which derive from a generalized quasi-potential U and the torsor of which vanishes (verifies the conditions (19.1.93), (19.1.93')) are those for which the quasi-potential is of the form (19.1.94). The motion of the mass centre is rectilinear and uniform and the theorem of areas can be applied in projection on the three planes of co-ordinates. We can also state Theorem 19.1.12 (Mayer). The necessary and sufficient condition of existence of the first integral of the mechanical energy, hence of the existence of a function of coordinates, velocities and time the total derivative with respect to time of which be n

n

∑ ∑ Fij

i = 1 j =1

⋅ vi

is U = 0 ; the internal forces must derive from a generalized potential which does not contain explicitly the time. The first integral is of the form T −U +

n

∂U

i =1

j

∑ v(j i ) ∂v( i )

= h , h = const .

(19.1.95)

If the potential U verifies the conditions of the Theorem 19.1.10 (is of the form (19.1.94) with U = 0 ), then the first integral reads T −U +

n

n

∂U

∑ ∑ rij ∂rij

i =1 j =1

= h , h = const .

(19.1.95')


158

19.1.3.5 Motion of the Rigid Solid In case of a rigid solid S , free or subjected to constraints, we can start from the considerations made in Sect. 18.3.2.6. For instance, in case of the motion of a heavy homogeneous rigid solid of rotation, of mass M , which slides without friction on a fixed horizontal plane, we start from the kinetic energy (18.3.62) and from the potential energy (18.3.62'); the mechanical system being scleronomic and observing that p1 = M ρ1′ , p2 = M ρ 2′ , pϕ = I 3 (ϕ + ψ cos θ ), pψ = J ψ sin θ + I 3 (ϕ + ψ cos θ ) cos θ , pθ = ⎡⎣ Mf ′2 ( θ ) + J ⎤⎦ θ ,

we can express Hamilton’s function in the form

( pψ − pϕ cos θ ) pϕ2 ⎤ pθ2 1 ⎡ p12 + p22 + + + + Mgf ( θ ) . ⎢ 2 2⎣ M I 3 ⎦⎥ Mf ′ ( θ ) + J J sin2 θ (19.1.96) 2

H = T ′ −U =

Hamilton’s equations read ρ1′ =

pϕ ( pψ − pϕ cos θ ) cos θ p1 p , ρ2′ = 2 , ϕ = − , M M I3 J sin2 θ ( pψ − pϕ cos θ ) pθ ψ = ,θ = , 2 2 J sin θ Mf ′ ( θ ) + J p1 = 0, p2 = 0, pϕ = 0, p ψ = 0, p θ =

+

Mf ′( θ ) f ′′( θ ) ′2

2

⎡⎣ Mf ( θ ) + J ⎤⎦

(19.1.96')

p02

1 ( pψ − pϕ cos θ )( pψ cos θ − pϕ ) − Mgf ′( θ ) . J sin 3 θ

We notice that ξ1′ , ξ2′ , ϕ and ψ are cyclic co-ordinates, four of the generalized momenta being thus constant; as well, H = h is the fifth first integral of the canonical system. In case of the plane motion of a rigid straight bar, we consider the formulation of the problem in Sect. 18.3.2.7. The kinetic energy is given by (18.3.64) and the potential of the forces is expressed by (18.3.64'), so that we can define the generalized momenta in the form p1 = M ρ1′ , p2 = M ρ2′ , pθ = I θ , obtaining Hamilton’s function H = T ′ −U =

1⎡ 1 1 p12 + p22 ) + pθ2 + k ( M ρ2′2 + I sin2 θ ) ⎤ . ( ⎢ ⎥⎦ 2 ⎣M I

We can write the canonical equation in the form

(19.1.97)


159

1 1 1 p , ρ ′ = p , θ = pθ , M 1 2 M 2 I p1 = 0, p 2 = − kM ρ2′ , p θ = − kI sin θ cos θ . ρ1′ =

(19.1.97')

We notice that ρ1′ is a cyclic co-ordinate, the corresponding generalized momentum being constant; the constraints are scleronomic, so that Hamilton’s function will be a first integral of this system of equations too.

19.1.3.6 Double Pendulum The problem of the double pendulum has been treated in Sect. 18.3.2.8. In the frame of Lagrangian mechanics. The generalized co-ordinates being θ1 and θ2 , the corresponding momenta are

p1 = (I 1 + M 2 I 12 )θ1 + M 2 H 2 θ2 cos( θ2 − θ1 ), p2 = (I 2 + M 2 I 22 )θ2 + M 2 H 2 θ1 cos( θ2 − θ1 ) .

(19.1.98)

Starting from these results, we can express Hamilton’s function H = T ′ − U as a function on the canonical co-ordinates, obtaining then Hamilton’s equations.

19.1.3.7 Sympathetic Pendulums Let us take again the case of the sympathetic pendulums, considered in Sect. 18.3.2.9 in the frame of Lagrangian mechanics, Let be two identical physical pendulums, each one of mass M , of mass centres C 1 and C 2 , respectively, so that O1C 1 = O2C 2 = l , where O1 and O2 , are the poles through which pass the respective axes of suspension; if Q1 and Q2 are on O1C 1 and O2C 2 , respectively, with O1Q1 = O2Q2 = a , then we consider that the two pendulum are linked by the elastic spring Q1Q2 of elastic constant k (see Fig. 18.13 too). Choosing as generalized co-ordinates the angles θ1 and θ2 made by O1C 1 and O2C 2 , respectively, with the vertical line, the formulae (18.3.69) and (18.3.69') lead to T =

1 1 I ( θ12 + θ22 ) , U = 2 MgI − ( MgI + ka 2 2 2

)( θ12

+ θ22 ) + ka 2 θ1 θ2 .

(19.1.99)

Having to do with a scleronomic system, the generalized momenta will be p1 = I θ1 , p2 = I θ2 , and Hamilton’s function reads H =

1 1 ( p 2 + p22 ) − 2MgI + 2 ( MgI + ka 2 2I 1

There result Hamilton’s canonical equations

)( θ12

+ θ22 ) − ka 2 θ1 θ2 .

(19.1.99')


160

1 1 θ1 = p1 , θ2 = p2 , I I p1 = − MgI θ1 + ka 2 ( θ2 − θ1 ), p2 = − MgI θ2 + ka 2 ( θ1 − θ2 ) .

(19.1.99'')

Eliminating the generalized momenta, we obtain Lagrange’s equations 2 θ1 + ( ω2 + ω02 ) θ1 − ( ω2 + ω02 ) θ2 = 0, 2 θ2 − ( ω2 + ω02 ) θ1 + ( ω2 + ω02 ) θ2 = 0.

(19.1.100)

with the notation (18.3.71); we find thus again the solutions (18.3.72).

19.2 The Hamilton–Jacobi method After some results with a general character, including the Hamilton-Jacobi theorem, one considers the problem of the systems of equations with separate variables, putting in evidence Stäckel’s studies; some applications for the motion of a particle or of a system of two particles are then given.

19.2.1 General Results To can obtain first integrals of the system of 2s canonical equations, one establishes the Hamilton–Jacobi theorem, which allows to build up the solution of that system, in the hypothesis of the determination of the complete integral of the corresponding partial differential equation; then some properties concerning the integration constants are put in evidence and one deals with the particular case of the scleronomic constraints, as well as with the cases which admit cyclic co-ordinates. One makes also other considerations on the Hamilton–Jacobi method (Hamilton, W.R., 1890; Jacobi, C.G.J., 1882; Nordheim, L. and Fues, E., 1927; Pars, L., 1965).

19.2.1.1 The Hamilton-Jacobi Partial Differential Equation Let be a function S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as , as + 1 ) ,

(19.2.1)

which depends on s generalized co-ordinates, on time and on s + 1 arbitrary constants (corresponding to the s + 1 independent variables); we calculate the partial derivatives ∂S S ≡ = p0 ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as , as + 1 ) , ∂t

(19.2.1')

∂S = p j ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as , as + 1 ) , j = 1, 2,..., s . ∂q j

(19.2.1'')

In this case, s + 1 of the s + 2 relations (19.2.1), (19.2.1'), and (19.2.1'') allow to determine the constants a j , j = 1, 2,..., s + 1 , as functions of qk , t , S , S and


161

∂S ∂qk , k = 1,2,..., s ; eliminating these constants between the mentioned relations, we

obtain one or more relations of the form ∂S ∂S ∂S ⎞ , ,..., = 0. f ⎛⎜ q1 , q2 ,..., qs ; t ; S , S , ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.2)

If there results only one relation, this one is a partial differential equation of first order, non-linear in general, with respect to the function S ; this function, of the form (19.2.1), represents the complete integral of the partial differential equation (19.2.2). Let be, e.g., a partial differential equation of the form ∂S ∂S ∂S ⎞ , ,..., = 0. f ⎛⎜ ∂qs ⎟⎠ ⎝ ∂q1 ∂q2

(19.2.3)

The corresponding complete integral will be S = a1q1 + a2q2 + ... + as −1qs −1 + Cqs + as + 1 ,

(19.2.3')

f ( a1 , a2 ,..., as −1 ,C ) = 0 .

(19.2.3'')

with

In case of Clairaut's equation q1

∂S ∂S ∂S ∂S ∂S ∂S ⎞ + q2 + ... + qs + f ⎛⎜ , ,..., − S = 0, ∂q1 ∂q2 ∂qs ∂qs ⎟⎠ ⎝ ∂q1 ∂q2

(19.2.4)

we get the complete integral S = a1q1 + a2q2 + ... + as qs + f ( a1 , a2 ,..., as ) .

(19.2.4')

In case of the equation (19.2.3) (which does not contain the function S ), the arbitrary constant as + 1 appears in an additive form in the complete integral (19.2.3'), while, in case of the equation (19.2.4) (which contains the function S ), no one additive constant intervenes in the complete integral (19.2.4'). Let be thus a partial differential equation ∂S ∂S ∂S ⎞ , ,..., = 0, f ⎛⎜ q1 , q2 ,..., qs ; t ; ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.5)

which does not contain the function S ; the complete integral is thus of the form S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as ) + as + 1 ,

(19.2.5')

one of the arbitrary constants being additive. We notice that in the relations (19.2.1'), (19.2.1'') intervene now only s arbitrary constants, called essential constants, which


162

can be determined univocally from the relations (19.2.1''), assuming that the Hessian with respect to the generalized co-ordinates and to the arbitrary constants is non-zero ( S is a function of class C 2 ) ⎡ ∂2 S ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂q j ∂ak ⎦

(19.2.6)

As a matter of fact, the integral (19.2.5') is complete if this necessary and sufficient condition is fulfilled. Replacing in (19.2.1'), we see that the corresponding partial differential equation can be written in the form ∂S ∂S ∂S ⎞ , ,..., = 0; S + ϕ ⎛⎜ q1 , q2 ,..., qs ; t ; ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.7)

the complete integral which contains s essential constants and verifies the condition (19.2.6) is given by S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as ) .

(19.2.7')

If we denote q 0 = t and if we take into account the notations (19.2.1'), (19.2.1''), then we can write the partial differential equation (19.2.5) in the form f ( q 0 , q1 , q2 ,..., qs ; p0 , p1 , p2 ,..., ps ) = 0 ;

(19.2.8)

the ordinary differential equations of the characteristics of the partial differential equation (19.2.8) read dq k ∂f dpk ∂f = =− , , q = qk ( τ ) , pk = pk ( τ ) , k = 0,1,2,..., s . ∂pk dτ ∂qk k dτ

(19.2.8')

We can say that the partial differential equation of first order (19.2.8) is associated to the system of ordinary differential equations of first order (19.2.8') and inversely. Let us take Hamilton’s function H = H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) as function ϕ in the equation (19.2.7); we obtain thus the partial differential equation ∂S ∂S ∂S ⎞ , ,..., ;t = 0 , S + H ⎛⎜ q1 , q2 ,..., qs , ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

where the generalized momenta

pj

(19.2.9)

are replaced by the partial derivative

∂S ∂q j , j = 1,2,..., s ; the complete integral of this equation, called the Hamilton–

Jacobi equation (or the equation in S ), is of the form (19.2.7'), with the condition (19.2.6). If we make τ = t in the system of ordinary differential equations (19.2.8') and use the notations which have been introduced, we get


dq k ∂H dpk ∂H = = − , , k = 1, 2,..., s , ∂pk dt ∂qk dt dq 0 dp ∂H = 1, 0 = − . ∂t dt dt

163

(19.2.9')

The equations in the first row are just Hamilton’s canonical equations; the first equation in the second row is an identity and the second one leads to the relation dH / dt = H , which takes place along the trajectory of the representative point in the space Γ 2 s . Hence, the Hamilton–Jacobi equation is associated to Hamilton’s equations, the study of this equation being thus equivalent to the study of the canonical equations; but sometimes, the integration of the Hamilton–Jacobi equation can be realized on a simpler way or can give interesting suggestions for the integration of the canonical equations, being useful to solve the mechanical problem thus put.

19.2.1.2 The Hamilton-Jacobi theorem In connection with the Hamilton–Jacobi equation we can state Theorem 19.2.1 (Hamilton–Jacobi). If S ∈ C 2 is the complete integral (19.2.7') (which verifies the condition (19.2.6)) of the Hamilton–Jacobi equation (19.2.9), then the sequences ∂S ∂S = bj , bj = const, = p j , j = 1, 2,..., s , ∂a j ∂q j

(19.2.10)

determine the solutions q j = q j (t ) and p j = p j (t ) , j = 1,2,..., s , of the system of canonical equations (19.1.14). Starting from the first sequence of relations (19.2.10) (which is, in fact, a sequence of first integrals), taking into account the condition (19.2.6), we can determine the generalized co-ordinates q j = q j ( t ;a1 , a2 ,..., as , b1 , b2 ,..., bs ) , j = 1, 2,..., s , by means of the theorem of implicit functions; replacing then in the second sequence of the relation (19.2.10), we obtain the generalized momenta p j = p j ( t ; a1 , a2 ,..., as , b1 , b2 ,..., bs ) . The integration constants a j and bj of same index j , j = 1, 2,..., s , are called conjugate constants. Because the relations (19.2.10) must take place along the trajectory of the representative point P in the space Γ 2 s at any moment t , their total derivatives with respect to time must vanish; we will have ∂2 S ∂2S ∂2 S ∂2 S qk + = 0, qk + − p j = 0 . ∂qk ∂a j ∂t ∂a j ∂qk ∂q j ∂t ∂q j

(19.2.11)

We notice that, by introducing the complete integral (19.2.7') in the equation (19.2.9), the latter one must be identically verified in generalized co-ordinates and essential constants; differentiating this equation, successively, with respect to a j and to q j , we obtain


164

∂2 S ∂H ∂2 S ∂2 S ∂H ∂2 S + = 0, + = 0. ∂a j ∂t ∂pk ∂a j ∂qk ∂q j ∂t ∂pk ∂q j ∂qk

Subtracting these relations one of the other and observing that S ∈ C 2 , we can write ∂2S ∂qk ∂a j ∂2S ∂q j ∂qk

∂H ⎞ ⎛ ⎜ qk − ∂p ⎟ = 0, ⎝ k ⎠

∂H ⎞ ⎛ ∂H ⎞ ⎛ ⎜ qk − ∂p ⎟ − ⎜ p j − ∂q ⎟ = 0, j = 1,2,..., s . ⎝ ⎠ j ⎠ k ⎝

(19.2.11')

Hence, these relations are equivalent to the relations (19.2.11). If the canonical equations take place, then – obviously – the relations (19.2.11') are also verified. Let us suppose now that the first sequence of the relations (19.2.11') holds; these relations can be considered as a homogeneous system of linear algebraic equations with a non-zero determinant (the Hessian (19.2.6)), which admits only trivial solutions, obtaining thus the first subsystem of the canonical equations. If we take into account this result, the second sequence of relations leads to the second subsystem of the canonical equations. The theorem enounced above is thus proved. Thus, the solution of the system of canonical equations (19.1.14) is reduced to the finding of a complete integral of the Hamilton–Jacobi equations (19.2.9). The two problems are equivalent, the Hamilton–Jacobi theorem being useful if we succeed to obtain such a complete integral. The respective method of calculation is known as the Hamilton-Jacobi method. We mention that the Hamilton–Jacobi theorem can be put in connection with the Theorem 19.1.19 of Liouville (see Sect. 19.1.2.4). We notice that a complete integral of the Hamilton–Jacobi equation is not the general integral of this equation; indeed a complete integral contains, in general, a smaller number of solutions than the general integral. But starting from a complete integral, we can find again the partial differential equation to which it corresponds, on the way shown in Sect. 19.2.1.1 for the function (19.2.5') with as + 1 = 0 . We mention that the equation S = S ( q1 , q2 ,..., qs ; t ; a1 , a2 ,..., as ) = 0

(19.2.12)

represents the equation of a movable or deformable hypersurface in the configuration space Λs ; this equation is, in fact, a wave surface and pays an important rôle in the undulatory mechanics. The trajectory of the representative point in this space is orthogonal, at each point, to the mentioned surface.

19.2.1.3 Properties of the Integration Constants As it is known (see Theorem 19.1.4), Lagrange’s brackets formed by to integration constants of the system of canonical equations are constants along the trajectory of the representative point in the space Γ 2 s (see Sect. 19.1.2.1). We try now to specify the integration constants which are introduced by the Hamilton-Jacobi method. Let thus be


165

[ a j , ak ] =

∂qi ∂pi ∂q ∂pi . − i ∂a j ∂ak ∂ak ∂a j

Differentiating the sequence (19.2.10) with respect to ak , we obtain ∂2 S ∂qi ∂2S + = 0, ∂qi ∂a j ∂ak ∂ak ∂a j

(19.2.13)

∂p j ∂2 S ∂ql ∂2S + = , j , k = 1, 2,..., s . ∂ql ∂a j ∂ak ∂ak ∂q j ∂ak

Inverting j by k we obtain other analogous relations. Replacing the partial derivative of the generalized momenta in Lagrange’s bracket considered above it results

[ a j , ak ] =

∂qi ∂ql ⎞ ∂2 S ⎛ ∂qi ∂ql ∂2 S ∂qi ∂2 S ∂ql . − + − ∂ql ∂qi ⎜⎝ ∂a j ∂ak ∂ak ∂a j ⎟⎠ ∂ak ∂qi ∂a j ∂a j ∂qi ∂ak

The first sum contains products of symmetric and antisymmetric factors with respect to the indices i and l , being thus equal to zero; taking into account the first relations (19.2.13) and observing that S ∈ C 2 , one sees that the other sums vanish too, so that

[ a j , ak ] = 0, j , k

= 1, 2,..., s .

(19.2.14)

Analogically,

[ bj ,bk ] =

∂qi ∂pi ∂q ∂p − i i . ∂bj ∂bk ∂bk ∂bj

Differentiating the sequences (19.2.10) with respect to bk , we get

∂p j ∂2 S ∂qi ∂2 S ∂ql , j , k = 1,2,..., s . = δ jk , = ∂qi ∂a j ∂bk ∂ql ∂q j ∂bk ∂bk

(19.2.13')

Inverting j by k , we obtain other relations, analogue to the above ones. Replacing the partial derivatives of the generalized momenta in this Lagrange bracket, we get

[ bj ,bk ] =

∂q ∂ql ⎞ ∂2 S ⎛ ∂qi ∂ql − i . ∂ql ∂qi ⎜⎝ ∂bj ∂bk ∂bk ∂a j ⎟⎠

This sum contains products of terms symmetric and antisymmetric with respect to the indices i and l , thus vanishing; it results

[ bj , bk ] = 0, j , k As well,

= 1, 2,..., s .

(19.2.14')


166

[ b j , ak ] =

∂qi ∂pi ∂q ∂pi . − i ∂bj ∂ak ∂ak ∂bj

Replacing the partial derivative of the generalized momenta in the above Lagrange bracket, one obtains

[ b j , ak ] =

∂q ∂ql ⎞ ∂2 S ⎛ ∂qi ∂ql ∂2 S ∂qi − i + . ⎜ ⎟ ∂ql ∂qi ⎝ ∂bj ∂ak ∂ak ∂bj ⎠ ∂qi ∂ak ∂bj

Considerations analogue to those above show that the first sum vanishes; concerning the second sum, we take into account the first relation (19.2.12'). We obtain

[ bj , kk ] = δ jk , j , k

= 1, 2,..., s .

(19.2.14'')

Thus, all the Lagrange brackets which can be set up have been put in evidence. 19.2.1.4 Generalized Conservative Mechanical Systems. Cyclic Co-ordinates In case of a generalized conservative mechanical system, hence for which H = 0 , one obtains a remarkable form of the Hamilton–Jacobi equation; indeed, in this case we can write the first integral H = h , so that the equation in S reads S + h = 0 , wherefrom the integral S = − ht + S ( q1 , q2 ,..., qs ) , one of the integration constants being the energy constant h . One obtains thus the reduced Hamilton–Jacobi equation ∂S ∂S ∂S ⎞ ⎛ , ,..., =h, H ⎜ q1 , q2 ,..., qs , ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.15)

while the complete integral is given by

S = − ht + S ( q1 , q2 ,..., qs ; a1 , a2 ,..., as −1 , h ) .

(19.2.15')

The condition (19.2.6) becomes ⎡ ∂2S ⎤ det ⎢ ⎥ ≠ 0, as = h . ⎣ ∂q j ∂ak ⎦

(19.2.15'')

The solutions of the system of canonical equations will be given by the sequences of relations ∂S ∂S ∂S = bj , j = 1, 2,..., s − 1, = t + bs , = pk , k = 1, 2,..., s . ∂a j ∂h ∂qk

(19.2.15''')

The first s − 1 relations give the trajectory of the representative point in the configuration space Λs (hence, q j = q j (qs ), j = 1, 2,..., s − 1 ), the subsequent


167

relation specifying the motion on this trajectory (after replacing the previous results, we get qs = qs (t ) and, returning, we obtain q j = q j (t ), j = 1,2,..., s − 1 ) . The last relations show that the function S is a simple potential for the generalized momenta, which are – in this case – conservative. As well, we see that the function S plays the rôle of a simple quasi-potential for the generalized momenta, which are thus quasi-conservative. Taking into account that the complete integral is not unique, it results that this quasi-potential is not uniquely determinate too (excepting an arbitrary constant, which appears in the usual case). In case of a generalized conservative mechanical system, the wave surface (19.2.12) is given by S = ht .

(19.2.16)

At a given moment t = t0 = const , one obtains S = const ; we have dS =

∂S dq j = p j dq j . ∂q j

If the constraints are scleronomic, the mechanical system being conservative (we have U = 0 too), then we can write 2T =

∂Tj ∂L q = q = − p j q j , ∂q j j ∂q j j

so that S =

P

∫P 2Tdt ,

(19.2.17)

0

the representative point P0 corresponding to the initial moment and the representative point P being an arbitrary point in the configuration space Λs . In this case, the function S is called action; it verifies the reduced Hamilton–Jacobi equation (19.2.15). In case of r cyclic co-ordinates q α , hence for which ∂H ∂q α = 0, α = 1, 2,..., r , the corresponding generalized momenta are constant ( pα = aα , α = 1, 2,..., r ); it results ∂S ∂q α = a α , so that the complete integral will be S =

r

∑ aαqα

α =1

+ S 0 ( qr + 1 , qr + 2 ,..., qs ; t ; a1 , a2 ,..., as ) ,

(19.2.18)

the conditions (19.2.6) being reduced to s

⎡ ∂2S 0 ⎤ ≠ 0. det ⎢ ⎥ ⎣ ∂q j ∂ak ⎦ j ,k =r + 1

(19.2.18')


168 The Hamilton–Jacobi equation reads

∂S 0 ∂S 0 ∂S S0 + H ⎛⎜ qr + 1 , qr + 2 ,..., qs , a1 , a2 ,..., ar , , ,..., 0 ; t ⎞⎟ = 0 , q q ∂ ∂ ∂qs ⎠ ⎝ r +1 r +2

(19.2.18'')

corresponding to a canonical system in 2(s − r ) variables (as it is known – as a matter of fact – from the study of Hamilton’s system of equations). The sequences of relations (19.2.10) become ∂S 0 ∂S = −q α + bα , α = 1, 2,..., r , 0 = bj , j = r + 1, r + 2,..., s , ∂aα ∂a j pα = a α , α = 1,2,..., r ,

∂S 0 = p j , j = r + 1, r + 2,..., s . ∂q j

(19.2.18''')

In case of a generalized conservative mechanical system, which has r cyclic co-ordinates, the function S will be of the form S = − ht +

r

∑ a αq α

α =1

+ S 0 ( qr + 1 , qr + 2 ,..., qs , a1 , a2 ,..., as −1 , h ) ,

(19.2.19)

and the reduced Hamilton–Jacobi equation becomes ∂S 0 ∂S 0 ∂S , ,..., 0 ⎞⎟ = h . H ⎛⎜ qr + 1 , qr + 2 ,..., qs , a1 , a2 ,..., ar , ∂qr + 1 ∂qr + 2 ∂qs ⎠ ⎝

(19.2.19')

There result ∂S 0 = −q α + bα , α = 1, 2,....r , ∂aα ∂S 0 = bj , j = r + 1, r + 2,..., s − 1, ∂a j ∂S 0 = t + bs , pα = aα , α = 1, 2,....r , ∂h ∂S 0 = p j , j = r + 1, r + 2,..., s . ∂q j

(19.2.19'')

19.2.1.5 Case of a Discrete System of Particles Let be a discrete system S of n particles, of masses mi , of velocities vi and of momenta Hi = m vi , i = 1, 2,..., n . Assuming that this system is not subjected to constrains, it results that s = 3n and that we can choose as space of configurations the representative space E 3n . Using the corresponding notations (see Sect. 18.1.1.2), we can introduce the generalized momenta Pk = M kVk = M k X k , k = 1, 2..., 3n .

(19.2.20)


169

Assuming that the generalized forces Qk derives from a simple quasi-potential U = U ( X1 , X 2 ,..., X 3 n ; t ) , the equation of motion (18.1.38) reads Pk +

∂P dX j ∂U = Qk = , k = 1, 2,..., 3n ; ∂ d t Xk j =1 3n

∑ ∂Xkj

we observe that this equation can be written also in the form Pk +

⎤ ∂ ⎡ 3n 1 Pj2 − U ⎥ + ∑ ⎢ ∂Xk ⎣ j =1 2 M j ⎦

3n

1

⎛ ∂P

∑ M j Pj ⎜⎝ ∂Xkj j =1

−

∂Pj ⎞ = 0. ∂Xk ⎟⎠

If the generalized momenta derive from a quasi-potential S = S ( X1 , X 2 ,..., X 3 n ; t ) , then we find ∂ ⎡ ⎢S + ∂Xk ⎣⎢

2 ⎤ 1 ⎛ ∂S ⎞ ∑ 2M j ⎜⎝ ∂X j ⎟⎠ − U ⎥ = 0, k = 1, 2,..., 3n . j =1 ⎦⎥ 3n

It results that the function between the square brackets depends only on the time t , being of the form f (t ) . If, instead of the quasi-potential S , we use the quasi-potential

S − ∫ f (t )dt , then we notice that the 3n generalized momenta are not influenced;

hence, we can write the equation of motion 2

1 3 n 1 ⎛ ∂S ⎞ S + ∑ −U = 0 . 2 j =1 M j ⎜⎝ ∂X j ⎟⎠

(19.2.21)

As a matter of fact, observing that Hamilton’s function H = T − U is given by a relation of the form (19.1.87'), we can also write ∂S ∂S ∂S ⎞ S + H ⎛⎜ X1 , X 2 ,..., X 3 n ; t ; = 0, , ,..., ∂X1 ∂X 2 ∂Xn ⎟⎠ ⎝

(19.2.21')

obtaining thus the corresponding Hamilton–Jacobi equation. Let us build up the Pfaff form of Hamilton ω =

3n

∑ Pj dX j j =1

− Hdt .

This form is a total differential if and only if the relations ∂Pj ∂Pk ∂H = = Pj , j .k = 1, 2,..., 3 n , ,− ∂Xk ∂X j ∂X j

hold. Taking into account the expression of Hamilton’s function, we can write

(19.2.22)


170 Pj +

3n

∂P

1

∑ M k Pk ∂Xkj

= Pj +

k =1

3n

1

∂Pj

∑ M k Pk ∂Xk

k =1

=

∂U . ∂X j

But Pk M k = dXk dt , so that dPj dV j ∂U = Mj = . ∂X j dt dt

(19.2.23)

Hence, the existence of the quasi-potential S is connected to the existence of the quasi-potential U ; but this one is only a necessary condition for the existence of the function S , because some conditions concerning the initial state must be satisfied too. Thus, at the initial moment t = t0 we must have X j (t0 ) = X j0 , Vj (t0 ) = Vj0 , wherefrom Pj (t0 ) = Pj0 , U (t0 ) = U 0 and H (t0 ) = H 0 ; because, in a problem of Cauchy type, the equation of motion (19.2.23) must hold also at the initial moment, it results that the Pfaff form of Hamilton ω0 =

3n

∑ Pj0 dX j0 j =1

− H 0 dt 0

(19.2.22')

must be a total differential at the initial moment too. If this condition is also fulfilled, then one can use the field description by means of the Hamilton–Jacobi equation. Observing that dS =

∂S

3n

∑ ∂X j dX j j =1

+ Sdt ,

we can write 3n

∑ Pj dX j

dS =

j =1

− Hdt

(19.2.24)

too, where we take into account the Hamilton–Jacobi equation. Then ⎛ 1 3n 1 2 dS = ⎜ ∑ Pj + U ⎝ 2 j =1 M j

⎞ ⎛ 1 3n 2 ⎟ dt = ⎜ 2 ∑ M jVj + U ⎠ ⎝ j =1

⎞ ⎟ dt = ( T + U ) dt . ⎠

Introducing the kinetic potential (18.2.34), we obtain dS =

L dt .

(19.2.25)

We can write S1 − S 0 =

t1

∫t

0

L dt

(19.2.25')


171

for a finite variation, integrating along the trajectory of the representative point in the space E 3 n . 19.2.1.6 Hamilton’s Principal Function Taking the final moment t1 as an arbitrary moment t , we can define the function S in the form S =

t

∫t

L (q1 , q2 ,..., qs , q1 , q2 ,..., qs ; τ )dτ .

(19.2.26)

0

As a matter of fact, this quantity can be called Lagrangian action (shortly, action), corresponding – in a non-conservative case – to the quantity (19.2.17). If we take into account the equations of motion at the moment t , written in the form q j = q j (t ; q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ), p j = p j (t ; q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ), j = 1,2,..., s ,

then we get S = S (t ; q10 , q20 ,..., qs0 , p10 , p20 ,..., ps0 ) . Hamilton proposed to eliminate the generalized momenta at the initial moment; there results the function S = S (t ; q1 , q2 ,..., qs , p1 , p2 ,..., ps ) ,

(19.2.27)

called the Hamilton principal function. We can calculate dS ∂S = q + S = L . dt ∂q j j

But ∂S ∂S ∂S ∂S q j − L = H (q1 , q2 ,..., qs , , ,..., ;t ) , ∂q j ∂q1 ∂q2 ∂qs

with the notation p j = ∂S ∂q j , j = 1, 2,..., s . Hence, Hamilton’s principal function must verify the equation in S . On the other hand, dS =

∂S dq j + Sdt = p j dq j − Hdt . ∂q j

(19.2.28)

If we take into account all the arguments in (19.2.27), then we can write dS = p j dq j − Hdt − p j0 dq j0 .

(19.2.28')

Putting the condition that this is a total differential, we find again the Hamilton-Jacobi equation, as well as the sequence ∂S ∂q j = p j , j = 1, 2,..., s ; the essential constants

172


being q10 , q20 ,..., qs0 , we can write ∂S ∂q j0 = p j0 , j = 1, 2,..., s . The first subsystem of relations determines the motion of the representative point in the space Λs ; associating the second subsystem of relations, we obtain the motion of the representative point in the space Γ 2 s . Thus, Hamilton showed, in 1834, the possibility to obtain the final equation of motion at the moment t (the canonical co-ordinates as functions of time), by means of a complete integral of the equation in S . But the complete integral used by Hamilton is not arbitrary; indeed, the essential constants which have been used are the initial values of the canonical co-ordinates. Thus, one obtains a vicious circle: to obtain the final equations of motion, Hamilton’s principal function is necessary, while to set up this function, one must know the final equations of motion. The essential merit of Jacobi consists in showing how one can avoid this vicious circle; indeed, Jacobi showed, in 1837, that the final equations of motion can be obtained by means of an arbitrary complete integral of the Hamilton–Jacobi equation (19.2.9).

19.2.2 Systems of Equations with Separate Variables The setting up of a complete integral of the Hamilton–Jacobi partial differential equation can be realized, in many cases, in the form of a sum of s functions of only one generalized co-ordinate; in this case, we say that we have to do with a system of s equations with separate variables, which has interesting properties. We mention that the separability is a property connected both to the system and to the chosen independent co-ordinates. One puts thus the problem to determine the conditions of separability, as well as the form of the system which verifies such conditions; we consider thus natural systems with s degrees of freedom, the motion of which is described by means of s Lagrangian co-ordinates. There are a few cases (e. g., the problem of small oscillations) in which the separation leads to s independent equations, each one containing only one generalized co-ordinate. In general, one cannot isolate a particular generalized co-ordinate, studying its variation as in the case of a system with only one degree of freedom; but, in a certain sense, we can consider the variation of a single co-ordinate, neglecting the behaviour of the other co-ordinates. One cannot give the general form of the systems of equations with separate (or with separable) variables. Therefore, we will limit ourselves to the catastatic (hence holonomic and scleronomic) systems for which the kinetic energy is a positive definite quadratic form in the generalized velocities. In what follows, we will present firstly the case of mechanical systems with two degrees of freedom. After some particular cases of separability, we will study – further – the general case of mechanical systems with s degrees of freedom, in the frame of Stäckel’s theory. 19.2.2.1 Mechanical Systems with Two Degrees of Freedom We notice first of all that any scleronomic mechanical system with a single degree of freedom is separable, the complete integral of the corresponding Hamilton–Jacobi equation being of the form


173 S (q ; t ; h ) = − ht + S (q ; h ) .

(19.2.29)

Passing to the scleronomic mechanical systems with two degrees of freedom, we will limit ourselves to the case of orthogonal systems, so that in the expression of the kinetic energy does not intervene the product of two generalized velocities; hence, T =

1 ( g q2 + g22q22 ) . 2 11 1

(19.2.30)

In this case, Hamilton’s function reads H =

1 11 2 ( g p1 + g 22 p22 ) − U , 2

(19.2.31)

where g 11 = 1/ g11 > 0, g 22 = 1/ g22 > 0 and U are functions of class C 1 in the generalized co-ordinates q1 and q2 ; we have supposed that the generalized forces derive from a simple potential. There results the Hamilton–Jacobi equation 2 2 1 ⎡ 11 ⎛ ∂S ⎞ 22 ⎛ ∂S ⎞ ⎤ g g + ⎜ ∂q ⎟ ⎜ ∂q ⎟ ⎥ − U = h , 2 ⎢⎣ ⎝ 1⎠ ⎝ 2 ⎠ ⎦

(19.2.32)

where the complete integral is of the form S = − ht + S (q1 , q2 ; a1 , h ) ; to realize a separation of the form S (q1 , q2 ,; a1 , h ) = S1 (q1 ; a1 , h ) + S2 (q2 ; a1 , h )

(19.2.33)

it is necessary to have g 11ϕ1 + g 22 ϕ2 = U + h , 1 ⎛ ∂Sk ⎞ , k = 1,2, a2 = h . 2 ⎜⎝ ∂qk ⎟⎠ 2

ϕk = ϕk (qk ; a1 , a2 ) =

(19.2.34)

Differentiating with respect to a1 and a2 , respectively, we obtain g 11ϕ1,1 + g 22 ϕ2,1 = 0, g 11ϕ1,2 + g 22 ϕ2,2 = 1 .

The coefficients g 11 and g 22 being positive, it results that the derivatives ϕ1,1 , ϕ2,1 and ϕ1,2 , ϕ2,2 , respectively, cannot be identically zero (for any generalized co-ordinates); as

well

⎡ ∂2S ⎤ ⎡ ∂ (ϕ1 , ϕ2 ) ⎤ ∂S1 ∂S2 ϕ1,1ϕ2,2 − ϕ1,2 ϕ2,1 = det ⎢ det = ⎢ ⎥ ≠ 0. ⎣ ∂ (a1 , a2 ) ⎦⎥ ∂q1 ∂q2 ⎣ ∂q j ∂ak ⎦


174 We can write

g 11 =

P1 P2 , g 22 = , Q1 + Q2 Q1 + Q2

the functions Pk = Pk (qk ), Qk = Qk (qk ), k = 1, 2 , being given by P1 =

ϕ1,2 ϕ2,2 1 1 . , P2 = − , Q1 = , Q2 = − ϕ1,1 ϕ2,1 ϕ1,1 ϕ2,1

(19.2.34')

The potential U must be given by U =

U1 + U 2 , Q1 + Q2

(19.2.35)

where the functions U k = U k (qk ), k = 1, 2 are U1 =

ϕ1 − hϕ1,2 ϕ2 − hϕ2,2 , U2 = − . ϕ1,1 ϕ2,1

(19.2.34'')

Resulting the kinetic energy T =

1 ( Q + Q2 2 1

) ⎛⎜

1 2 1 2 q1 + q P2 2 ⎝ P1

⎞, ⎟ ⎠

(19.2.35')

it is necessary that Hamilton’s function be of the form H =

1 P1 p12 + P2 p22 U 1 + U 2 . − 2 Q1 + Q2 Q1 + Q2

(19.2.36)

The corresponding reduced Hamilton–Jacobi equation (19.2.15) is given by 1 Q1 + Q2

⎡ 1 ⎛ ∂S1 ⎞2 1 ⎛ ∂S2 ⎞2 ⎤ ⎢ P1 ⎜ ⎟ + 2 P2 ⎜ ∂q ⎟ − (U 1 + U 2 ) ⎥ = h . ∂ 2 q ⎝ 1 ⎠ ⎝ 2 ⎠ ⎣⎢ ⎦⎥

Choosing the function S in the form (19.2.33), this equation is decomposed in two equations of only one variable 2 2 1 ⎛ ∂S1 ⎞ 1 ⎛ ∂S2 ⎞ = + − = hQ2 + U 2 + a1 , P1 ⎜ hQ U a , P 1 1 1 2 ⎝ ∂q1 ⎟⎠ 2 2 ⎜⎝ ∂q2 ⎟⎠

where a1 is an arbitrary constant. Hence, the complete integral is given by


175

S (q1 , q 2 ; t ;a1 , h ) = − ht +

2 ( hQ1 + U 1 − a1 ) dq1 + P1

∫

∫

2 ( hQ2 + U 2 + a1 ) dq 2 , P2 (19.2.37)

the condition of separability imposed above being sufficient too. Applying the Hamilton–Jacobi theorem, we obtain −

∂S = ∂a1

dS = dh p1 =

∫

dq1 − R1

∫

dq 2 = −b1 , R2

Q1 dq1 Q2 dq2 ∫ R1 + ∫ R2 = t − t0 ,

∂S 1 = ∂q1 P1

R1 , p2 =

∂S 1 = ∂q2 P2

R2 ,

(19.2.38)

(19.2.38')

where R1 (q1 ) = 2 P1 ( hQ1 + U 1 − a1 ) , R2 (q2 ) = 2 P2 ( hQ2 + U 2 + a1 ) .

(19.2.38'')

The equations (19.2.38) specify the motion of the representative point in the space Λ2 , being sufficient to solve the Lagrangian problem; by differentiation, these equations read d q1 dq 2 dt . = = Q1 + Q2 R1 R2

(19.2.39)

dt = dτ , Q1 + Q2

(19.2.39')

We denote

where τ can be considered as a conventional time in the motion of the representative point on the corresponding trajectory in the space Λ2 ; we notice that Q1 + Q2 > 0 , so that τ increases together with t . We obtain the equations 2

2

⎛ dq1 ⎞ = R , ⎛ dq2 ⎞ = R , ⎜ ⎟ ⎟ 1 ⎜ 2 ⎝ dτ ⎠ ⎝ dτ ⎠

(19.2.39'')

which can be considered each one as in the case of a mechanical system with only one degree of freedom (see Sect. 7.1.1.1); but the relation between t and τ depends on both generalized co-ordinates. However, the motion can be studied independently, in a certain measure. Obviously, the two motions would be independent if the sum Q1 + Q2 would be constant (certainly, we should, simultaneously, have Q1 = const and


176

Q2 = const ); the motion would be only approximately independent if the sum Q1 + Q2 would vary not very much.

Let us assume that the sum Q1 + Q2 is bounded for all the values of the generalized co-ordinates (at least for all the values which are taken by these co-ordinates during the considered motion), so that 0 < Q1 + Q2 < N , N finite for any t . In this case τ → ∞ together with t → ∞ and we can make an appreciation of the general nature of the motion with respect to each generalized co-ordinate. If the generalized co-ordinate q1 , e.g., is – at the beginning – between two consecutive simple real zeros

of R1 (q1 ) ( α1 ≤ q1 ≤ β1 ) , then takes place an oscillatory motion in time, between these values, for any moment t ; the motion is periodical in τ , but is not periodical in t , yet it will be called motion of libration. Th sign of the radical will be taken + or – as q1 is increasing or decreasing, respectively. If q1 → γ1 , where γ1 is a double root of R1 , for t (and τ ) tending to infinite, then one can speak about an asymptotic (limit)

motion. If the sum Q1 + Q2 is not bounded, e.g. if Q1 + Q2 → ∞ together with t (which can take place if q1 and q2 tend to infinite with t ), then it can happen that the integral ∞

∫t

0

dt Q1 + Q2

be convergent; in this case lim τ = τ 0 for t → ∞ . If, in this case, we have α1 ≤ q1 ≤ β1 at the initial moment, then we have no more a motion of libration and lim q1 = l1 , α1 ≤ l ≤ β1 , for t → ∞ . As well, if q1 → γ1 , γ1 double root, then the

motion is pseudoasymptotic. Obviously, one can make analogous considerations for the generalized co-ordinate q2 . In the case in which librations can take place for each of the two generalized co-ordinates, the motion of the mechanical system is periodical if ν

v∫ dq2 R2

=

v∫ dq1 , ν ∈ _ . R1

(19.2.40)

Thus, if ν = n2 n1 , where n1 , n2 ∈ ` , have no one common factor, then, after n1 librations corresponding to the generalized co-ordinate q1 and after n2 librations corresponding to the generalized co-ordinate q2 , the mechanical system returns to the previous situation (the representative point P ∈ Λ2 returns to the same position), as well as in what concerns the position and the velocities. Obviously, the period will be T = n1 v∫

Q1 dq1 Q dq + n2 v∫ 2 2 . R1 R2

(19.2.40')


177

If ν is an irrational number, then the motion is not periodical, while the trajectory of the representative point P ∈ Λ2 is contained in a rectangular interval ( αk ≤ qk ≤ βk , k = 1, 2 ), being an open curve; in this case the motion is quasi-periodical. We mention that we can choose the generalized co-ordinates corresponding to the mechanical system with two degrees of freedom in various modes; e.g., in case of the motion of a particle subjected to the action of a central force of Newtonian attraction one can use, in the plane of the motion, polar or parabolic or confocal co-ordinates etc. Correspondingly, one can obtain a separation of the variables in several modes for the same problem.

19.2.2.2 Mechanical Systems with s Degrees of Freedom. Particular Cases of Hamiltonians In case of a generalized conservative mechanical system with s degrees of freedom we search a complete integral (19.2.15'), trying to obtain a decomposition of the function S (q1 , q2 ,..., qs ; a1 , a2 ,..., as −1 , h ) in the form s

∑ S j ( q j ;a1 , a2 ,..., as −1 , h ) .

(19.2.41)

∂S j ∂S ∂S = = , j = 1, 2,..., s . ∂q j ∂q j ∂q j

(19.2.41')

S =

j =1

In this case pj =

The reduced Hamilton–Jacobi equation becomes ∂S ∂S ∂Ss ⎞ H ⎛⎜ q1 , q2 ,..., qs , 1 , 2 ,..., =h. ∂q1 ∂q2 ∂qs ⎟⎠ ⎝

(19.2.41'')

p j = f j ( q j ; a1 , a2 ,..., as −1 , h ) , j = 1, 2,..., s ,

(19.2.42)

Observing that

we get S = − ht +

s

∑ ∫ fj ( q j ;a1 , a2 ,..., as −1 , h ) dq j j =1

(19.2.43)

and the condition (19.2.6) reads ⎡ ∂f j ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂ak ⎦

(19.2.43')


178

The generalized co-ordinates will be given by the system of algebraic equations ∂f j

s

∑ ∫ ∂ak j =1

dq j = bk , k = 1, 2,..., s − 1,

(19.2.42')

∂f j ∑ ∫ ∂ak dq j = t + bks , j =1 s

so that the generalized momenta will result in the form (19.2.42). The above integrals can be considered between the limits c j and q j , where c j are fixed constants, independent on the integration constants ak and bk . The most simple case in frame of the above considerations is that in which the variables are separated in the structure of Hamilton’s function, so that H = H ( g1 , g2 ,..., gs ) , g j = g j (q j , p j ),

∂g j ≠ 0, j = 1, 2,..., s . ∂p j

(19.2.44)

Imposing the conditions g j (q j , p j ) = a j , j = 1, 2,..., s , we can apply the theorem of implicit functions; it results p j = f j (q j ; a j ), j = 1, 2,..., s .

(19.2.45)

The reduced Hamilton–Jacobi equation leads to h = H (a1 , a2 ,..., as ) , so that the complete integral will be of the form S = − H ( a1 , a2 ,..., as ) t +

s

∑ fj (q j ;a j )dq j

.

j =1

We notice that ∂q j ∂p j

(

wherefrom ∂f j ∂a j = ∂g j ∂p j

)p = f

p j = fj

−1 j

∂f j = 1, ∂a j

≠ 0 ; but ∂f j ∂ak = 0, j ≠ k , so that j

⎡ ∂f j ⎤ det ⎢ ⎥ = ⎣ ∂ak ⎦

s

∂f j

∏ ∂a j

≠ 0,

j =1

the condition (19.2.43') being fulfilled. Hence, the sequences of algebraic relations which give the canonical co-ordinates will be (19.2.45) and


179

−1 ∂f j ∂H ⎛ ∂g j ⎞ q t , j = 1, 2,..., s , d = dq j = b j + ∫ ∂a j j ∫ ⎜⎝ ∂p j ⎟⎠ aj ∂ p j = fj ( q j ;a j )

(19.2.45')

the problem being thus reduced to quadratures. If we choose Hamilton’s function in the form H = gs , g j = g j ( g j − 1 ; q j , p j ) ,

∂g j ≠ 0, j = 1, 2,..., s ; g 0 = 0 , ∂p j

(19.2.46)

then we can impose the conditions g j (a j −1 ; q j , p j ) = a j , j = 1, 2,..., s , a 0 = 0, as = h ; the theorem of implicit functions allows to write p j = f j (q j , a j −1 , a j ), j = 1, 2,..., s , a 0 = 0, as = h .

(19.2.47)

The reduced Hamilton–Jacobi equation is identically verified because we have taken gs = as = h ; the complete integral will be of the form S = − ht +

s

∑ ∫ f j ( g j ; a j − 1 , a j ) dq j . j =1

Taking into account considerations analogue to those in the preceding case, it results

(

∂f j ∂a j = ∂g j ∂p j

)p = f −1 j

≠ 0 ; we notice that ∂f j ∂ak = 0, j < k , too. As above, the j

condition (19.2.43') is verified, the value of the determinant being reduced to the product of its elements (all non-zero) of the principal diagonal. We notice that ∂g j + 1 ∂p j + 1

p j +1 = f j +1

∂f j + 1 ∂g j + 1 + = 0. ∂a j ∂a j

The trajectory of the representative point P ∈ Λs will be specified by the equations −1 ∂f j + 1 ∂f j ⎛ ∂g j ⎞ q + q = d d dq j ∫ ∂a j j ∫ ∂a j j +1 ∫ ⎜⎝ ∂p j ⎟⎠ p j = f j ( q j ;a j −1 ,a j )

−∫

∂g j + 1 ∂a j

−1

⎛ ∂g j +1 ⎞ dq j + 1 = bj , j = 1, 2,..., s − 1 , ⎜ ∂p ⎟ ⎝ j + 1 ⎠ p j +1 = fj +1 ( g j +1 ;a j ,a j +1 )

(19.2.47')

and the motion on this trajectory by the equation −1

⎛ ∂qs ⎞ ∫ ⎜⎝ ∂ps ⎟⎠ps = fs (qs ;a ,h ) dqs = t + bs . s −1

(19.2.47'')


180

The generalized momenta are given by the relations (19.2.47), the problem being thus completely solved. Another interesting case is that in which the Hamiltonian is of the form H =

g1 + g2 + ... + gs g1 + g2 + ... + gs

, g j = g j ( q j , pj ) ,

(19.2.48)

∂g j ∂g j g j = g j ( q j , pj ) , −h ≠ 0, j = 1, 2,..., s . ∂p j ∂p j

The reduced Hamilton–Jacobi equation (19.2.15) reads ⎡

s

⎛

∂S ⎞ ⎟ − hg j ⎠

∑ ⎢⎣ g j ⎝⎜ q j , ∂q j j =1

∂S ⎞ ⎤ ⎛ ⎜ q j , ∂q ⎟ ⎥ = 0 . j ⎠⎦ ⎝

We can denote g j ( q j , pj

) − hg j ( q j , p j ) = a j , j

= 1, 2,..., s ,

s

∑aj j =1

= 0,

wherefrom, applying the theorem of implicit functions, we get p j = f j ( q j ; a j , h ) , j = 1, 2,..., s .

(19.2.49)

The complete integral of the Hamilton–Jacobi equation will be S = − ht +

s

∑ ∫ f j ( q j ; a j , h ) dq j . j =1

We notice that (we take into account the relation as = −(a1 + a2 + ... + as −1 ) , the s th essential constant being h ) ∂gk ⎞ ∂fk ∂f ∂g ∂gs ⎞ ∂fs ⎛ ∂gk = 1, ⎛⎜ 0 − h = −1, k = 0, k ≠ l , ⎟ ⎜ ∂p − h ∂p ⎟ ∂ps ⎠ ps = fs ∂ak ∂al ⎝ ∂ps ⎝ k k ⎠ pk = fk ∂ak ∂g j ⎞ ∂f j ⎛ ∂g j − g j = 0, j = 1, 2,..., s , k , l = 1, 2,..., s − 1 ; ⎜ ∂p − h ∂p ⎟ j ⎠ p = f ∂h ⎝ j j j

it results s ∂g j ⎞−1 ⎛ ∂g j ⎡ ∂f j ⎤ det ⎢ Δs , ⎥ = − ∏ ⎜ ∂p − h ∂p ⎟ ⎣ ∂ak ⎦ j j ⎠p = f j =1 ⎝ j j

where


181

Δs =

1

0

0 ... 0

0

g1

0

1

0 ... 0

0

g2

0

0

1 ... 0

0

g3

... ... ... ... ... ...

...

.

0

0

0 ... 0

1

gs − 1

1

1

1 ... 1

1

− gs

Expanding after the first column (the second minor after the first line) we obtain Δs = 1 ⋅ Δs −1 + ( −1)s −1 ( −1)s − 2 g1 = Δs −1 − g1 ; analogously, we get Δs −1 = Δs − 2 − g2 a.s.o. Finally, we can write Δs = −(g1 + g2 + ... + gs ) ≠ 0 , the denominator of

the Hamiltonian cannot vanish, so that

det ⎡⎣ ∂f j ∂ak ⎤⎦ ≠ 0 . The generalized

co-ordinates will be given by the relations ∂fk ∂gk ⎞ −1 ∂fs ⎛ ∂gk q + q = − h d d s ⎜ k ∫ ∂ak ∫ ∂ak ∫ ⎝ ∂pk ∂pk ⎟⎠p = f (q ;a ,h ) dqk k k k k ∂gs ∂gs ⎞ −1 − ∫ ⎛⎜ −h dqs = bk , k = 1, 2,..., s − 1, ⎟ ∂ps ⎠ps = fs (qs ;as ,h ) ⎝ ∂ps

(19.2.49')

the motion on the trajectory of the representative point P ∈ Λs being specified by s

∂f j

∑ ∫ ∂h j =1

dq j =

⎛ ∂g j

s

∑ ∫ g j ⎜⎝ ∂p j j =1

−h

∂g j ⎞ −1 dq j = t + bs , ∂p j ⎟⎠ p j = fj ( q j ;a j ,h )

(19.2.49'')

where as = −(a1 + a2 + ... + as −1 ) ; the relations (19.2.47) allow then the calculation of the generalized momenta. As a particular case, we can state Theorem 19.2.2 (Liouville). If the kinetic energy and the potential energy of the mechanical system are of the form s

T =

s

s

qk2

∑U k

1 Q j ∑ , V = −U = − k s=1 2∑ j =1 k = 1 Pk

∑Qj

,

(19.2.50)

j =1

respectively, where Q j = O j (q j ) , Pk = Pk (qk ) , U k = U k (qk ), j , k = 1, 2,..., s are functions of only one variable, then the equations of motion can be obtained by quadratures. Indeed, in this case Hamilton’s function is of the form


182 s

H =

s

s

∑ Pk pk2 ∑U k 1 2

k =1 s

−

∑Qj j =1

k =1 s

=

∑ Qj

∑ ⎡⎢⎣ 2 Pk (qk )pk2 1

k =1

− U k (qk ) ⎤ ⎥⎦

s

∑ Qj (q j )

j =1

,

(19.2.51)

j =1

hence a particular case of the function (19.2.48). We obtain thus the reduced Hamilton– Jacobi equation s ⎡ 1 ⎛ ∂S ⎞2 ⎤ ∑ ⎢ 2 Pk ⎜⎝ ∂qk ⎟⎠ − U k (qk ) ⎥ = h ∑ Qj (q j ), h = as , k =1 ⎣ j =1 ⎦ s

where we took into account (19.2.15'). If S = S1 (q1 ) + S2 (q2 ) + ... + Ss (qs ) , then, by separation of variables, we can decompose this system in the form 2

1 ⎛ ∂Sk ⎞ P = hQk (qk ) + U k − ak , k = 1, 2,..., s , a1 + a2 + ... + as = 0 . 2 k ⎜⎝ ∂qk ⎟⎠

We get, in this case, the complete integral (ensured by the above proof) S = − ht +

s −1

∑∫

k =1

2 ( hQk + U k − ak ) dq k Pk

2 ( hQs + U s + a1 + a2 + ... + as −1 ) dqs . Ps

+∫

The trajectory of the representative point P in the configuration space Λs will be given by the equation −

∂S = ∂ak

∫

dq k − Rk

∫

dqs = −bk , k = 1, 2,..., s − 1 , Rs

(19.2.52)

while the motion on the trajectory will be specifies by

∂S = ∂h

s

∑∫ j =1

Q j dq j Rj

= t − t0 ,

(19.2.52')

where we have denoted Rk (qk ) = 2 Pk (hQk + U k − ak ), k = 1, 2,..., s − 1, Rs (qs ) = 2 Ps (hQs + U s + a1 + a2 + ... + as −1 ).

There result then the generalized momenta

(19.2.52'')


183

pj =

Rj ∂S = , j = 1, 2,..., s . Pj ∂q j

(19.2.52''')

As in the case s = 2 , the differential equations of motion may be written in a compact form dq1 = R1 (q1 )

dq 2 = ... = R2 (q 2 )

dqs dt = = dτ , Rs (qs ) Q1 (q1 ) + Q2 (q 2 ) + ... + Qs (qs ) (19.2.52iv)

where τ is a conventional measure of the time. In connection to these results one can make a discussion analogue to that in the case s = 2 . The mechanical systems characterized by the kinetic energy and the potential energy (19.2.50) are called Liouville’s systems. In the case s = 2 , as we have seen in the preceding subsection, the mechanical systems, which lead to systems of equations with separate variables, are just of such a form; but if s > 2 , then not any mechanical system, separable from the point of view of the canonical co-ordinates, is characterized by the formulae (19.2.50). 19.2.2.3 Stäckel’s Theorem As in the case s = 2 , we consider the orthogonal, scleronomic mechanical systems, studied by P. Stäckel in 1893–1895; in this case (Stäckel, P., 1908), T =

1 s 1 s g jj q2j = ∑ g jj p 2j , g jj = g jj (q1 ,q2 ,..., qs ) ≥ 0, ∑ 2 j =1 2 j =1 g jj =

1 , U = U (q1 ,q2 ,..., qs ) , g jj

(19.2.53)

where g jj , j = 1, 2,..., s , and U are functions of class C 1 . We can state Theorem 19.2.3 (Stäckel). The Hamilton–Jacobi equation built up by means of the kinetic energy and of the potential energy V = −U (19.2.53) is with separate variables if and only if there exists a non-singular square matrix λ ≡ ⎡⎣ λjk ⎤⎦ , λjk = λjk (q j ) , and a column matrix μ ≡ [ μj

]T ,

s

∑ g jj λjk

μj = μj (q j ), j , k = 1, 2,...s , so that to have

= δks , k = 1, 2,..., s ,

j =1

s

∑ g jj μj j =1

=U .

(19.2.54) (19.2.54')

Indeed, if the reduced Hamilton–Jacobi equation 2

1 s jj ⎛ ∂S ⎞ g ⎜ ⎟ − U = h , h = as , 2∑ ⎝ ∂q j ⎠ j =1

(19.2.55)


184

is separable, then we can write a complete integral for which S =

s

∑ S j (qj ; a1 , a2 ,..., as ) .

(19.2.55')

j =1

Introducing in (19.2.55), this equation must be identically satisfied with respect to the essential constants a j ; differentiating with respect to these constants, we can write s

∂S

∂2S = 0, k = 1, 2,..., s − 1 , ∂ak ∂q j

∑ g jj ∂q j j =1

∂S

s

∑ g jj ∂q j j =1

∂2 S = 1. ∂as ∂q j

(19.2.56) (19.2.56')

Denoting λjk =

∂S j (q j ) ∂2 S j ∂S ∂2 S , = ∂q j ∂ak ∂q j ∂q j ∂ak ∂q j

we notice that the relations (19.2.56), (19.2.56') correspond to the relations (19.2.54); on the other hand, λjk = λjk (q j ), j , k = 1,2,..., s , and ⎡ ∂S j ∂2 S j ⎤ ∂S1 ∂S 2 ∂Ss ⎡ ∂2S j ⎤ = det ⎡⎣ λjk ⎤⎦ = det ⎢ ... det ⎢ ⎥ ⎥ ≠ 0, ⎣ ∂q j ∂ak ∂q j ⎦ ∂q1 ∂q2 ∂qs ⎣ ∂ak ∂q j ⎦

because the function S leads to a complete integral. Observing that s

∑ g jj λjs j =1

= 1,

the equation (19.2.55) allows to write U =

⎡ 1 ⎛ ∂S ⎞ ∑ g jj ⎢ 2 ⎝⎜ ∂q j ⎠⎟ − as λjs ⎢⎣ j =1 s

2

⎤ ⎥. ⎥⎦

Denoting 1 ⎛ ∂S ⎞ 1 ⎡ ∂S j (q j ) ⎤ μj = ⎜ − as λjs = ⎢ − as λjs , 2 ⎝ ∂q j ⎟⎠ 2 ⎣ ∂q j ⎦⎥ 2

2

we see that the relation (19.2.57) corresponds to the relation (19.2.54'), while μj = μ j (q j ), j = 1, 2,..., s . Hence, the conditions (19.2.54), (9.2.54') are necessary conditions of separability.


185

Taking into account the relations of conditions (19.2.54), (19.2.54'), we can write the Hamilton–Jacobi reduced equation in the form 2

s s 1 s jj ⎛ ∂S ⎞ g ⎜ P ak − μ = j j ∑ ∑ ∑ 2 j =1 ⎝ ∂q j ⎟⎠ j =1 k =1

s

∑ g jj λjk j =1

or in the form s ⎡ 1 ⎛ ∂S ⎞2 ⎤ ∑ g ⎢ 2 ⎜⎝ ∂q j ⎟⎠ − ∑ ak λjk − μj ⎥ = 0 , ⎥⎦ j =1 k =1 ⎣⎢ s

jj

where a1 , a2 ,..., as −1 are arbitrary constants. To calculate a complete integral we use the sum (19.2.55'), where the functions of only one variable are given by the equations 2 ⎛ ∂S j ⎞ ⎛ s ⎞ ϕ j (q j ) = ⎜ = 2 ⎜ ∑ ak λjk + μj ⎟ , j = 1, 2,..., s . ⎟ ∂ q ⎝ k =1 ⎠ ⎝ j ⎠

(19.2.58)

It results

∫

S j (q j ) =

ϕ (q j ) dq j , j = 1, 2,..., s .

(19.2.58')

We obtain easily ⎡ ∂2 S j ⎤ ⎡ ∂2 S ⎤ = det ⎢ det ⎢ ⎥ = ⎥ ⎣ ∂ak ∂q j ⎦ ⎣ ∂ak ∂q j ⎦

s

∏ j =1

1 det ⎡⎣ λjk ⎤⎦ ≠ 0 , ϕj

so that the integral obtained on this way for the equation in S is a complete one. Hence, the conditions (19.2.54), (19.2.54') are sufficient conditions of separability too, Stäckel’s theorem being completely proved. The solution of Lagrange’s problem (the trajectory of the representative point P in the space of configurations Λs and the motion on this trajectory) is given by the Hamilton-Jacobi theorem in the form ∂S = ∂ak

s

∂S j

∑ ∂ak j =1

=

s

∑ ∫ λjk j =1

∂S = ∂as

s

∑ ∫ λjs j =1

dq j ϕ j (q j )

= bk , k = 1, 2,..., s − 1,

dq j ϕ j (q j )

(19.2.59) = t + bs .

The solution of Hamilton’s problem in the phase space Γ 2 s is then specified by the generalized momenta

pj =

ϕ j (q j ), j = 1,2,..., s .

(19.2.59')


186

Thus, the considered problem is solved by quadratures. The obtained solution depends on the integration constants a j

and bj ,

j = 1, 2,..., s ; as one can notice, the constant as plays a special rôle, being the energy constant h . The formula (19.2.59') shows that the generalized momentum p j depends

only on the generalized co-ordinate q j ; in exchange, the generalized velocities

q j = g jj ϕ j (q j ), j = 1, 2,..., s ,

(19.2.60)

depend on all the generalized co-ordinates. The relation (19.2.60) allows to write

dq j

ϕj (q j )

= g jj dt = dτ j , j = 1, 2,..., s ,

(19.2.60')

too, where τ j , j = 1, 2,..., s , is a local measure of time, different for each co-ordinate; the sign of the radical is positive or negative as q j increases or decreases, respectively, during time, because g jj ≥ 0 . If g jj ≥ g jj > 0 and if

ϕ j (q j ) is a continuous

function, then the general character of the motion, corresponding to the co-ordinate q j , is specified by (19.2.60') It is interesting to see that, by differentiating the relation (19.2.59), we obtain s

∑ λjk j =1

q j ϕ j (q j )

= δks , k = 1, 2,..., s .

(19.2.61)

Comparing with the relation (19.2.54), we find again the equations (19.2.60'). The conventional time τ j tends to infinite together with t , the nature of the variation of the generalized co-ordinate q j depending on the zeros of ϕ j (q j ) . Thus, if qj

is between two consecutive simple zeros ϕ j (q j ) (with ϕ j (q j ) > 0 for

αj < q j < β j ), then we have a motion of libration in q j , while if q j → γ j for t → ∞ , where γ j is a double zero for ϕ j (q j ) , then we have an asymptotic (limit)

motion. We must notice that if a motion of libration takes place indefinitely between α j and β j , then this one is not, in general, periodical with respect to time. As a matter of fact, as in the case s = 2 , the discussion of the motion with respect to the generalized co-ordinates is of a particular interest. If Pj → 0 for t → ∞ , then it is possible that τ j does tend to a finite limit (not to an infinite one); in this case, the motion with respect to q j is pseudoasymptotic. If αj ≤ q j ≤ β j at the initial moment, then we do not obtain a motion of libration, while (eventually after a finite number of oscillations) lim q j = l j , α j ≤ l j ≤ β j for t → ∞.


187

We mention that Stäckel’s theorem leads. at the same time, to s first integrals of Lagrange’s equations; thus, the relations (19.2.60) lead to 2 1 q j − βj = 2 Pj2

s

∑ ak λjk , j

k =1

= 1, 2,..., s ,

(19.2.62)

so that s

⎛ 1 q2

∑ λ jk ⎜⎝ 2 Pk2

k =1

k

⎞ − βk ⎟ = ak , j = 1, 2,..., s , ⎠

(19.2.62')

where λ jk is the normalized algebraic complement corresponding to the element λjk in the matrix λ , with λjk λlj = δkl . One see easily that for g jj =

Pj s

∑Ql

, Pj = Pj (q j ), Ql = Ql (ql ), j , l = 1, 2,..., s ,

(19.2.63)

l =1

and ⎡− 1 ⎢ P1 ⎢ ⎢ 0 ⎢ ⎢ ⎡⎣ λjk ⎤⎦ = ⎢ 0 ⎢ ⎢ ... ⎢ ⎢ 1 ⎢⎣⎢ P s Uj μj = ,Uj Pj

−

0

0

1 P2

0

...

1 P3 ...

1 Ps

1 Ps

0

−

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ Qs ⎥ Ps ⎥⎦⎥

Q1 P1 Q2 ... 0 P2 Q3 ... 0 P3 ... ... ... ...

...

0

1 Ps

= U j (q j ), j = 1, 2,..., s ,

(19.2.63')

(19.2.63'')

we find again Liouville’s system; taking into account the expansion of Δs , from the previous subsection, one can show that s

s

l =1

l =1

det ⎣⎡ λjk ⎦⎤ = ( −1)s −1 ∏ Pl −\1 ∑ Ql ≠ 0 .

(19.2.63''')

The conditions in Stäckel’s theorem can be satisfied if we take g 11 = ψ2 ψ3 ...ψs , g 22 = ψ3 ψ4 ...ψs ,..., g s −1,s −1 = ψs , g ss = 1 ,

(19.2.64)


188

with ψl = ψl (ql ) , l = 1, 2,..., s ; in this case ⎡ −1 0 0 ⎢ ⎢ ψ2 −1 0 ⎢ 0 ψ −1 3 ⎢ ⎣⎡ λjk ⎤⎦ = ⎢ ... ... ... ⎢ 0 0 ⎢ 0 ⎢ 0 0 0 ⎢⎣

0⎤ ⎥ ... 0 0 ⎥ ... 0 0 ⎥ ⎥, ... ... ... ⎥ ⎥ ... −1 0 ⎥ ... ψs 1 ⎥⎥⎦ ...

0

(19.2.64')

with det ⎡⎣ λjk ⎦⎤ = ( −1)s −1 ≠ 0 . The system is separable if the potential U is of the form U = μ1 ψ2 ψ3 ...ψs + μ2 ψ3 ψ4 ...ψs + μs −1 ψs + μs .

(19.2.64'')

Formally, taking into account the relations (19.2.54), (19.2.54'), we can express the kinetic energy and the simple potential in the form s

T =

s

s

1 g jj λjs ∑ gkk qk2 = 2∑ j =1 k =1

∑ g kk pk2

k =1 s

2 ∑ g jj λjs k =1

s

U =

∑ gkk μk

k =1 s

∑ g jj λjs

,

(19.2.65)

,

k =1

which has an affinity with Liouville’s formulation. But we must notice that Stäckel’s theorem does not cover the conditions of separability for an arbitrary Hamilton–Jacobi equation; such conditions could not yet be found. Let λ jk be the algebraic complement of the element λjk in the determinant Δ = det ⎡⎣ λjk ⎤⎦ ; in this case, by an expansion after the last column, we can write Δ = λjs λ js , so that λ js = ∂Δ ∂λjs . The normalized algebraic complement being

λ jk = λ jk Δ , j , k = 1, 2,..., s , it results

λjk λ js = δks , so that we can take

g jj = λ js , j = 1, 2,..., s in Stäckel’s theorem.

19.2.2.4 Levi-Civita’s Conditions Another attempt to the problem of the separation of variables in the Hamilton-Jacobi equation has been made by T. Levi-Civita in 1904. To do this, we will calculate the


189

total derivative of the Hamilton–Jacobi equation (19.2.41'') with respect to q j in the form dH ∂H ∂H ∂pk = + = 0. dq j ∂q j ∂pk ∂q j

But the generalized momenta are given by (19.2.41'), so that pk = pk (qk ) , obtaining ∂H ∂H ∂p j + = 0(!), j = 1, 2,..., s . ∂q j ∂p j ∂q j

We must assume that ∂H ∂p j ≠ 0 ; otherwise, we should have ∂H ∂q j = 0 too, but we suppose that no one co-ordinate is cyclic. It results ∂H ∂q j dp j dp = − (!), k = 0, k ≠ j . ∂H dq j dq j ∂p j

(19.2.66)

We obtain thus the necessary and sufficient conditions which must be fulfilled by Hamilton’s function, so that the function (19.2.41) does correspond to a complete integral of the equation (19.2.41''). Other developments in this directions have been made by V. G. Imshenetskiĭ in 1869, by Burgatti in 1911, by M. S. Yarov-Yarovoi in 1963 and by others. Imposing also the conditions of integrability d ⎛ ∂pr ⎞ d ⎛ ∂pr ⎞ = , dqk ⎜⎝ ∂q j ⎟⎠ dq j ⎜⎝ ∂qk ⎟⎠

we are led to Levi-Civita’s conditions ⎛ d ⎜ ⎜ dq k ⎜ ⎜ ⎝

∂H ∂q j ∂H ∂p j

⎞ ⎟ ⎟ = 0, j ≠ k , j , k = 1,2,..., s . ⎟ ⎟ ⎠

(19.2.67)

Taking into account (19.2.66), we find s (s − 1)/ 2 relations (we notice that the relations are symmetric with respect to the indices j and k ) of the form ∂2 H ∂H ∂H ∂2 H ∂H ∂H − ∂qk ∂q j ∂p j ∂pk ∂q j ∂qk ∂p j ∂qk −

∂2 H ∂H ∂H ∂2 H ∂H ∂H + = 0, j ≠ k , j = 1, 2,..., s . ∂qk ∂p j ∂q j ∂pk ∂p j ∂pk ∂q j ∂qk

(19.2.67')


190

We mention that these conditions, which must be fulfilled by Hamilton’s function so that the Hamilton-Jacobi equation be with separate variables, have a general character; but, unlike the conditions which appear in the less general case considered by Stäckel, these conditions can hardly be manipulate. As a matter of fact, they could be used only in the cases of two or three degrees of freedom; we mention that the case of two degrees of freedom has been considered by G. Morera in 1887. In 1944, Forbat completes Levi-Civita’s conditions with s condition of the form

∂H ∂2 H ∂H ∂2 H − = 0, j = 1,2,..., s ; ∂p j ∂qk ∂t ∂p j ∂p j ∂t

(19.2.67'')

the results can be thus applied to the case of rheonomous mechanical systems too. 19.2.2.5 A New Formulation of Hamilton–Jacobi Type One sees easily that, in the canonical system of equations (19.1.14), the generalized co-ordinates q j and the generalized momenta p j play a symmetric rôle, being sufficient to replace Hamilton’s function H by −H . In this order of ideas, the Hamilton–Jacobi partial differential equation (19.2.9) can be replaced by the partial differential equation ∂W ∂W ∂W , ,..., , p1 , p2 ,..., ps ; t ⎞⎟ = 0 . W − H ⎛⎜ ∂ p ∂ p ∂ p s ⎝ 1 ⎠ 2

(19.2.68)

If W ( p1 , p2 ,..., ps , a1 , a2 ,..., as ; t ) is a complete integral of this equation, the condition

⎡ ∂2W ⎤ det ⎢ ⎥ ≠0 ⎣ ∂p j ∂ak ⎦

(19.2.68')

taking place, then the canonical co-ordinates q j = q j (t ), p j = p j (t ), j = 1, 2,..., s , will be given by the sequences of relations ∂W ∂W = −bj , = q j , j = 1,2,..., s , ∂a j ∂p j

(19.2.68'')

where a j are essential constants, while bj are constants conjugate to these ones. Making a transformation of Legendre type and taking into account Donkin’s theorem (the formula (19.1.8)), we can write S + W = pj q j ;

(19.2.69)

thus, the new variable p j and the new function W will be given by pj =

∂S ∂S ,W = qj −S. ∂q j ∂q j

(19.2.69')


191

Remaking the modality to pass from Lagrange’s equations to the canonical ones (Legendre’s transformation), we obtain qj =

∂S ∂W , S = pj − W , S = −W . ∂q j ∂q j

(19.2.69'')

We can write, as well, ∂S ∂W , =− ∂a j ∂a j

(19.2.70)

being thus led to the above results. Obviously, the modality to approach the problem is equivalent to the previous one, that we can make an analogous study concerning the possibility to separate the variables.

19.2.3 Applications First of all, we will consider the case of motion of a single particle; we pass then to the application of the Hamilton–Jacobi method in case of a system of two particles, as well as in case of a rigid solid. 19.2.3.1 Motion of a Particle The canonical co-ordinates are q j = x j , p j , j = 1,2, 3 in case of a free particle P (x1 , x 2 , x 3 ) (see Sect. 19.1.3.1); the Hamilton-Jacobi equation, which corresponds to the Hamiltonian (19.1.79'), reads 2 2 2 1 ⎡ ⎛ ∂S ⎞ ⎛ ∂S ⎞ + ⎛ ∂S ⎞ ⎤ = 2m (U + h ) . + S + ⎜ ∂x ⎟ ⎜ ∂x ⎟ ⎥ 2m ⎢⎣ ⎜⎝ ∂x1 ⎟⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ ⎦

(19.2.71)

Assuming that the quasi-potential U is a potential (U = 0 ), we can take S = − ht + S ( x1 , x 2 , x 3 ) , h being the energy constant; the reduced Hamilton–Jacobi equation reads

( grad S )

2

2

2

2

⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ =⎜ ⎟ + ⎜ ∂x ⎟ + ⎜ ∂x ⎟ = 2m (U + h ) . x ∂ ⎝ 1⎠ ⎝ 2 ⎠ ⎝ 3 ⎠

(19.2.71')

If S = − ht + S (x1 , x 2 , x 3 ; a1 , a2 , h ) is a complete integral, then we obtain the sequences of relations ∂S ∂S ∂S = b1 , = b2 , = t + b3 , ∂a1 ∂a2 ∂h

(19.2.72)

∂S ∂S ∂S . ,p = ,p = ∂x1 2 ∂x 2 3 ∂x 3

(19.2.72')

p1 =


192

The first two equations of the sequence (19.2.67) specify the trajectory of the representative point P ∈ Λ3 , while the last equation gives the motion on this trajectory; the sequence of relations (19.2.67') completes the solution for the representative point P ∈ Γ 6 . If we take into account (19.1.79), we obtain the components of the velocity (we have p = grad S = grad S = m v ) x1 =

1 ∂S 1 ∂S 1 ∂S , , x = , x = m ∂x1 2 m ∂x 2 3 m ∂x 3

(19.2.72'')

the reduced Hamilton–Jacobi equation corresponding to the first integral of the mechanical energy. We notice that, at each point, the velocity v , of components x j , j = 1, 2, 3 , is normal to the wave surface S ( x1 , x 2 , x 3 ) = const ; hence, the trajectory of the representative point is normal to this surface. If we make the change of function S = K ln ψ in the reduced Hamilton–Jacobi equation (19.2.71'), then it results

(

K2 grad ψ 2m

)

2

− (U + h ) ψ2 = 0 ,

(19.2.71'')

getting thus a first form of Schrödinger’s equations. This result has been obtained in the frame of the research made to establish a differential equation from which must result the energetical levels of the electron of the atom of hydrogen; one takes U = e 2 / r , where m , e , h are the mass. the charge and the energy of the electron, respectively, and r is the distance electron-nucleus. Starting from the Hamilton function (19.1.80'), one obtains the Hamilton–Jacobi equation

( )

1 ⎡ ∂S S = 2m ⎢⎣ ∂r

2

+

( ) ( ) ⎤⎥⎦ − U (r , θ, z ;t ) = 0

1 ∂S r 2 ∂θ

2

+

∂S ∂z

2

(19.2.73)

in cylindrical co-ordinates ( q1 = r , q2 = θ , q 3 = z ); as well, if U = 0 , then we have S = − ht + S (r , θ , z ) , the reduced Hamilton–Jacobi equation being given by 2

2

2

1 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂r ⎟ + 2 ⎜ ∂θ ⎟ + ⎜ ∂z ⎟ = 2m (U + h ) . r ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(19.2.73')

A complete integral is S = − ht + S (r , θ , z ; a1 , a2 , h ) ; we can write a sequence of relations of the form (19.2.72), as well as the sequence ∂S ∂S ∂S = pr , = pθ , = pz . ∂r ∂θ ∂z

(19.2.74)


193

The components of the velocity will be 1 ∂S 1 ∂S 1 ∂S . ,θ = , z = 2 ∂θ m ∂r m ∂z mr

r =

(19.2.74')

As well, in spherical co-ordinates ( q1 = r , q2 = θ , q 3 = ϕ ) we obtain the Hamilton– Jacobi equation

( )

1 ⎧ ∂S S + ⎨ 2m ⎩ ∂r

2

+

( )

1 ⎡ ∂S ⎢ r 2 ⎣ ∂θ

2

+

1 ⎛ ∂S ⎞2 ⎤ ⎫ ⎜ ⎟ ⎥ ⎬ − U (r , θ , ϕ ; t ) = 0 . sin2 θ ⎝ ∂ϕ ⎠ ⎦ ⎭

(19.2.75)

We can take S = − ht + S (r , θ , z ) if U = 0 ; it results the reduced Hamilton–Jacobi equation 2 2 2 1 ⎡ ⎛ ∂S ⎞ 1 ⎛ ∂S ⎞ ⎤ ⎛ ∂S ⎞ ⎥ = 2m (U + h ) . ⎜ ∂r ⎟ + 2 ⎢ ⎜ ∂θ ⎟ + ⎜ ⎟ sin2 θ ⎝ ∂ϕ ⎠ ⎦ r ⎣⎝ ⎝ ⎠ ⎠

(19.2.75')

The complete integral S = − ht + S (r , θ , ϕ ; a1 , a2 , h ) leads to a sequence of the form (19.2.72) , to the sequence ∂S ∂S ∂S = pr , = pθ , = pϕ ∂r ∂θ ∂ϕ

(19.2.76)

and to the components of the velocity 1 ∂S 1 ∂S 1 ∂S . ,θ = , ϕ = 2 2 2 m ∂r mr ∂θ mr sin θ ∂ϕ

(19.2.76')

U (x 1 , x 2 , x 3 ) = U 1 ( x1 ) + U 2 (x 2 ) + U 3 (x 3 ) ,

(19.2.77)

r =

If

then the reduced Hamilton–Jacobi equation (19.2.71') correspond to a Hamilton’s function of the form (19.2.44), hence being with separate variables. We notice that g j ( x j , p j ) = p j2 / 2m − U j = a j , j = 1, 2, 3 ; there result the generalized momenta pj =

2m [U j ( x j ) + a j ], j = 1, 2, 3 .

Because H (a1 , a2 , a 3 ) = a1 + a2 + a 3 , while ∂g j / ∂p j = p j / m =

(19.2.77') 2(U j + a j )/ m ,

the co-ordinates x1 , x 2 , x 3 are given by (19.2.45'); we have

∫

dx j

U j (x j ) + a j

=

2 ( t + bj ) , j = 1, 2, 3 , m

(19.2.77'')


194

the problem being thus reduced to three quadratures. Obviously, Hamilton’s first equations (19.2.79'') lead to the same result. If 1 U (r , θ , z ) = U r (r ) + U θ ( θ ) + U z ( z ) , 2

(19.2.78)

then Hamilton’s function (19.1.80') is of the form (19.2.46), with p 2θ − U θ ( θ ) = a1 , 2m g p2 gr = gr (g θ ; r , pr ) = θ2 − U r (r ) + r = a2 , 2m r pz2 H = gz = gz (gr ; z , pz ) = gr − U z (z ) + = a3 = h . 2m g θ = g θ ( θ , pθ ) =

One obtains the generalized momenta pθ =

2m [U θ ( θ ) + a j ],

a 2m ⎡U r (r ) − 12 + a2 ⎤ , ⎢⎣ ⎥⎦ r pz = 2m [U z (z ) + h − a2 ].

(19.2.78')

pr =

Taking into account the partial derivatives ∂g θ p = θ = m ∂pθ

2 (U θ + a1 ) ∂gr pr , = = m m ∂pr

pz ∂gz = = m ∂pz

(

)

a1 + a2 r2 , m

2 Ur −

2 (U z + h − a2 ) ∂gr 1 ∂gz , = 2, = 1, m ∂a1 r ∂a2

the formulae (19.2.47') give the trajectory of the particle P in the form

∫ ∫

dθ dr 2 b , −∫ = a1 m 1 2 U θ (θ ) + a j r U r (r ) − 2 + a2 r dr dz 2 b , −∫ = a1 m 2 U z ( z ) + h − a2 U r (r ) − 2 + a2 r

(19.2.78'')

while the formula (19.2.47'') specifies the motion on this trajectory by the equation

∫

dz = U z ( z ) + h − a2

2 b . m 2

(19.2.78''')


195

The problem is thus reduced to quadratures. In case of a potential of the form (corresponding to the potential (18.3.46)) U ( r , θ , ϕ ) = U r (r ) +

1 r2

⎡U ( θ ) + 1 U ϕ (ϕ ) ⎤ , ⎢⎣ θ ⎥⎦ sin2 θ

(19.2.79)

Hamilton’s function (19.1.81') is of the form (19.2.46) too, with pϕ2 − U ϕ (ϕ ) = a1 , 2m pθ2 gϕ g θ = g θ ( g ϕ ; θ , pθ ) = + − U θ ( θ ) = a2 , sin2 θ 2m g pr2 H = gr = gr ( g θ ; r , pr ) = θ2 + − U r (r ) = a 3 = h . 2m r gϕ = gϕ (ϕ , pϕ ) =

The generalized momenta will be 2m [U ϕ (ϕ ) + a1 ],

pϕ =

a1 + a2 ⎤ , 2m ⎡U θ ( θ ) − ⎣⎢ ⎦⎥ sin2 θ a pr = 2m ⎡U r (r ) + 22 + h ⎤ . ⎣⎢ ⎦⎥ r

pθ =

(19.2.79')

We calculate the partial derivatives p ∂g θ = θ = m ∂pθ

p 2 (U θ + a1 ) ∂g θ , = θ = m m ∂pθ

pr ∂gr = = m ∂pr

(

(

2 Uθ −

)

)

a1 + a2 sin2 θ , m

a2 +h ∂g 1 1 ∂gr r2 , θ = , = 2. 2 m a ∂a1 ∂ r sin θ 2

2 Ur −

As in the preceding case, the trajectory of the particle will be given by

∫ ∫

dϕ dθ − U ϕ (ϕ ) + a1 ∫ sin2 θ U ( θ ) − θ dθ U θ (θ ) −

a1 + a2 sin2 θ

−

∫

r2

= a1 + a 2 sin2 θ dr = a U r (r ) − 22 + h r

2 b , m 1 2 b , m 2

(19.2.79'')

the motion on this trajectory being specifies by

∫

dr a U r (r ) − 22 + h r

=

2 ( t + b3 ) . m

(19.2.79''')


196

In this case too, the problem is reduced to quadratures. If U = k / r , r > 0 (case of forces of Newtonian attraction), then Hamilton’s function is given by (19.1.81') and the Hamilton–Jacobi equation reads

( )

1 ⎡ ∂S S + 2m ⎣⎢ ∂r

2

+

( )

1 ∂S r 2 ∂θ

2

+

2 1 ⎛ ∂S ⎞ ⎤ − k = 0 . ⎜ ⎟ ⎥ r 2 sin2 θ ⎝ ∂ϕ ⎠ ⎦ r

(19.2.80)

As a matter of fact, this case is contained in the previously considered one. The generalized momenta will be (U ϕ = U θ = 0, U r = k / r , while ϕ is a cyclic co-ordinate) pϕ = pθ = pr =

2ma1 = const,

( sina θ ), k a 2m ( h + − . r r )

2 m a2 −

1 2

(19.2.80')

2 2

The trajectory of the particle is given by dθ

2 ϕ = − b , a1 a1 m 1 sin θ a2 − sin2 θ dθ dr 2 −∫ = b2 , a1 m + − r hr kr a 2 a2 − sin2 θ

∫

∫

2

(19.2.80'')

the motion on this trajectory being specified by

∫

rdr hr + kr − a2 2

=

2 ( t + b3 ) . m

(19.2.80''')

In the case of elliptic co-ordinates (see Sects. 18.3.2.1 and 19.1.3.1) let us suppose that the simple potential U is given by U =

U 1 (q1 ) U 2 (q2 ) U 3 (q 3 ) + + . (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

(19.2.81)

Taking into account Hamilton’s function (19.1.83) and the notations (18.2.37'), (18.3.39'), we can write the reduced Hamilton-Jacobi equation in the form 2

2

2

2 2 2 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ − U1 − U2 f (q1 ) ⎜ f (q2 ) ⎜ f (q 3 ) ⎜ ⎟ ⎟ ⎟ − U3 m m m ⎝ ∂q1 ⎠ ⎝ ∂q2 ⎠ ⎝ ∂q 3 ⎠ + + =h. (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

The identity


197

2a1 + 2a2q1 + hq12 2a1 + 2a2q2 + hq22 2a1 + 2a2q 3 + hq 32 + + =h (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

is easily verified for any a1 and a2 ; denoting 2S j (q j ) = 2a1 + 2a2q j + hq 2j , j = 1, 2, 3 ,

(19.2.82)

the Hamilton–Jacobi equation takes the form 2

2

2

2 2 2 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎛ ∂S ⎞ f (q1 ) ⎜ ⎟ − 2S1 m f (q2 ) ⎜ ∂q ⎟ − 2S 2 m f (q 3 ) ⎜ ∂q ⎟ − 2S 3 m ⎝ ∂q1 ⎠ ⎝ 2 ⎠ ⎝ 3 ⎠ + + = 0. (q1 − q2 )(q1 − q 3 ) (q2 − q 3 )(q2 − q1 ) (q 3 − q1 )(q 3 − q2 )

We find thus the complete integral

∫

S (q1 , q2 , q 3 ; a1 , a2 , h ) =

S1 (q1 ) dq1 + f (q1 )

∫

S 2 (q2 ) dq 2 + f (q2 )

∫

S 3 (q 3 ) dq 3 , f (q 3 )

which annuls each of the three ratios. The trajectory of the particle will be given by the equations

∫ ∫

d q1 S1 (q1 ) f (q1 ) q1 d q1 S1 (q1 ) f (q1 )

+

∫

+∫

dq 2 S2 (q2 ) f (q2 ) q 2 dq 2 S 2 (q2 ) f (q2 )

+

+

∫ ∫

dq 3 S 3 (q 3 ) f (q 3 ) q 3 dq 3 S 3 (q 3 ) f (q 3 )

= b1 ,

(19.2.82') = b2 ,

the motion on the trajectory being specified by the equation

∫

q12 dq1 S1 (q1 ) f (q1 )

+

∫

q22 dq2 S2 (q2 ) f (q2 )

+

∫

q 32 dq 3 S 3 (q 3 ) f (q 3 )

= 2 ( t + b3 ) .

(19.2.82'')

If U 1 = U 2 = U 3 = 0 , then one obtains the equations of a straight line in elliptic co-ordinates. In case of an electrized particle in an electromagnetic field (see Sect. 19.1.3.3), the formula (19.2.88) leads to the Hamilton–Jacobi equation 1 S + ( S,i − qAi m

)( S,i

− qAi

) + qA0

= 0.

(19.2.83)

19.2.3.2 Mechanical Systems with Two degrees of freedom In the case of a mechanical system with two degrees of freedom (e.g., the motion of a particle on a plane or on a surface – a holonomic constraint) the Hamilton–Jacobi equations reads


198

∂S ∂S ⎞ S + H ⎛⎜ q1 , q2 , , ;t = 0 . ∂q1 ∂q2 ⎟⎠ ⎝

(19.2.84)

By means of a complete integral S = S (q1 , q2 ; a1 , a2 ; t ) , the integrals of the canonical equations will be given by ∂S ∂S ∂S ∂S = b1 , = b2 , = p1 , = p2 . ∂a1 ∂a2 ∂q1 ∂q2

(19.2.84')

If the constraint is scleronomic, then we ca take S = − ht + S (q1 , q2 ; a1 , h ) , the reduced Hamilton–Jacobi equation being ∂S ∂S ⎞ ⎛ H ⎜ q1 , q 2 , , =h. ∂ q1 ∂q2 ⎟⎠ ⎝

(19.2.85)

The motion of the mechanical system is specified by ∂S ∂S ∂S ∂S = b1 , = t + b2 , = p1 , = p2 . ∂a1 ∂h ∂q1 ∂q2

(19.2.85')

The trajectory of the representative point (for various values of the constant b1 ) are normal to the curves S = const . In case of a heavy particle P , which is moving in a vertical plane, the reduced Hamilton-Jacobi equation reads 2 2 m ⎡ ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎤ +⎜ ⎢ ⎥ − gx 2 = h , ⎜ ⎟ ⎟ 2 ⎣ ⎝ ∂x1 ⎠ ⎝ ∂x 2 ⎠ ⎦

the Ox 1 -axis being horizontal, while the Ox 2 -axis is along the descendent vertical; because x1 is a cyclic co-ordinate, we search a solution of the form S = ax1 + ϕ (x 2 ) , being led to the equation m 2 ⎡a + ϕ ′2 (x 2 ) ⎤⎦ − gx 2 = h , 2⎣ 1

wherefrom S ( x1 , x 2 ; a1 , h ) = a1x1 +

∫

2 h − a12 + 2 gx 2 dx 2 .

The trajectory of the particle is given by the equation x1 − a1 ∫

dx 2 2h −

a12

+ 2 gx 2

= x1 −

a1 g

2 h − a12 + 2 gx 2 = b1 ,

(19.2.86)


199

being an are of parabola, tangent at the vertex to the straight line x 2 = (a12 − 2h )/ 2 g , and the motion on this trajectory is specified by

∫

dx 2 2h −

a12

+ 2 gx 2

=

1 2 h − a12 + 2 gx 2 = t + b2 . g

(19.2.86')

From (19.2.86), (19.2.86') one obtains x1 = a1 (t + b2 ) + b1 , the motion of the projection of the particle on the horizontal being uniform (see Sect. 7.1.2.1 too). The curves S = const are the semi-cubical parabolas a1x 1 +

1 3g

( 2h − a12

+ 2 gx 2

)3

= const ,

(19.2.86'')

which can be obtained one from the other by a translation along the Ox 1 -axis; these curves are normal to the straight line x 2 = (a12 − 2h )/ 2 g on which they have a cuspidal point. Varying the constant b1 , one obtains a family of parabolic trajectories (by a translation along the Ox1 -axis too), normal to the curves S = const , the general theoretical result being thus verified. In case of a plane motion of a particle, the Hamilton–Jacobi equation will of the form 2 2 ∂S ⎞ ⎤ 1 ⎡ ⎛ ∂S ⎞ S + + ⎛⎜ ⎜ ⎟ ⎟ ⎢ ⎥ − U (x1 , x 2 ; t ) = 0 , 2m ⎣ ⎝ ∂x1 ⎠ ⎝ ∂x 2 ⎠ ⎦

(19.2.87)

in Cartesian co-ordinates; if U = 0 , then we take S = − ht + S (x1 , x 2 ) , being led to the reduced Hamilton–Jacobi equation 2

2

⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂x ⎟ + ⎜ ∂x ⎟ = 2 m (U + h ) . ⎝ 1⎠ ⎝ 2 ⎠

(19.2.87')

A complete integral S = − ht + S (q1 , q2 ; a1 , h ) leads to the sequences of relations ∂S ∂S ∂S ∂S = b1 , = t + b2 , = p1 , = p2 . ∂a1 ∂h ∂x 1 ∂x 2

(19.2.87'')

The first equation specifies the trajectory of the particle in the plane Λ2 and the second equation corresponds to the motion on this trajectory; the last two equations complete the solution of the problem in the space Γ 4 . Let, e.g., be an elliptic oscillator, to which corresponds Hamilton’s function H =

1 k ( p 2 + p22 ) + 2 ( x12 + x 22 ) , 2m 1

(19.2.88)


200

where k is an elastic constant. We notice that this function is of the form (19.2.44), where g j = p 2j / 2m + kx 2j / 2, j = 1, 2 ; denoting g j = a j , a j = const it results kx 2j ⎞ ⎛ 2m ⎜ a j − ⎟ , j = 1, 2 . 2 ⎠ ⎝

pj =

(19.2.88')

We obtain the complete integral 2

∑∫

S = − ( a1 + a2 ) t +

j =1

kx 2j ⎞ ⎛ 2m ⎜ a j − ⎟ dx j . 2 ⎠ ⎝

The Hamilton–Jacobi theorem allows to write the relations dx j 1 = t + bj , ω = ∫ ω Aj2 − x 2j

k ,A = m j

2a j , j = 1, 2 . k

Effecting the quadrature, we get x j = Aj sin ω ( t + bj ) , j = 1, 2 ,

(19.2.88'')

remaining to put the initial conditions. We obtain the Hamilton–Jacobi equation

( )

1 ⎡ ∂S S + 2m ⎢⎣ ∂r

2

+

( ) ⎤⎥⎦ − U ( r , θ;t ) = 0 ,

1 ∂S r ∂θ

2

(19.2.89)

in polar co-ordinates ( q1 = r , q2 = θ ). If U = 0 , then we have S = − ht + S (r , θ ) and the reduced Hamilton–Jacobi equation is given by 2

2

1 ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂r ⎟ = 2 ⎜ ∂θ ⎟ = 2m (U + h ) . r ⎝ ⎝ ⎠ ⎠

(19.2.89')

Starting from the complete integral S = − ht + S (r , θ ; a1 , h ) , the trajectory and the motion on it will be given by ∂S ∂S = b1 , = t + b2 , ∂a1 ∂h

(19.2.89'')

and the generalized momenta by pr =

∂S ∂S ,p = . ∂r θ ∂θ

(19.2.89''')


201

If U (r , θ ) = U r (r ) +

1 U θ (θ ) , r2

(19.2.90)

then Hamilton’s function is of the form (19.2.46), with pθ2 − U θ ( θ ) = a1 , 2m g pr2 H = gr = gr (g θ ; r , pr ) = θ2 − U r (r ) + = a2 = h . 2m r g θ = g θ ( θ , pθ ) =

One obtains the generalized momenta pθ = pr =

2m [U θ ( θ ) + a1 ],

(19.2.90')

a 2m ⎡U r (r ) + 12 + h ⎤ . ⎢⎣ ⎥⎦ r

Taking into account the partial derivatives ∂g θ p = θ = ∂pθ m pr ∂gr = = m ∂pr

(

2 (U θ + a1 ) , m

)

a1 +h ∂g 1 r2 , 1 = 2, m ∂a1 r

2 Ur −

the formulae (19.2.47') give the trajectory of the particle P in the form

∫

dθ dr −∫ = a1 2 U θ ( θ ) + a1 r U r (r ) − 2 + h r

2 b , m 1

(19.2.90'')

and the formula (19.2.47'') specifies the motion on this trajectory by the equation

∫

dr a U r (r ) − 12 + h r

=

2 ( t + b2 ) . m

(19.2.90''')

The problem is thus reduced to quadratures. If U = k / r , k > 0 (case of forces of Newtonian attraction), then the Hamilton-Jacobi equation is written in the form

( ) + ( r1 ) ( ∂∂Sθ ) ⎤⎥⎦ − kr = 0 .

1 ⎡ ∂S S + 2m ⎢⎣ ∂r

2

2

2

(19.2.91)


202

As a matter of fact, this case is contained in the previously considered one. The generalized momenta are (U θ = 0,U r = k / r , k > 0 , and θ is a cyclic co-ordinate) pθ =

2ma1 ,

(

(19.2.91')

)

k a 2m h + − 12 . r r

pr =

The trajectory of the particle is given by

∫r

dr

hr + kr − a1 2

=

θ − a1

2 b , m 1

(19.2.91'')

the motion on the trajectory being specified by

∫

rdr hr + kr − a1 2

=

2 ( t + b2 ) . m

(19.2.91''')

Let us suppose that the particle P lies on a surface S for which the element of arc is given by a relation of the form

⎛ dq 2 dq 2 ⎞ ds 2 = ( Q1 + Q2 ) ⎜ 1 + 2 ⎟ , Pj = Pj (q j ),Q j = Q j (q j ), j = 1, 2 . P2 ⎠ ⎝ P1

(19.2.92)

In this case, one obtains the kinetic energy T =

1 ( Q + Q2 2 1

⎛ q12 q22 ⎞ P1 p12 + P2 p22 + . ⎟= ⎝ P1 P2 ⎠ 2 ( Q1 + Q2 )

)⎜

(19.2.92')

If the particle is subjected to no one given force (U = 0 ), then it results H = T and we can apply Liouville’s theorem. By separation of variables, hence by quadratures, one can obtain thus the possible trajectories (depending on the integration constants); in fact, thus are specified the geodesic lines of the surface. The surfaces which have this property are called Liouville’s surfaces. 19.2.3.3 Problem of Two Particles. Motion of Planets The problem of two particles has been considered in Sects. 19.3.2.3 and 18.1.3.2, assuming the action of forces of Hamiltonian attraction. The generalized system has three degrees of freedom and we can choose the generalized co-ordinates q1 = r , q2 = θ , q 3 = ϕ ; assuming that we study the relative motion of a particle with respect to the other one, we take the latter one as origin of the frame of reference. We can thus use the results obtained in Sect. 19.2.3.1 for the motion of a free particle (in spherical co-ordinates – the formulae (19.2.75'), (19.2.75''), and (19.2.75''')).


203

We assume that the initial position of the particle in on the line of nodes ON , in the plane Ox1x 2 (Fig. 19.1); hence, for t = t0 we have θ = θ0 = π / 2, ϕ = ϕ0 . Imposing these conditions, it results b3 = −t0 , b1 =

m ϕ . 2a1 0

(19.2.93)

As far as we know, the trajectory of the particle is an ellipse of semi-major axis a and eccentricity e ; in this case, rmin = a (1 − e ) and rmax = a (1 + e ) . By differentiation of the equation (19.2.75'''), we obtain m dr r = 2 dt

hr 2 + kr − a2 .

The extreme values of the vector radius r must annul the quantity under the radical. From Viète’s relations, it results rmin + rmax = 2a = − k / h , rmin rmax = a 2 (1 − e 2 ) = −a2 / h , wherefrom h =

k −k < 0, a2 = a ( 1 − e 2 2a 2

)=

k p > 0. 2

(19.2.93')

Fig. 19.1 Motion of the Sun and of a planet

We notice that for a given colatitude θ one must have a2 − a1 / sin2 θ ≥ 0 , hence

θ ≥ arcsin a1 / a2 ; but θmin is just the complement of the angle i made by the plane of the orbit with the plane Ox1x 2 . Hence, a1 = a2 cos2 i =

k p cos2 i . 2

(19.2.93'')


204

The trajectory will be given by (for θ we take the inferior limit π / 2 , corresponding to a point on the line of nodes, while for r we take the inferior limit rmin = a (1 − e ) , corresponding to the pericentre π ) θ

cos i dϑ

π /2

sin ϑ sin2 ϑ − cos2 i

∫

1 p − a

= ϕ − ϕ0 ,

sin ϑ dϑ

θ

∫

π / 2 sin ϑ

(19.2.94)

sin2 ϑ − cos2 i

r

dρ = ρ (1 ) ρ ][ ρ − a (1 − e ) ] a e + − [ a (1 −e )

∫

k b , m 2

while the motion on the trajectory will be specified by r

a

ρ dρ = [ a (1 + e ) − ρ ][ ρ − a (1 − e ) ]

∫

a (1 −e )

k ( t − t0 ) , m

(19.2.94')

At the moment in which the particle attains the pericentre π , the second equation (19.2.94) becomes θπ

∫

π /2

sin ϑ dϑ

sin ϑ dϑ

θπ

sin2 ϑ − cos2 i

=

∫

π /2

sin2 i − cos2 ϑ

= −χπ .

with χ π = − p k / mb2 . By integrating, we get cos θπ = χπ , sin i

(19.2.93''')

cos θπ = sin i sin χ π .

(19.2.93IV)

arcsin

wherefrom

Considering the relations between the elements of the spherical triangle N ππ ′ on a sphere with the centre O and of unit radius (the point π is the piercing point of the p π . These results correspond to those radius with this sphere), we see that χ = N π

obtained in Sect. 9.2.2.4 in the study of the orbit of a satellite. As a matter of fact, we can imagine that at the pole O is situated the Sun, the particle P being a planet; in this case, the plane Ox1x 2 is the plane of the ecliptic, while the point π is the perihelion. Thus, all six parameters ( a , e , i , ϕ0 , t0 and χ π ) which define the orbit of the planet are determined. We notice that


205

dρ ∫ ρ [ a (1 + e ) − ρ ][ ρ − a (1 − e ) ] = a (1 −e ) r

ap

dρ

r

ap

∫

a (1 −e ) ρ

− ρ + 2a ρ − ap 2

r

= arcsin

−2ap + a ρ π r −p r −p = arcsin = − − arcsin er er 2 ρ 4a (a − p ) a (1 −e ) =−

( π2 − arcsin per− r ) ,

as well as sin ϑ dϑ

θ

∫

π /2

sin ϑ − cos i 2

2

= arcsin

cos θ = −χ , sin i

p corresponds to an arbitrary position on the orbit (the point P is where the arc χ = NP the intersection of the radius OP with the sphere of unit radius); hence,

χ − χπ = −

( π2 − arcsin per− r ) ,

so that cos ( χ − χ π ) = cos

( π2 − arcsin per− r ) = per− r .

Finally, we find the known equation of the trajectory in its plane r =

p , 1 + e cos ( χ − χ π )

(19.2.95)

corresponding to the equation (9.2.6). Starting from the equation (19.2.94') and using the constant h given by (19.2.93'), we find again the integral given in Sect. 9.2.1.3, which leads to Kepler’s equation (9.2.13). The first equation (19.2.94) specifies the motion in the plane of the ecliptic. Knowing that the trajectory is a plane curve, the problem can be treated in another way too, choosing the plane of the trajectory as plane Ox1x 2 . Considering, further, the relative motion of a planet with respect to the Sun, we can use polar co-ordinates as in Sect. 19.2.3.2 (the mechanical system formed by the two particles has two degrees of freedom in the plane of the motion); the Hamilton-Jacobi equation (19.2.91) leads thus to the classical equation of the trajectory. But we can make also a study in orthogonal Cartesian co-ordinates, after Jacobi. The reduced Hamilton–Jacobi equation will be 2

2

(

)

k ⎛ ∂S ⎞ ⎛ ∂S ⎞ ⎜ ∂x ⎟ + ⎜ ∂x ⎟ = 2m r + h . ⎝ 1⎠ ⎝ 2 ⎠

(19.2.96)


206

Taking an arbitrary point P0 on the Ox 1 -axis, so that OP0 = r0 , and denoting

P0 P = ρ , we have r =

x12 + x 22 , ρ =

( x1 − r0 )2 + x 22 .

(19.2.97)

We will show that the function S =

σ

∫σ ′

k h + ds , s + r0 2

(19.2.96')

where σ ′ = r − ρ, σ = r + ρ

(19.2.97')

is a complete integral of the equation (19.2.96), r0 being an essential constant. Indeed, observing that σ and σ ′ depend on x1 and x 2 by means of r and ρ , it results x x −r ⎞ x x −r ⎞ h⎞ h⎞ ∂S ⎛ k ⎛ k m⎜ m⎜ = ⎜⎛ 1 + 1 + ⎟ − ⎜⎛ 1 − 1 + ⎟, ⎟ ⎟ ′ 2⎠ ⎝ r 2⎠ ρ ⎠ ρ ⎠ ∂x1 ⎝ r ⎝ σ + r0 ⎝ σ + r0 x x x x h⎞ h⎞ ∂S ⎛ k ⎛ k = ⎜⎛ 2 + 2 ⎟⎞ m ⎜ + ⎟ − ⎜⎛ 2 − 2 ⎟⎞ m ⎜ + ⎟, ′ r r r r 2 2⎠ ρ σ ρ σ ∂x 2 + + ⎝ ⎠ ⎠ ⎝ ⎠ ⎝ ⎝ 0 0

wherefrom, using the notation (19.2.97), we find (the terms which contain the product of the two radicals disappear) 2 x1 (x1 − r0 ) + 2 x 22 ⎤ ⎛ k h ⎡ ⎛ ∂S ⎞ ⎛ ∂S ⎞ m 2 + = + + ⎞⎟ ⎜ ⎜ ∂x ⎟ ⎜ ∂x ⎟ ⎢ ⎥ rρ ⎝ 1⎠ ⎝ 2 ⎠ ⎣ ⎦ ⎝ σ + r0 2 ⎠ 2 2 x (x − r0 ) + 2 x 2 ⎤ ⎛ k h⎞ ⎡ +m ⎢ 2 − 1 1 ⎥ ⎜ σ′ + r + 2 ⎟ . r ρ ⎠ 0 ⎣ ⎦⎝ 2

2

Taking into account the identities 2+

2 x1 (x1 − r0 ) + 2 x 22 σ 2 − r02 2r ρ + r 2 + ρ2 − r02 = = , rρ rρ rρ

2−

2 x1 ( x1 − r0 ) + 2 x 22 2r ρ − r 2 − ρ 2 + r02 σ ′2 − r02 = = , rρ rρ rρ

one sees that the equation (19.2.96) is verified. The equation of the trajectory and the motion on it will be given by ∂S / ∂r0 = K , K = const, ∂S / ∂h = t − t0 . We notice that x − r0 ∂σ ′ x − r0 ∂ρ ∂ρ ∂σ = =− 1 =− = − 1 , , ∂r0 ∂r0 ∂r0 ∂r0 ρ ρ


207

so that the first equation leads to −

x1 − r0 ρ

x − r0 k h m ⎛⎜ + ⎞⎟ − 1 ρ ⎝ σ + r0 2 ⎠ mk ds σ

−∫

σ′

k h m ⎛⎜ + ⎞⎟ ⎝ σ ′ + r0 2 ⎠

k h 2 ( s + r0 ) m ⎛⎜ + ⎞⎟ s r 2⎠ + ⎝ 0

=K.

Effecting the quadrature, we can also write x1 − r0 ⎞ x1 − r0 ⎞ h⎞ ⎛ h⎞ ⎛ ⎛ k ⎛ k ⎜1 − ⎟ m⎜ σ + r + 2 ⎟ − ⎜1 + ⎟ m ⎜ σ′ + r + 2 ⎟ = K . ρ ρ ⎝ ⎠ ⎠ ⎝ ⎠ ⎝ ⎝ ⎠ 0 0

Because 2r0 (x1 − r0 ) = r 2 − r02 − ρ2 , it results 1−

x1 − r0 2r ρ − r 2 + r02 + ρ2 ( σ + r0 ) ( σ ′ − r0 ) = 0 =− , ρ 2 ρr0 2 ρr0

1+

x1 − r0 2r ρ + r 2 − r02 − ρ 2 ( σ − r0 ) ( σ ′ + r0 ) = 0 = − , ρ 2 ρr0 2 ρr0

obtaining thus the equation of the trajectory in bipolar co-ordinates (of poles O and P0 ) in the form

( σ + r0 ) ( σ ′ − r0 )

k h k h + + ( σ − r0 ) ( σ ′ + r0 ) + = K′, σ + r0 2 σ ′ + r0 2 (19.2.98)

where K ′ = 2 ρr0 K / m is a new integration constant. We have ρ = 0 , r = r0 for the point P0 , hence σ = σ ′ = r0 ; taking into account the form of the constant K ′ , we see that the trajectory, which is a conic, passes through the point P0 . The genus of the conic depends on the sign of the constant h ; in case of the Solar system we have h < 0 , the conic being an ellipse. The motion on the orbit will be specified by t − t0 =

1σ 4 σ∫′

ds , k h⎞ ⎛ + ⎟ m⎜ ⎝ s + r0 2 ⎠

(19.2.98')

beginning from the point P0 for which t = t0 . If the orbit is parabolic ( h = 0 ), then the integration is immediate, obtaining Euler’s formula t − t0 =

1 ⎡ ( r + ρ + r0 )3 − 6 mk ⎣

( r − ρ + r0 )3 ⎤⎦ .

(19.2.98'')


208

In the case in which h ≠ 0 (elliptic or hyperbolic orbits), analogous formulae have been given by Gauss. 19.2.3.4 Problem of the Two Centres We take again the problem of two centres, considered in Sect. 18.3.2.4. The kinetic energy is given by (18.3.55) and the simple potential reads (18.3.55'), obtaining Hamilton’s function μ2 ⎞ m 2 ⎛ λ2 λ − μ2 ) ⎜ 2 + 2 ( ⎟ 2 2 c − μ2 ⎠ ⎝λ −c 1 − 2 [ ( k1 + k2 ) λ − ( k1 − k2 ) μ ] , λ − μ2

H =

(19.2.99)

where λ and μ are elliptic co-ordinates in the plane; m is the mass of the particle, k1 and k2 characterize the centres of attraction (in inverse proportion to the squares of the distances), the distance between these centres being 2c . We find the generalized momenta p1 = m

( λ2

− μ2 ) λ

λ2 − c 2

, p2 = m

( λ2

− μ2 ) μ

c 2 − μ2

,

(19.2.100)

so that Hamilton’s function becomes H =

1 m ( λ2 − μ2 ) ⎣⎡ ( λ2 − c 2 ) p12 + ( c 2 − μ2 ) p22 ⎦⎤ 2 1 − 2 [ ( k1 + k2 ) λ − ( k1 − k2 ) μ ] . λ − μ2

(19.2.99')

This Hamiltonian is of the form (19.2.51), hence of Liouville’s type, and one can make the separation of the variables. Thus, we obtain – obviously – the same relations (18.3.56), (18.3.56'), the trajectory and the motion on the trajectory being obtained by quadratures (elliptic integrals). The relations (19.2.100) complete then the solution in the phase space. This problem has been considered for the first time by Euler (in the plane case); it has been taken again in a three-dimensional treatment by Lagrange and then by Jacobi, who integrated it by his general method of calculation. This research has been an important impetus to enlarge the theory of elliptic integrals. We mention also the results obtained by Serret, Desboves and Andrade in this direction. G. Dincă studied, in 1965, the general motion of a particle subjected to the action of n central forces (which pass through n fixed centres), in a resistant medium; the problem is reduced to the above considerations, in some interesting particular cases. These results can be seen also as a first step in the study of the problem of three particles, which can be solved – in general – only by approximate methods of calculation; besides, the results can be used in the general problem of n particles too.


209

19.2.3.5 Motion of the Rigid Solid The Hamilton–Jacobi method can be applied, as well, in the case of the free or the constrained rigid solid S . As in Sect. 19.1.3.5, we will consider a heavy, homogeneous rigid solid of rotation, of mass M , which slides frictionless on a fixed horizontal plane, having five degrees of freedom. Starting from Hamilton’s function (19.1.96), one obtains the canonical equations (19.1.96'). The generalized co-ordinates are ρ1′ , ρ2′ (which specify the position of the mass centre) and Euler’s angles ψ, θ and ϕ ; the generalized momenta p1 , p2 , pψ , pθ and pϕ are corresponding to them, I 3 and J are the moments of inertia with respect to the axis of rotation and to an axis normal to this one (with respect to the movable frame of reference R ), respectively, while the function f ( θ ) specifies the applicate of the mass centre with respect to the fixed plane.

The Hamilton–Jacobi equation reads 1⎧ 1 S + ⎨ 2 ⎩M

( )

⎡ ⎛ ∂S ⎞2 ⎛ ∂S ⎞2 ⎤ ∂S 1 ⎢ ⎜ ∂ρ ′ ⎟ + ⎜ ∂ρ ′ ⎟ ⎥ + 2 ⎝ 2 ⎠ ⎦ Mf ′ ( θ ) + J ∂θ ⎣⎝ 1 ⎠ +

2

+

1 ⎛ ∂S ⎞2 ⎜ ⎟ I 3 ⎝ ∂ϕ ⎠

2 1 ⎛ ∂S − cos θ ∂S ⎞ ⎫ + Mgf ( θ ) = 0 . ⎬ ⎜ ⎟ ∂ϕ ⎠ ⎭ J sin2 θ ⎝ ∂ψ

(19.2.101)

Because the constraints are scleronomic and the generalized co-ordinates ρ1′ , ρ2′ , ψ and ϕ are cyclic, we can choose a complete integral of the form S = − ht + a1 ρ1′ + a2 ρ2′ + a 3 ψ + a 4 ϕ + F ( θ ) ,

where h , a1 , a2 , a 3 and a 4 are integration constants. Replacing in the equation (19.2.101), we get the equation −h + +

( )

1⎡ 1 2 1 ∂F ( a + a22 ) + ′2 2 ⎢⎣ M 1 Mf ( θ ) + J ∂θ

2

1 2 1 a + ( a − a 4 cos θ )2 ⎥⎤ + Mgf ( θ ) = 0 , I 3 4 J sin2 θ 3 ⎦

which determines the function F ( θ ) ; we obtain thus the complete integral S = − ht + a1 ρ1′ + a2 ρ2′ + a 3 ψ + a 4 ϕ +

∫

Mf ′2 ( θ ) + J Θ ( θ ) dθ ,

(19.2.102)

where we have denoted Θ ( θ ) = 2 [ h − Mhf ( θ ) ] −

a12 + a22 a 42 ( a 3 − a 4 cos θ )2 − − , M I3 J sin2 θ

the five constants being essential constants.

(19.2.102')


210

Differentiating the function S with respect to the initial constants, we find the equations of the trajectory of the representative point P ∈ Λ5 in the form ρ1′ −

a1 M

∫

Mf ′2 ( θ ) + J Θ (θ ) ψ−

ϕ−

∫

dθ = b1 , ρ2′ −

a2 M

Mf ′2 ( θ ) + J

∫

Θ(θ )

dθ = b2 ,

Mf ′2 ( θ ) + J a 3 − a 4 cos θ 1 dθ = b3 , ∫ J sin2 θ Θ(θ )

(19.2.103)

Mf ′2 ( θ ) + J ⎡ a 4 cos θ − ( a 3 − a 4 cos θ ) ⎤⎥ dθ = b4 , 2 ⎢ I Θ(θ ) ⎣ 3 J sin θ ⎦

the motion along this trajectory being specified by the equation t − t0 =

∫

Mf ′2 ( θ ) + J Θ(θ )

dθ .

(19.2.103')

Replacing the integral (19.2.103') in the first two equations (19.2.103), we obtain ρ ′ − b1 =

a1 a ( t − t0 ) , ρ ′ − b2 = 2 ( t − t0 ) ; M M

(19.2.103'')

we find thus again a known result, in accordance to which the projection of the mass centre on the fixed plane has a rectilinear and uniform motion. The components of the generalized momentum will be p1 = a1 , p2 = a2 , pψ = a 3 , pϕ = a 4 , pθ =

Mf ′2 ( θ ) + J

Θ(θ ) .

(19.2.103''')

Let be also the problem of the plane motion of a rigid straight bar, of mass M , attracted by elastic forces (of elastic constant k ) by the Ox1 -axis, considered in Sects. 18.3.2.7 and in 19.1.3.5. Hamilton’s function is given by (19.1.97), where ρ1′ and ρ2′ (the co-ordinates of the mass centre) and the angle of rotation θ are the generalized co-ordinates, while I is the central moment of inertia with respect to an axis normal to the plane of motion (see Fig. 18.12 too). The Hamilton-Jacobi equation will be

( )

1 ⎧ 1 ⎡ ∂S ⎞2 ⎛ ∂S ⎞2 ⎤ 1 ∂S S + ⎨ ⎢ ⎛⎜ +⎜ ⎟ ⎥+ 2 ⎩ M ⎣ ⎝ ∂ρ1′ ⎟⎠ ⎝ ∂ρ2′ ⎠ ⎦ I ∂θ

2

⎫ + k ( M ρ2′2 + I sin2 θ ) ⎬ = 0. ⎭ (19.2.104)

The constraints are scleronomic and ρ1′ is a cyclic co-ordinate, so that the complete integral is of the form S = − ht + a1 ρ1′ + S1 ( ρ2′ ) + S 2 ( θ ) .


211

Replacing in the equation (19.2.104), we obtain −h +

1⎧ 1 ⎨ 2 ⎩M

⎡ 2 ⎛ ∂S1 ⎞2 ⎤ 2 ⎫ 2 ⎢ a1 + ⎜ ∂ρ ′ ⎟ ⎥ + kM ρ2′ ⎬ = −a2 , ⎝ ⎠ 2 ⎣ ⎦ ⎭ 1 ⎡ 1 ⎛ ∂S 2 ⎞2 2 ⎤ 2 ⎜ ⎟ + kI sin θ ⎥ = a2 , 2 ⎢⎣ I ⎝ ∂θ ⎠ ⎦

where a2 is the third essential constant; there results thus the complete integral S = − ht + a1 ρ1′ +

∫

R ( ρ2′ ) dρ2′ +

∫

Θ ( θ ) dθ ,

(19.2.105)

where we have denoted

R ( ρ2′ ) = 2 M ( h − a22 ) − a12 − kM 2 ρ2′2 , Θ ( θ ) = I ( 2a22 − kI sin2 θ ) .

(19.2.105')

The trajectory of the representative point P ∈ Λ3 is given by the equation dρ2′

ρ1′ − a1 ∫

∫

dρ2′ R ( ρ2′ )

−

I M

R ( ρ2′ )

∫

= b1 ,

(19.2.106)

b dθ = − 2 , 2 Ma2 Θ(θ )

while the motion on the trajectory is specified by 2M ∫

dρ2′ R ( ρ2′ )

= t − t0 .

(19.2.106')

Eliminating the integral between the first equation (19.2.106) and the equation (19.2.106'), we get ρ1′ =

a1 ( t − t0 ) + b1 , 2M

(19.2.107)

hence the projection of the mass centre on the Ox1 -axis has a uniform motion. The integral with respect to ρ2′ leads to ρ2′ = A cos ω ( t − t0 ) , A =

1 M k

2 M ( h − a22 ) − a12 , ω =

1 k; 2

(19.2.107')

the constants a1 and a2 must verify the positiveness of the quantity under the radical (we have k > 0, h > 0 too). The integral with respect to θ is an elliptic one. Obviously, these results correspond to those in Sect. 19.1.3.5, where one eliminates the

212


generalized momenta p2 and pθ between (19.1.97) and (19.1.97') ( θ verifies the equation of the mathematical pendulum); moreover, the respective formulae allow then to calculate these generalized momenta.

Chapter 20 Variational Principles. Canonical Transformations We will present first the main variational principles, which allow a formulation of the motion problems of mechanical systems in the configuration space Λs , as well as in the phase space Γ 2s . Then, we consider the canonical transformations which let invariant the equations of motion in the space Γ 2s , as well as the variational principles from which they are deduced (their canonical form). A special attention is given to Noether’s theorem, obtaining thus first integrals for the equations of motion.

20.1 Variational Principles The differential principles of mechanics (with a local character) have been presented in Sect. 6.2.1.4 for a particle and in Sect. 11.1.2.10 for a discrete mechanical system of particles. New differential principles, in a global form, have been introduced in Sects. 12.1.1.2 and 20.1.2 for continuous mechanical systems and in Sects. 14.1.1.10 and 20.1.2.1 for the free rigid solid and for the rigid solid with constraints, respectively. In Sect. 18. the principles of Newton, d’Alembert, Gauss and Hertz are stated in the space E 3n , as well as various forms of the principle of virtual work (d’Alembert-Lagrange, Jourdain, Gauss a.s.o.) in the space Λs (Newton, I., 1686–1687; Alembert d’, 1743; Hertz, H., 1894). Unlike the differential principles, the variational principles of mechanics have a global character – being integral principles; they do not reply to certain causality conditions and do not correspond to an actual (present) state of the mechanical system, but contain in them the past as well the future of this one. In connection with the variational principles, we can mention the names of W.R. Hamilton, M.V. Ostrogradskiĭ, J.-L. Lagrange, P.-L. Moreau de Maupertuis, K.-G.-J. Jacobi, O. Hölder, A. Voss etc (Hamilton, W.R., 1890; Ostrogradskiĭ, M.V., 1946; Lagrange, J.-L., 1788; Jacobi, C.G.J., 1882; Voss, A., 1901). We notice that the basic principles of Newtonian mechanics (see Sect. 1.1.2.5) have a physical, mechanical meaning, their formulation being thus adequate. In the frame of Lagrangian and of Hamiltonian mechanics we are led to an analytical formulation of those principles, with a tendency of universality, even if we are situated – sometimes – in the frame of particular mechanical systems; obviously, this analytical formulation has also a pregnant mechanical substratum, even this one is not evident from the very beginning. The analytical principles of mechanics try, in general, to contain – besides the laws of Newtonian mechanics – also other natural laws (corresponding to a thermal

P.P. Teodorescu, Mechanical Systems, Classical Models, © Springer Science+Business Media B.V. 2009

213


214

or electromagnetic field, or to some optical and acoustical phenomena a.s.o.); in fact, this is one of the tendencies of actual physics: unification of mechanical and physical phenomena in a unitary analytical formulation. The model used for these formulations is that of Fermat’s principle, governing geometrical optics, the variational method of calculus being essential. These analytical principles can have a differential character (as the principle of virtual work) or an integral character; in the last case, we study the behaviour of the mechanical system as a whole. We apply certain variations to the trajectories of the particles which constitute the mechanical system (or to the trajectory of the representative point), defining thus varied (admissible) paths; to a point on the true (real) trajectory will correspond, univocally, a point on the varied path. If the chronology of the motion is the same on both trajectories, then the variation is synchronous; otherwise, it is asynchronous. We pass in review some results having a general character (from a mathematical point of view, as well as concerning variational principles). We present then Hamilton’s and Maupertuis’s principles, and some applications of them to continuous mechanical systems; we consider also other variational principles in the frame of a general theory.

20.1.1 Mathematical Preliminaries We present in what follows some preliminaries having a mathematical character, concerning variational calculus, useful for the general form which can have the corresponding principles; the synchronous form as well as the asynchronous form can be put thus into evidence. 20.1.1.1 General Considerations In Sect. 7.2.1.4 we have made concise considerations concerning equations leading to the extremal curves in case of some particular functionals. In case of variational principles of mechanics, we take again the problem for the functionals of the form I ( x1 , x 2 ,..., xn ) ≡

t1

∫t

F ( t ; x1 ( t ) , x 2 ( t ) ,..., xn ( t ) , x1 ( t ) , x 2 ( t ) ,..., xn ( t ) ) dt ,

0

(20.1.1) where x j = x j ( t ) , j = 1, 2,..., n , are the parametric equations of a curve C in \n ; the co-ordinates x j can be, in particular, the co-ordinates of a particle in E 3 , the coordinates of a discrete mechanical system in E 3n , or the co-ordinates of a representative point in Λs or in Γ 2s . Corresponding to the definition of a functional, to every point P of C corresponds a real number I ( x1 , x 2 ,..., xn ) , by the integral (20.1.1); we admit that the functions x j ( t ) are of class C 2 on the interval [ t0 , t1 ] , the function F being also of class C 2 with respect to its arguments. In the basic problem one must determine the functions x j ( t ) for which I realizes an extremum (maximum or minimum). By convention, the geometric image of the functions x j ( t ) is called a real (true) curve C; starting from the point P of the curve C, we will define the point P ′ , hence

Variational Principles. Canonical Transformations

215

the varied path C ′ , using functions of the form x j′ = x j′ ( t ; ε ) , t ∈ [ t0 , t1 ] , of class

C 2 , the parameter ε varying in an interval containing the origin. We suppose that, along these varied paths, the conditions imposed in the problem, corresponding to the real (natural) curve, are not violated; if the latter one is obtained for ε = 0 , then we take x j′ ( t ;0 ) = x j ( t ) , so that x j′ ( t ) = x j ( t ) + εδx j ( t ) , j = 1, 2,..., n ,

(20.1.2)

where the variations δx j ( t ) are functions of class C 2 , while ε is a small parameter. It is convenient to assume that all varied paths C ′ are with fixed ends (they have the same origin P 0 and the same extremity P 1 ( x j′ ( t0 ) = x j ( t0 ) , x j′ ( t1 ) = x j ( t1 ) ) (see Fig. 20.1); hence, δx j ( t0 ) = δx j ( t1 ) = 0 , j = 1, 2,..., n .

(20.1.2')

If the points P and P ′ are in motion on the curve C and on the path C ′ , respectively, with the same chronology (the same time t ), we say that we have to do with a synchronous variation (denoted by δx j ); otherwise, the variation is asynchronous (denoted by Δx j ), the corresponding chronology being specified by the time variable

Fig. 20.1 Varied paths with fixed ends

t ′ = t + εΔt ,

(20.1.2'')

where Δt is a function of class C 2 , of time t ; we can write

x j′ ( t ′ ) = x j ( t ) + εΔx j ( t ) , j = 1, 2,..., n .

(20.1.2''')

Obviously, for Δt = 0 we obtain a synchronous variation. 20.1.1.2 Case of Synchronous Variations Introducing the variation of the function F , we can write F ( t ; x j′ , x j′

)

= F ( t ; x j , x j ) + εδF in case of a synchronous variation; but a development into a Taylor series allows us to write


216 F ( t ; x j′ , x j′

n

⎛ ∂F

) = F ( t ; x j , x j ) + ε ∑ ⎜ ∂x j =1 ⎝

δx j +

j

∂F ⎞ δx + O ( ε2 ) , ∂x j j ⎟⎠

so that, using both expressions, we obtain

δF =

∂ F ( t ; x j′ , x j′ ) . ∂ε ε =0

(20.1.3)

It results also

δI ( x1 , x 2 ,..., xn ) =

∂ I ( x1′ , x 2′ ,..., xn′ ) . ∂ε ε =0

(20.1.3')

Observing that δ I ( x1 , x 2 ,..., xn ) = δ ∫ F ( t ; x j ( t ) , x j ( t ) ) dt t1

t0

=

∂ t1 F ( t ; x j′ ( t ) , x j′ ( t ) ) dt ∂ε ∫ t0

ε =0

=

t1

∫t

0

∂ F ( t ; x j′ ( t ) , x j′ ( t ) ) dt , ∂ε ε=0

we have δ I ( x1 , x 2 ,..., x n ) =

t1

∫ δF ( t ; x j ( t ) , x j ( t ) ) dt .

(20.1.3'')

t0

If the functional

I ( x1 , x 2 ,..., x n )

has an extremum, then the function

I ( x1′ , x 2′ ,..., x n′ ) , considered to be a function of the parameter ε (denoted f ( ε ) ), takes an extreme value for ε = 0 . If f (0) is an extremum, then – necessarily – f ′(0) = ∂I ( x1′ , x 2′ ,..., xn′ ) / ∂ε ε = 0 = 0 for any variation δx j ; taking into account

(20.1.3'), there results δ I ( x1 , x 2 ,..., x n ) = 0 .

(20.1.4)

We can thus state Theorem 20.1.1 If the functional I ( x1 , x 2 ,..., xn ) has an extremum for x j = x j ( t ) , then – necessarily – the relation (20.1.4) takes place for any virtual variation δx j ,

j = 1, 2,..., n , of the accepted class (20.1.2). If the condition (20.1.4) is verified, then we will say – by definition – that the functional I ( x1 , x 2 ,..., xn ) is stationary on the varied paths obtained starting from the functions x j = x j ( t ) , j = 1, 2,..., n . To can associate a sufficient condition of extremum, one takes into account also the variation of second order δ 2 I ( x1 , x 2 ,..., xn ) = ∂ 2 I ( x1′ , x 2′ ,..., x n′ ) / ∂ε2 ε = 0 . One can


217

prove that to realize a maximum or a minimum for the functional I ( x1 , x 2 ,..., xn ) it is sufficient to have δ 2 I ( x1 , x 2 ,..., xn ) < 0 or, δ 2 I ( x1 , x 2 ,..., xn ) > 0 , respectively, for any variation δx j ( t ) , j = 1, 2,..., n . If we differentiate the relation (20.1.2) with respect to time, then we obtain x j′ = x j + εdδx j / dt , so that δx j =

d δx ; dt j

(20.1.5)

there results the operator relation δ

d d = δ, dt dt

(20.1.5')

which takes place only in case of synchronous variations. This permutability relation is replaced by a transitivity relation ( δ d − dδ ≠ 0 ) in case of asynchronous variations. Let f ( t ) be a continuous function on the interval [ t0 , t1 ] , with real values. One supposes that there exists τ ∈ ( t0 , t1 ) , so that f ( τ ) ≠ 0 (to fix the ideas, f ( τ ) > 0 ); there exists an interval ( τ 0 , τ1 ) ⊂ [ t0 , t1 ] for which τ ∈ ( τ 0 , τ1 ) , and f ( t ) > 0 if t ∈ [ τ 0 , τ1 ] . Let also be a function η ( t ) of class C 1 on the interval [ t0 , t1 ] , with real values, defined by ⎧ 0 ⎪⎪ η ( t ) = ⎨ ( t − τ 0 )2 ( t − τ1 )2 ⎪ ⎪⎩ 0

for t0 ≤ t < τ 0 , for τ 0 ≤ t ≤ τ1 , for τ1 < t ≤ t1 .

We have t1

∫t

f ( t ) η ( t ) dt =

0

τ1

∫τ

f ( t ) ( t − τ 0 )2 ( t − τ1 )2 dt > 0 ;

0

by reducing ad absurdum, we can state Theorem 20.1.2 (the basic lemma of variational calculus; J.-L. Lagrange). Let f : [ t0 , t1 ] → \ be a continuous function. If t1

∫t

f ( t ) η ( t ) dt = 0

(20.1.6)

0

for any function η : [ t0 , t1 ] → \ of class C 1 , vanishing at the ends of the interval ( η ( t0 ) = η ( t1 ) = 0 ), then f ( t ) = 0 on the interval [ t0 , t1 ] . This result takes place also for η ∈ C 2 . In case of synchronous variations, the relation (20.1.3'') leads to


218 δI ( x1 , x 2 ,..., x n ) =

t1 n

⎛ ∂F

δx j ⎜ ∫t ∑ j = 1 ⎝ ∂x j

+

0

∂F ⎞ δx dt , ∂x j j ⎟⎠

while the relation (20.1.5) allows to write (integration by parts) δI ( x1 , x 2 ,..., x n ) =

∂F ∑ ∂x j δx j j =1 n

t1

+

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j

−

0

t0

d ⎛ ∂F ⎞ ⎤ δx d t ; dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.7)

taking into account the conditions (20.1.2'), there results δI ( x1 , x 2 ,..., x n ) =

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j 0

−

d ⎛ ∂F ⎞ ⎤ δx dt . dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.7')

Imposing the stationarity condition (20.1.4), taking a non-zero variation δx j (all other variations vanishing), and using the basic lemma of the variational calculus, we can show that the parenthesis multiplying δx j must vanish; using this reasoning for any j = 1, 2,..., n , we obtain the Euler–Lagrange equations (denomination which puts in evidence the contribution of Euler (1744) and of Lagrange (1760) to the development of variational calculus) (Euler, L., 1955–1974)

∂F d ⎛ ∂F ⎞ − = 0 , j = 1, 2,..., n . ∂x j dt ⎜⎝ ∂x j ⎟⎠

(20.1.7'')

Corresponding to the results in Sect. 18.2.3.4, we notice that for F = 0 these equations admit the Jacobi’s first integral n

∂F

∑ ∂x j x j

−F =h,

(20.1.8)

j =1

where h is an integration constant. In general, we observe that in case of arbitrary synchronous varied paths (without fixed ends), we have δI ( x1 , x 2 ,..., x n ) =

∂F ∑ ∂x j δx j j =1 n

t1

,

(20.1.7''')

t0

along the integral curves of Euler–Lagrange equations. The problem of a brachistochrone curve, formulated by J. Bernoulli in 1696 and studied in Sect. 7.2.1.5, is the first problem of a variational nature which has been considered. 20.1.1.3 Case of Functionals Which Depend on Functions of Several Variables The results obtained above can be used successfully in case of discrete mechanical systems. To study continuous mechanical systems one must introduce a functional of the form


I (u ) =

t1

∫t

219

Fd t ,

(20.1.9)

0

where F =

∫Ω F ( t ; x1 , x 2 , x 3 , u , u , u1 , u2 , u3 ) dΩ ,

(20.1.9')

u = u ( x1 , x 2 , x 3 ; t ) being a function of class C 2 . Here Ω is a three-dimensional domain (eventually the support of a continuous mechanical system), which can be also two- or one-dimensional, dΩ = dx1 dx 2 dx 3 is the support element, while F is a density (corresponding to the unit of support); one has denoted u = ∂u / ∂t , ui = ∂u / ∂x i , i = 1, 2, 3 . Passing to a varied path, we introduce the admissible functions u ′ ( x1 , x 2 , x 3 ; t ) = u ( x1 , x 2 , x 3 ; t ) + δu ( x1 , x 2 , x 3 ; t ) , where δu are

functions of class C 2 , vanishing on the space frontier ∂Ω , as well as on the time one t0 , t1 . As in the previously considered synchronous case, we admit that the independent variables x1 , x 2 , x 3 , and t as well as their limits do not have variations. As in the previously subsection, we can prove Theorem 20.1.3 (the second basic lemma of the variational calculus). Let be a function f ( x1 , x 2 , x 3 ) , continuous on the domain Ω and with real values. If

∫ Ω f ( x1 , x 2 , x 3 ) η ( x1 , x 2 , x 3 ) dΩ = 0

(20.1.10)

for any function η of class C 1 , vanishing on Ω , then f ( x1 , x 2 , x 3 ) = 0 on the domain Ω . This result is valid also for η ∈ C 2 . Expanding into a Taylor series, we can write (we use permutability relations of the form (20.1.5')) δF =

∂F ∂F ∂ ∂F ∂ δu + δu + δu , ∂u ∂u ∂t ∂ui ∂x i

where a summation convention from 1 to 3 is used. We notice that t1

∫t

0

∂F ∂ ∂F ( δu ) dt = δu ∂u ∂t ∂u

∂F ∂ ( δu ) dΩ = ∂x i

∫Ω ∂ui

t1 t0

−

t1

∫t

0

∂ ⎛ ∂F ⎞ ⎜ ⎟ δudt , ∂t ⎝ ∂u ⎠

∂ ⎛ ∂F δu ⎞⎟ dΩ − ⎠

∫Ω ∂xi ⎜⎝ ∂ui

∂ ⎛ ∂F ⎞ ⎟ δudΩ . ⎠

∫Ω ∂xi ⎜⎝ ∂ui

If we apply a formula of flux-divergence type (see App., Subsec. 2.3.3) to the first integral in the second member of the second relation and if we notice that δu ∂Ω = δu t0 = δu t1 = 0 , then, finally, it follows


220 δI =

t1

t1

∫ δF dt =

t1

⎡ ∂F − d ⎛ ∂F ⎞ − d ⎛ ∂F ⎞ ⎤ δudΩdt ; ⎜ ⎟ ∂u dt ⎝ ∂u ⎠ dx i ⎝⎜ ∂ui ⎠⎟ ⎦⎥

∫ ∫ δF dΩdt = ∫t ∫Ω ⎣⎢

t0 Ω

t0

0

we have introduced the derivatives d / dt and d / dxi , i = 1, 2, 3 , to notice that the differentiation is made taking into account the implicit dependence (by the agency of the function u ) of F on the independent variables x1 , x 2 , x 3 , t . Applying the second basic lemma of the variational calculus and by a reasoning similar to that of the previous subsection, we obtain the Euler–Ostrogradskiĭ equations in the form ∂F d ⎛ ∂F ⎞ d ⎛ ∂F ⎞ − ⎜ ⎟− ⎜ ⎟ = 0. ∂u dt ⎝ ∂u ⎠ dxi ⎝ ∂ui ⎠

(20.1.11)

Obviously, in case of a n-dimensional domain one obtains a similar result (summation from 1 to n). Also, in case of several dependent functions u α , α = 1, 2,...,m , there result the equations ∂F d ⎛ ∂F ⎞ d ⎛ ∂F ⎞ − = 0 , α = 1, 2,...,m . ⎜ ⎟− ∂u α dt ⎝ ∂u α ⎠ dx i ⎜⎝ ∂uiα ⎟⎠

(20.1.11')

20.1.1.4 Case of Asynchronous Variations We suppose that the variations which lead from the motion on the true curve to the motion on a varied path are such that the two motions are not synchronous; we will thus consider the functional (20.1.1), using the notations introduced in Sect. 20.1.1.1 for asynchronous variations. The quantities Δx j ( t ) play the rôle of possible displacements, of class C 2 , which satisfy linear constraint relations of the form n

∑ Akj Δx j j =1

+ Ak 0 Δt = 0 , k = 1, 2,..., m ,

(20.1.12)

where Akj = Akj ( x1 , x 2 ,..., xn ; t ) , j = 1, 2,..., n , and Ak 0 = Ak 0 ( x1 , x 2 ,..., xn ; t ) are functions of class C 1 . We consider also the finite form of these relations n

∑ Akj x j j =1

+ Ak 0 = 0 , k = 1, 2,..., m ;

(20.1.12')

eliminating the parameter Ak 0 , we find constraint relations of the form n

∑ Akj δx j j =1

= 0 , k = 1, 2,..., m , δx j = Δx j − x j Δt , j = 1, 2,..., n ,

(20.1.12'')

where δx j are virtual displacements. We notice that, expanding into a Taylor series, one can write


221

x j′ ( t + εΔt ) = x j′ ( t ) + εx j′ ( t ) Δt + O ( ε2

= x j ( t ) + εδx j ( t ) + εx j ( t ) Δt + O ( ε

2

)

),

taking into account (20.1.2); using also the relations (20.1.2''), (20.1.2''') and neglecting O ( ε2 ) , we find again the second relation (20.1.12''). The velocity on the varied path is given by d x j + ε Δx j dx j′ d d t d x j′ = = = x j + ε Δx j 1 − ε Δt + ... d dt ′ dt dt 1 + ε Δt dt d d = x j + ε Δx j − x j Δt + O ( ε2 ) . dt dt

(

)(

(

)

)

Observing that x j′ = x j + εΔx j and neglecting O ( ε2 ) , we obtain the operator relation Δx j =

d d Δx j − x j Δt , dt dt

(20.1.13)

which, in the asynchronous case, replaces the relation (20.1.5); this result corresponds to d ∂ x j + ε dt Δx j Δx j = d ∂ε 1 + ε Δt dt

(20.1.13') ε =0

and can be written also in the form Δx j =

d δx + xj Δt = δx j + xj Δt , dt j

(20.1.13'')

if we take into account (20.1.5) and (20.1.12''). By a change of variable, we may calculate the asynchronous variation of the functional (20.1.1) in the form ( t0′ = t0 + εΔt0 , t1′ = t1 + εΔt1 ) ∂ t1′ F ( t ; x j′ , x j′ ) dt ′ ∂ε ∫t0′ ε=0 ∂ ⎡ d dt F ( t + εΔt ; x j + εΔx j , x j + εΔx j ) 1 + ε Δt ⎤ dt ∂ε ⎣⎢ ⎦⎥ ε = 0 t1 ∂ d = ∫ dt , [ F ( t ) + εΔF ( t ) ] 1 + ε Δt t0 ∂ε dt ε=0 ΔI ( x1 , x 2 ,..., x n ) =

=

t1

∫t

0

{

(

(

)}

)


222 so that ΔI ( x1 , x 2 ,..., x n ) =

t1

∫t

0

( ΔF + F ddt Δt ) dt .

(20.1.14)

This relation is reduced to the relation (20.1.3'') if Δt = 0 . Further, we may write ΔI ( x1 , x 2 ,..., x n ) =

=

n

∂F

∑ ∂x j Δx j j =1

t1 t0

t1

∫t

0

⎛ ⎜ F Δt + ⎝

∂F

n

j =1

j =1

t1

⎛ ⎞ ∂F x j − F ⎟ Δt −⎜∑ ⎝ j =1 ∂x j ⎠ n

+∫

⎡ ∂F

t1 n

t0

∑ ⎣⎢ ∂x j j =1

∂F

n

∑ ∂x j Δx j + ∑ ∂x j Δx j

−

+

⎡

n

+F

d ⎛ ∂F

x j ⎜ ∫t ⎣⎢ ∑ j = 1 dt ⎝ ∂x j t1 0

t0

⎞ d Δt ⎟ dt dt ⎠

⎤ ⎞ − F ⎟ + F ⎥ Δtdt ⎠ ⎦

d ⎛ ∂F ⎞ ⎤ Δx dt , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

where we took into account the operator relation (20.1.13). But n

d ⎛ ∂F

∑ dt ⎜⎝ ∂x j x j j =1

⎞ − F ⎟ + F = ⎠

⎡ d ⎛ ∂F ⎞ ∂F ⎟− ⎠ ∂x j j =1

⎤ ⎥ x j , ⎦

n

∑ ⎣⎢ dt ⎝⎜ ∂x j

and we are thus led to t

⎡ n ∂F ⎛ n ∂F ⎞ ⎤1 ΔI ( x1 , x 2 ,..., x n ) = ⎢ ∑ Δx j − ⎜ ∑ x j − F ⎟ Δt ⎥ ⎣ j =1 ∂x j ⎝ j =1 ∂x j ⎠ ⎦t 0 +∫

t1 n

t0

⎡ ∂F

∑ ⎣⎢ ∂x j

−

j =1

d ⎛ ∂F ⎞ ⎤ δx d t , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.15)

on the basis of the relation (20.1.13); the second relation (20.1.12'') allows us to write t

⎡ n ∂F ⎤1 ΔI ( x1 , x 2 ,..., x n ) = ⎢ ∑ δx j + F Δt ⎥ + ⎣ j =1 ∂x j ⎦t 0

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j

−

0

d ⎛ ∂F ⎞ ⎤ δx d t . dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.15') If the functions Δx j (or δx j ), j = 1, 2,..., n , and Δt vanish at the ends of the interval (varied paths with fixed ends, for which the time is the same at both ends), it follows ΔI ( x1 , x 2 ,..., x n ) =

t1 n

⎡ ∂F

⎢ ∫t ∑ j =1 ⎣ ∂x j 0

−

d ⎛ ∂F ⎞ ⎤ δx d t , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥ j

(20.1.15'')

as in the synchronous case. Taking into account the constraint relations (20.1.12'') and using Lagrange’s multipliers λk , k = 1, 2,..., m , we may write


ΔI ( x1 , x 2 ,..., x n ) =

t1 n

⎡ ∂F

⎢ ∫t ∑ j = 1 ⎣ ∂x j

−

0

223

d ⎛ ∂F ⎞ + dt ⎝⎜ ∂x j ⎠⎟

⎤

m

∑ λk Akj ⎦⎥δx j dt .

k =1

By the usual method, one determines the multipliers λk , k = 1, 2,..., m , m < n , so that the m square brackets vanish; there remain n − m brackets which multiply n − m virtual displacements, now independent. As in the synchronous case, we prescribe the stationarity condition ΔI ( x1 , x 2 ,..., xn ) = 0 ;

(20.1.16)

applying the basic lemma of the variational calculus, we obtain the Euler–Lagrange equations with multipliers ∂F d ⎛ ∂F ⎞ − + dt ⎜⎝ ∂x j ⎟⎠ ∂x j

m

∑ λk Akj

= 0 , j = 1, 2,..., n ,

(20.1.16')

k =1

corresponding to the true curve C. In general, in case of arbitrary asynchronous varied curves (without fixed ends and for which the time is not the same at the ends of the interval), we have t

⎡ n ∂F ⎛ n ∂F ⎞ ⎤1 x j − F ⎟ Δt ⎥ ΔI ( x1 , x 2 ,..., x n ) = ⎢ ∑ Δx j − ⎜ ∑ ⎣ j =1 ∂x j ⎝ j =1 ∂x j ⎠ ⎦t 0 t1

⎡ n ∂F ⎤ δx j + F Δt ⎥ = ⎢∑ ⎣ j =1 ∂x j ⎦t0

(20.1.15''')

along the integral curves of the Euler–Lagrange equations with multipliers. Starting from the relations (20.1.7) and (20.1.15''), we obtain the relation connecting the asynchronous to the synchronous variation of a functional in the form

ΔI = δI + F Δt

t1 t0

.

(20.1.17)

If Δt0 = Δt1 = 0 (or, more general, if F ( t0 ) = F ( t1 ) and Δt0 = Δt1 ), there results ΔI = δI ,

(20.1.17')

the asynchronous and synchronous variations being equal; thus, the difference between the effects of those variations disappears. We notice also that, in case of varied paths with fixed ends ( Δx j ( t0 ) = Δx j ( t1 ) = 0 , j = 1, 2,..., n ) for which F = 0 (the function F does not depend explicitly on time), the Euler–Lagrange equations entail also Jacobi’s first integral (20.1.18). The relation (20.1.15) shows that one can write


224

ΔI ( x1 , x 2 ,..., x n ) = − h ( Δt1 − Δt0 )

(20.1.18)

along the integral curves of Euler–Lagrange equations; to obtain ΔI = 0 , one must have Δt0 = Δt1 , hence the motion on the varied paths must have the same duration as on the true curve. This last condition concerning the asynchronous variations is somewhat restrictive; we can also write ΔI ( x1 , x 2 ,..., xn ) + h ( Δt1 − Δt0 ) t1

= Δ ∫ [ F ( x1 ( t ) , x 2 ( t ) ,..., x n ( t ) , x1 ( t ) , x2 ( t ) ,..., xn ( t ) ; t ) + h ] dt = 0 , t0

(20.1.18') a relation which can be used in various particular cases. We will estimate t1

∫t

ΔFdt =

0

t1

∫t

0

⎛ n ∂F ⎜ ∑ ∂x Δx j + ⎝ j =1 j

∂F x j Δt ∂ j = 1 x j n

−∑

t1

+

n

−

d ⎛ ∂F ⎞ ⎤ Δx j dt , dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥

d ⎛ ∂F

⎞

t1

t0

⎤

⎜ ∂x x j ⎟ + F ⎥ Δtdt ∫t ⎢⎣ ∑ d t j ⎝ ⎠ j =1 ⎦ t1

⎡ ∂F

⎢ ∫t ∑ j =1 ⎣ ∂x j

+

⎡

∂F ∑ ∂x j Δx j j =1 n

0

t0

t1 n

⎞ ∂F ∑ ∂x j Δx j + F Δt ⎟ dt = j =1 ⎠ n

0

using the variation (20.1.14) without the term Fd ( Δt ) / dt . We can write t1

∫t

0

⎛ ⎜ ΔF + ⎝

⎞ ∂F d ∑ ∂x j x j dt Δt − F Δt ⎟ dt = j =1 ⎠ n

+∫

⎡ ∂F

t1 n

t0

∑ ⎣⎢ ∂x j j =1

−

∂F ∑ ∂x j Δx j j =1 n

t1

t0

d ⎛ ∂F ⎞ ⎤ Δx j dt dt ⎝⎜ ∂x j ⎠⎟ ⎦⎥

(20.1.19)

too. It follows t1

∫t

0

⎛ ⎜ ΔF + ⎝

⎞ ∂F d ∑ ∂x j x j dt Δt − F Δt ⎟ dt = j =1 ⎠ n

∂F ∑ ∂x j Δx j j =1 n

t1

(20.1.19') t0

along the integral curves of Euler–Lagrange equations. If the varied paths are with fixed ends ( Δx j ( t0 ) = Δx j ( t1 ) = 0 ), then we can state Theorem 20.1.4 (O. Hölder). In the case of asynchronous varied paths with fixed ends, the functional (20.1.1) verifies the relation t1

∫t

0

⎛ ⎜ ΔF + ⎝

n

∂F

d

⎞

∑ ∂x j x j dt Δt − F Δt ⎟ dt j =1

⎠

=0

(20.1.19'')


225

along the integral curves of the corresponding Euler–Lagrange equations. In case of some constraint relations of the form (20.1.12), one uses, as above, the methods of Lagrange’s multipliers.

20.1.2 The General Integral Principle Following the ideas of the previous section, we will state the general form of a variational principle and of the general integral principle; these results will be given for synchronous as well as for asynchronous variations. 20.1.2.1 Preliminary Considerations Let be a discrete mechanical system S of n particles and Pi a particle of this system, of position vector ri with respect to an inertial frame of reference, subjected to the action of the resultant Fi of the given forces (by given forces we intend external as well as internal ones); it is obvious that i = 1, 2,..., n . In the time interval [ t0 , t1 ] , the particle Pi describes the true curve C i of equation ri = ri ( t ) . Let us suppose that the particle Pi is displaced by εΔri , where ε is a small parameter, till the point Pi ′ of position vector ri′ = ri + εΔri , i = 1, 2,..., n .

(20.1.20)

Fig. 20.2 Varied paths

From the set of possible displacements Δri ( t ) of class C 2 , we choose those which, univocally, lead to the point Pi ′ , the locus of which is a varied path C i′ (see Fig. 20.2); the motion on this path corresponds to the proper time t ′ , given by (20.1.2''). One obtains thus an infinity of varied paths, the motion on those ones being, in general, asynchronous ( Δt ≠ 0 ). If Δt = 0 , the variations are synchronous, and ri′ = ri + εδ ri , i = 1, 2,..., n ,

(20.1.20')


226

where δri are virtual displacements (synchronous variations). We assume that the possible displacements verify the linear constraint relations n

∑ αki

i =1

⋅ Δri + αk 0 Δt = 0 , k = 1, 2,..., m ,

(20.1.21)

where αki = αki ( r1 , r2 ,..., rn ; t ) , i = 1, 2,..., n , and αk 0 = αk 0 ( r1 , r2 ,..., rn ; t ) are functions of class C 1 . Considering also the finite form of the constraint relations n

∑ αki

i =1

⋅ ri + αk 0 = 0 , k = 1, 2,..., m ,

(20.1.21')

and eliminating the parameter αk 0 , we find constraint relations of the form n

∑ αki

i =1

⋅ δ ri = 0 , k = 1, 2,..., m , δri = Δri − vi Δt , i = 1, 2,..., n ,

(20.1.21'')

where, obviously, vi = ri are real velocities, while δ ri are virtual displacements. The velocity on the varied path is given by d dri′ vi + ε dt Δri d d = = vi + ε Δri 1 − ε Δt + ... vi′ = d ′ dt dt dt 1 + ε Δt dt d d = vi + ε Δr − vi Δt + O ( ε2 ) . dt i dt

(

(

)(

)

)

We notice that vi′ = vi + εΔvi and we neglect O ( ε2 ) ; the asynchronous variation of the velocity is thus given by Δ vi =

d d Δri − vi Δt , i = 1, 2,..., n . dt dt

(20.1.22)

Because d ( vi Δt ) / dt = vi dΔt / dt + ai dt , where ai is the acceleration of the particle Pi , we can write this relation also in the form Δ vi =

d δ r + ai Δt , i = 1, 2,..., n , dt i

(20.1.22')

where we took into consideration the second relation (20.1.21''). If Δt = 0 , then one obtains the synchronous variation δ vi =

d δ r , i = 1, 2,..., n , dt i

(20.1.23)


227

and the operator relation (20.1.5'). Hence, the relation between the two variations (asynchronous and synchronous) will be of the form dvi , i = 1, 2,..., n . dt

Δvi = δ vi + ai Δt , ai =

(20.1.22'')

The synchronous variation of the kinetic energy will be given by

1 n mi ( vi′2 − vi2 2 i∑ =1

εδT = T ′ − T =

n

1

) = 2 ∑ mi ⎡⎣ ( vi i =1

+ εδ vi )2 − vi2 ⎤⎦

n

= ε ∑ mi vi ⋅ δ vi + O ( ε2 ) , i =1

so that δT =

n

∑ mi vi

i =1

⋅ δ vi ;

(20.1.24)

analogously, we obtain the asynchronous variation ΔT =

n

∑ mi vi

i =1

⋅ Δ vi .

(20.1.24')

Performing a scalar multiplication of the relation (20.1.22'') by mi vi summing for all values of the index i , taking into account (20.1.24), (20.1.24'), and observing that d ( vi2 / 2 ) / dt = vi ⋅ ai , we can write the relation between the asynchronous and the synchronous variation of the kinetic energy in the form ΔT = δT +

dT Δt . dt

(20.1.25)

In case of quasi-conservative given forces, which derive from a quasi-potential U = U ( r1 , r2 ,..., rn ; t ) , we obtain the asynchronous variation ΔU =

n

∑ ∇iU ⋅ Δri

+ U Δt , U =

i =1

∂U ; ∂t

(20.1.26)

the synchronous variation is of the form δU =

n

∑ ∇iU ⋅ δri .

(20.1.26')

i =1

Observing that dU = dt

n

∑ ∇iU ⋅ vi

i =1

+ U


228

and taking into account the second relation (20.1.21''), one obtains the relation between the asynchronous and the synchronous variations of the quasi-potential in the form ΔU = δU +

For the mechanical energy analogously,

dU Δt . dt

(20.1.27)

E = T − U = T + V , V = −U , one obtains,

ΔE = δE +

dE Δt , dt

(20.1.28)

while for Lagrange’s kinetic potential L = T + U = T − V we can write ΔL = δL +

dL Δt . dt

(20.1.28')

Generically, we denote by E a quantity of energetic nature, e.g., T ,U , E , L or any other linear combination (with numerical coefficients) of them; as a matter of fact, two of these quantities can be considered as basic ones (for instance T and L ), while the other quantities are expressed by means of them. We will thus have ΔE = δE +

dE Δt . dt

(20.1.28'')

Let us introduce the functional

A =

t1

∫t

Edt ,

(20.1.29)

0

called action; the quantity Edt will be the elementary action. We have thus to do with the kinetic action AT , corresponding to E = T , with the potential action AU , for E = U , with the Lagrangian action AL , corresponding to E = L , or with the mechanical action AE , for E = E . We mention that the magnitude 2 AT corresponds to the classical notion of action. Using the formula (20.1.14), we can express the asynchronous variation of the action A in the form t1

Δ A = Δ ∫ E dt = t0

t1

∫t

0

( ΔE + E ddt Δt ) dt ;

(20.1.29')

analogously, the corresponding synchronous variation is given by t1

δ A = δ ∫ E dt = t0

t1

∫t

δE dt ,

(20.1.29'')

0

where a permutation between the operator δ and the integral operator is possible. Taking into account (20.1.28''), we can express the asynchronous variation (20.1.29') in the form


Δ A = EΔt

t1 t0

+

t1

∫t

229

δE dt ;

(20.1.30)

0

hence, the relation between the asynchronous and the synchronous variations of an action is given by Δ A = δ A + EΔt

t1 t0

.

(20.1.30')

If Δt0 = Δt1 = 0 (or, more general, if E ( t0 ) = E ( t1 ) and Δt0 = Δt1 , the motion having the same duration on all varied paths), we obtain ΔA = δ A ,

(20.1.30'')

the asynchronous variation being equal to the synchronous one.

20.1.2.2 Two Forms of a General Integral Principle Starting from the principle of virtual work, which can be applied in case of ideal constraints (see Sect. 11.1.2.10), written in the form n

∑ Φi

i =1

⋅ δri = 0 , Φi = Fi − mi ai ,

(20.1.31)

where Fi are given forces (external and internal), and integrating with respect to time, between the limits t0 and t1 , we obtain t1 n

∫t i∑=1 Φi

⋅ δri dt = 0 .

(20.1.31')

0

Inversely, starting from (20.1.31') and using the basic lemma of Lagrange (the virtual displacements δri are independent or verify the constraint relations (20.1.21''), case in which one uses Lagrange’s multipliers λk , k = 1, 2,..., m , leading to the constraint Ri =

forces

m

∑ λk αki ,

i = 1, 2,..., n ),

we

find

k =1

Φi = 0

or

Φi + Ri = 0 ,

i = 1, 2,..., n , hence Newton’s equations of motion. The relations (20.1.31) and (20.1.31') are thus equivalent. We can calculate t1 n

mi ai ∫ t i∑ =1

⋅ δri dt =

0

=

n

t1

i =1

t0

∑ mi vi ⋅ δri

−

n

∑ ∫t

t1

i =1

mi

0

t1 n

∫t

0

t1 n d d ( vi ⋅ δri ) dt − ∫ ∑ mi vi ⋅ ( δ ri )dt t dt dt 0 i =1

∑ mi vi ⋅ δ vi dt =

i =1

n

t1

i =1

t0

∑ mi vi ⋅ δri

−

t1

∫t

where we took into account (20.1.23) and (20.1.24); on the other hand

0

δTdt ,


230 n

∑ Fi

i =1

⋅ δri = δW ,

(20.1.32)

where δW is the virtual work of the external and internal given forces. Finally, the relation (20.1.31') leads to t1

∫t

( δW + δT ) dt =

n

∑ mi vi

i =1

0

⋅ δri

t1 t0

;

(20.1.33)

using only asynchronous variations, one can also write t1

∫t

0

(

)

t

1 n ⎡ ΔW + ΔT − P + dT Δt ⎤ dt = ⎡ m v ⋅ Δr − 2T Δt ⎤ , i ⎢∑ i i ⎥ ⎥⎦ dt ⎣⎢ ⎣ i =1 ⎦t 0

(20.1.33')

where we took into account (20.1.21'') and (20.1.25), where ΔW = δW + P Δt =

n

∑ Fi

i =1

⋅ Δri

(20.1.32')

is the asynchronous variation of the work of external and internal forces and where P =

dW = dt

n

∑ Fi

i =1

⋅ vi

(20.1.34)

is the power of the same given forces. In case of synchronous varied paths with fixed ends, the relation takes the form (20.1.33), hence t1

∫t

( δW + δT ) dt = 0 ,

(20.1.35)

0

equivalent to the relation (20.1.31'), that is to the principle of virtual work (20.1.31). Following V. Vâlcovici, we will call this relation the general integral principle, because – starting from (20.1.35) – we can obtain the most important variational (integral) principles, in case of synchronous variations (Vâlcovici, V., 1969, 1973). In the asynchronous case, we start from the relation (20.1.33') and consider asynchronous varied paths with fixed ends, for which Δt0 = Δt1 = 0 ; we obtain thus t1

∫t

0

(

)

⎡ ΔW + ΔT − P + dT Δt ⎤ dt = 0 . ⎥⎦ dt ⎣⎢

(20.1.33'')

We notice that the relation (20.1.33') can be written also in the form t1

∫t

0

(

)

⎡ ΔW + ΔT + dT − P Δt + 2T d Δt ⎤ dt = ⎢⎣ ⎥⎦ dt dt

n

∑ mi vi

i =1

⋅ Δri

t1

; t0

(20.1.33''')


231

one can obtain in this case a general integral principle, corresponding to asynchronous varied paths with fixed ends ( Δri ( t0 ) = Δri ( t1 ) = 0 , i = 1, 2,..., n ) in the form t1

∫t

0

(

)

⎡ ΔW + ΔT + dT − P Δt + 2T d Δt ⎤ dt = 0 . dt dt ⎣⎢ ⎦⎥

(20.1.35')

As it can be easily seen, in the above calculus an important rôle is played by the operator Δ = δ + Δt

d , dt

(20.1.36)

which is applied for the position vector ri and the velocity vi , as well as for all the quantities of energetical nature or of the nature of an action which occur (even the relation (20.1.30') can be obtained by applying this operator). As well, the operator can be used to pass from a synchronous to an asynchronous form of an integral or of a variational principle. The permutation relation of the form (20.1.5') does no more take place in this case; one must use the transitivity relation (see relation (20.1.22)) d d d d ( Δt ) . Δ−Δ = dt dt dt dt

(20.1.36')

20.1.3 Hamilton’s Principle We will present, in what follows, Hamilton’s, Hölder’s and Voss’s principles, corresponding to synchronous or asynchronous variations of true trajectories, respectively; some methods of calculus are thus put into evidence.

20.1.3.1 Case of Synchronous Variations We can write δW =

n

∑ ∇iU ⋅ δri

= δU

(20.1.37)

i =1

in case of quasi-conservative forces and of synchronous variations; taking into account the definition relation of the kinetic potential ( L = T + U ), the general integral principle (20.1.35) leads to t1

∫t

δL dt = 0 .

(20.1.38)

0

Using the permutability property (20.1.3'') between the operator δ and the integral, we have t1

δ AL = δ ∫ L dt = 0 . t0

(20.1.38')


232

We can thus state that along the varied paths of a discrete mechanical system subjected to ideal constraints and to quasi-conservative given forces, the Lagrangian action AL is stationary (necessary condition) in case of synchronous variations; we notice that the mechanical system must be holonomic, corresponding to this type of variations. Let us assume that the motion of the mechanical system is described by means of generalized co-ordinates, so that L = L ( q1 , q 2 ,..., qs , q1 , q2 ,..., qs ; t ) ; in this case, starting from the relation (20.1.38) and using the results of Sect. 20.1.1.2 (Euler– Lagrange equations (20.1.7'')), we find again Lagrange’s equations of second kind (18.2.38). If the given forces do not derive from a quasi-potential, then n

∑ Fi

i =1

⋅

∂ri = Q j , j = 1, 2,..., s , ∂q j

and, starting from (20.1.35), we obtain (we use again the results of Sect. 20.1.1.1 and the summation convention for dummy indices) t1

∫t

0

d ⎛ ∂T ⎞ ⎡ ∂T ⎢ ∂q − dt ⎜ ∂q ⎟ + Q j ⎣ j ⎝ j ⎠

⎤ ⎥ δq j dt = 0 ; ⎦

(20.1.39)

applying the basic lemma of Lagrange, we find again the equations (18.2.29). In case of non-holonomic constraints, we use Lagrange’s multipliers method as in Sect. 20.1.1.4; there results (we assume to have the constraint relations (18.2.10)) t1

∫t

0

d ⎛ ∂T ⎞ ⎡ ∂T ⎢ ∂q − dt ⎜ ∂q ⎟ + Q j + ⎣ j ⎝ j ⎠

m

⎤

∑ λk akj ⎦⎥ δq j dt

k =1

= 0.

(20.1.39')

By the same reasoning as in the above mentioned subsection, we find again Lagrange’s equations with multipliers (18.2.32), (18.2.33). Hence, the relation (20.1.38') is sufficient (not only necessary) to specify the motion in the frame of the mentioned conditions. So, we state Theorem 20.1.5 (W.R. Hamilton). Among all possible motions of a mechanical system subjected to holonomic and ideal constraints and acted upon by quasi-conservative given forces on synchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) for which the Lagrangian action is stationary takes place. We mention that the most times – not always – the Lagrangian action realizes a minimum for the actual motion (on the true curve). We have obtained this result in the form of a theorem, starting from the principle of the virtual work; but we can interpret this result also as a variational principle (Hamilton’s principle), which leads to Lagrange’s equations of motion. This principle was stated in 1834 by W.R. Hamilton for scleronomic constraints; it was extended by M.V. Ostrogradskiĭ in 1848 to the case of rheonomous constraints. One observes that, unlike differential principles (in which, to establish the motion at a given moment, one considers only the motion in a vicinity of this one), in case of variational principles the


233

motion of the mechanical system at a given moment is specified by its motion in the whole (finite) interval of time. Often, Hamilton’s principle is considered to be the basic principle of mechanics; indeed, classical mechanics as well as the relativistic one can be deduced starting from such a principle. As well, the theory of the electromagnetic field can be obtained from an analogous principle, as other theories of physical nature. Hence, the importance of those principles consists also in their expression in a unified form, corresponding to the tendency of extremum of phenomena in the nature. So, the variational principles of Hamilton’s type lead to “equations of motion” which can be Newton’s, Maxwell’s or Schrödinger’s equations; using a variational principle as basis of formulation, all these domains of physical nature have, in a certain measure, a structural analogy. If experimental results impose the change of physical matter in the theory corresponding to a certain domain, then this analogy can show similar changes in another domain. For instance, experiments made at the beginning of this century have put into evidence the necessity to quantify the electromagnetic radiation as well as the elementary particles; indeed, starting from the development of quantification methods for the mechanics of particles, one could construct a quantum electrodynamics. Although its importance from the point of view of the philosophy of mechanical systems, we must notice that Hamilton’s principle can – sometimes – lead to more intricate calculations from a practical point of view as a direct study based on Newton’s equation. If the given forces are not quasi-conservative, then one cannot introduce the function L , and if the mechanical system is not holonomic, then the operator δ cannot permute with the integral. In this case, in general, the varied paths are no more possible curves, and we must use the relation (20.1.35); the problem is no more a problem of variational calculus. We observe that we may distinguish between an integral principle and a variational one, the first one having a more general character. Indeed, by convention, an integral principle is deduced from the general integral principle (20.1.35), without the intervention of the variation of a functional (the variation of an action), this aspect appearing only under the integral sign; if the variation of a functional occurs, then we have to do with a variational principle. Perhaps, this distinction is conventional; however, we must mention that a variational principle is linked to a problem of extremum of a functional, allowing thus to use a rich methodology of calculus. If the constraints are non-holonomic but the given forces are quasi-conservative, then Hamilton’s principle is an integral one and has the form (20.1.38) (it is true that the virtual displacements are not independent), while if the constraints are holonomic, the given forces being quasi-conservative or not, then we find again the general integral principle (20.1.35), which – in fact – represents a more general integral form of Hamilton’s principle in a synchronous case. Thus, as one can see, Hamilton’s principle has a sufficient general character, even in the synchronous case. However, the difference (given by the non-holonomic or holonomic character of the ideal constraints) between the forms (20.1.38) and (20.1.38') of Hamilton’s principle is clearly put in evidence. We notice also that, in case of quasi-conservative forces deriving from a generalized potential, the virtual work of the given forces can be expressed in the form


234 δW =

=

n

∑ Fi

i =1

d ⎛ ∂U ⎞ ⎤ ⎡ ∂U δq ⋅ δri = Q j δq j = ⎢ − q t ⎝⎜ ∂q j ⎠⎟ ⎦⎥ j d ∂ ⎣ j

∂U d ⎛ ∂U d ⎛ ∂U ⎞ ∂U d ⎞ δq − δq + δq = δU − δq ; ∂q j j dt ⎝⎜ ∂q j j ⎠⎟ ∂q j dt j dt ⎜⎝ ∂q j j ⎟⎠

in case of synchronous variations, for which the operator relation (20.1.5') is valid, the general integral principle becomes t1

∫t

( δU + δT ) dt =

0

∂U δq ∂q j j

t1

. t0

Further, assuming variations with fixed ends ( δq j ( t0 ) = δq j ( t1 ) = 0 , j = 1, 2,..., s ), there results that Hamilton’s variational principle maintains its form for any natural system (we can permute the operator δ with the integral sign). The property of extremum expressed by Hamilton’s principle is an intrinsic property; by way of consequence, it can be expressed in any system of co-ordinates. We will thus pass from the co-ordinates q j , u j = q j , j = 1, 2,..., s , in the space Λs , to the coordinates q j , p j , j = 1, 2,..., s , in the space Γ 2s ; a Legendre transformation permits it (see Sects. 19.1.1.2 and 20.1.1.3). But we must have some care, because the variations δq j and δu j are not independent; indeed, to given variations δq j there correspond variations of the generalized velocities of the form δu j =

In

this

case,

let

us

d δq , j = 1, 2,..., s . dt j

consider

the

Lagrangian

(20.1.40) action

for

which

L = L ( q1 , q 2 ,..., qs , u1 , u2 ,..., us ; t ) ; corresponding to Hamilton’s theorem, this

action must be stationary along the curves satisfying the differential equations q j = u j , j = 1, 2,..., s .

(20.1.40')

The co-ordinates q j are fixed at the ends, while the co-ordinates u j do not possess this property. Because we deal only with arcs of curve which verify the differential equations (20.1.40'), the variations being synchronous, it is sure that the conditions (20.1.40) are verified. Using the method of Lagrange’s multipliers, we obtain δ∫

t1 t0

[L

( q1 , q 2 ,..., qs , u1 , u2 ,..., us ; t ) + λj ( q j − u j

) ] dt

= 0,

where the functions λj = λj ( t ) , j = 1, 2,..., s , remain to be determined. The equations of the extremal curves (the Euler–Lagrange equations) will be given by dλ j ∂L ∂L − = 0, − + λj = 0 , j = 1, 2,..., s ; ∂q j ∂u j dt


235

obviously, by eliminating the multipliers λj , we find again Lagrange’s classical equations. We denote H ( q1 , q 2 ,..., qs , p1 , p2 ,..., ps ; t ) =

∂q L o o u −L , ∂u j j

where the linear transformation p j = ∂L / ∂u j , j = 1, 2,..., s , allows to pass from the 2s-dimensional space of co-ordinates q j , u j , j = 1, 2,..., s , to the phase space Γ 2s , the cap specifying the fact that the transformation has been effected (as in Sect. 19.1.1.3); we assume that the condition ⎡ ∂2L ⎤ det ⎢ ⎥ ≠0 ⎣ ∂u j ∂qk ⎦

is verified. We can thus state Theorem 20.1.6 (G.H. Livens). The stationarity condition of Theorem 20.1.5 (Hamilton’s theorem) is written in the form δ ALΓ = δ ∫

t1

t0

[ p j q j

− H ( q1 , q2 ,..., qs , p1 , p2 ,..., ps ; t ) ] dt = 0 ,

(20.1.41)

in the phase space. The form taken by Hamilton’s principle in the phase space (given by Livens in 1919) is called also the canonical form of Hamilton’s principle. We notice that δ∫

t1 t0

( p j q j

− H ) dt =

=

t1

∫t

0

∫ t δ ( pj q j t1 0

∂H ⎡⎛ ⎢ ⎜ q j − ∂p j ⎣⎝

− H ) dt =

t1

∫t

0

∂H ∂H ⎛ ⎜ p j δq j + q j δ p j − ∂q δq j − ∂p δ p j j j ⎝

∂H ⎞ ⎛ ⎟ δ p j − ⎜ p j + ∂q j ⎠ ⎝

⎞ ⎟ δq j ⎠

⎤ ⎥ dt + p j δq j ⎦

t1 t0

⎞ ⎟ dt ⎠

= 0,

where we took into account the operator relation (20.1.5'). In the hypothesis of fixed ends (for generalized co-ordinates) the integrated part vanishes. Applying the basic lemma of the variational calculus and observing that the variations δq j , δ p j , j = 1, 2,..., s , are arbitrary, we find again Hamilton’s canonical equations (19.1.14); however, if we write Euler-Lagrange equations for (20.1.41), hence the equations d (p dt j

⎛

∂H ⎞ ⎛ ∂H ⎞ ⎟ = 0 , −q j − ⎜ − ∂p ⎟ = 0 , j = 1, 2,..., s , j ⎠ j ⎠ ⎝

) − ⎜ − ∂q ⎝

we find again the same equations (19.1.14). The time and the generalized co-ordinates are fixed at the ends of the interval ( δq j ( t0 ) = δq j ( t1 ) = 0 ), but the generalized momenta do not verify a similar


236

property ( δ p j ( t0 ) ≠ 0 , δ p j ( t1 ) ≠ 0 ); indeed, the conditions at the ends (bilocal) for

p j do not furnish any information, taking into account that the derivatives p j do not appear in the integrand expression. Of course, the two integrals which occur in (20.1.38') and (20.1.41) have the same value in the actual motion. But Hilbert observed that, in the formulation (20.1.38'), one can attain a minimum, while in the formulation (20.1.41) – for the same problem – it is possible to obtain neither a maximum nor a minimum (hence, a saddle point). We can put in evidence this observation by a simple example: the rectilinear motion of a particle subjected to its own weight (a field of uniform acceleration g ), for which x = gt 2 / 2 + vt , t0 = 0 , x (0) = 0 , x ( t1 ) = gt12 / 2 + vt1 . In this case,

L = mx 2 / 2 + mgx , while H = p 2 / 2m − mgx . Let us compare the actual motion with a motion on a varied path, given by x = gt 2 / 2 + vt + α ( t ) , α ( t ) ∈ C 2 , α ( 0 ) = α ( t1 ) = 0 ; we have

AL + δ AL = =

t1

∫0

( m2 x

2

)

+ mgx dt

(

)

⎡ m ( v + gt + α )2 + mg 1 gt 2 + vt + α ⎤ dt . ⎢⎣ 2 ⎥⎦ 2

t1

∫0

There results ⎡ m α 2 + m ( v + gt ) α + mg α ⎤ dt ⎣2 ⎦ t1 m 2 t1 m 2 t1 = ∫ α dt + m ( v + gt ) α 0 = ∫ α dt ; 0 2 0 2 δ AL =

t1

∫0

we state that δ AL > 0 , excepting the case in which α = 0 , hence α ( t ) = const , so that α ( t ) ≡ 0 (taking into account the conditions at the ends of the interval). Hence, the motion on the true curve corresponds to a minimum of the Lagrangian action. Using the canonical form of Hamilton’s principle, we can calculate

ALΓ =

t1

∫0

( − 21m p

2

)

+ mgx + px dt =

t1

∫0

⎡ − 1 ( p − mx )2 + m x 2 + mgx ⎤ dt . 2 ⎣⎢ 2m ⎦⎥

For the actual motion we have x = gt 2 / 2 + vt , p = m ( v + gt ) , while for the varied motion

we

can

take

x = gt 2 / 2 + vt + α ( t ) ,

p = m ( v + gt ) + β ( t ) ,

α ( t ) , β ( t ) ∈ C , α ( 0 ) = α ( t1 ) = 0 , without any conditions for β ( t ) at the ends of the interval. Analogously, we obtain 2

δ ALΓ =

t1

∫0

(

m⎡ 2 β − α α − 2 ⎢⎣ m

) ⎤⎥⎦ dt . 2

We observe that δ ALΓ > 0 for β = m α and α non-identical zero, while δ ALΓ < 0 for α ≡ 0 and β non-identical zero; the functional ALΓ is stationary on the true curve and has neither a maximum, nor a minimum (saddle point).


237

Let us remember that we have introduced the Lagrangian action in Sect. 19.2.1.6 in the form of a function S (t ) which verifies the Hamilton-Jacobi equation, as one can see by a total differentiation with respect to time; starting from this function, one has obtained Hamilton’s principal function. So, we can write the relation

dS = p j dq j − H ( q1 , q 2 ,..., qs , p1 , p2 ,..., ps ; t ) dt , which corresponds to Hamilton’s Pfaff form (19.1.45).

20.1.3.2 Case of Asynchronous Variations. Hölder’s Principle. Voss’s Principle. Another Form of Hamilton’s Principle In case of asynchronous variations, occurs the time variation Δt as a function of time. If Δt0 = Δt1 = 0 it follows that the initial and the final times are the same, the duration of the motion being also the same on all varied paths; besides, we can make the latter affirmation also in the more general case in which Δt0 = Δt1 ≠ 0 . We mention that these conditions are sufficiently restrictive. Corresponding to the results of Sect. 20.1.1.4, we can write an identity of the form t1

∫t

0

∂L d ∂L ⎛ ⎞ ⎜ ΔL + q j ∂q dt Δt − LΔt ⎟ dt = ∂q Δq j j j ⎝ ⎠

t1 t0

t1 ⎡ ∂L d ⎛ ∂L ⎞ ⎤ +∫ ⎢ − Δq dt t0 ∂q j dt ⎝⎜ ∂q j ⎠⎟ ⎦⎥ j ⎣ (20.1.42)

for Lagrange’s kinetic potential L = L ( q1 , q 2 ,..., qs , q1 , q2 ,..., qs ; t ) , where asynchronous variations are put into evidence. Along the integral curves of Lagrange’s equations there results t1

∫t

0

∂L d ∂L ⎛ ⎞ ⎜ ΔL + q j ∂q dt Δt − LΔt ⎟ dt = ∂q Δq j j j ⎝ ⎠

t1

;

(20.1.42')

t0

in case of varied curves with fixed ends ( Δq j ( t0 ) = Δq j ( t1 ) = 0 ) and in case of catastatic constraints (the set of possible displacements coincides with the set of virtual displacements), we can state Theorem 20.1.7 (Hölder). Among all possible motions of a mechanical system subjected to holonomic, catastatic, ideal constraints, and to quasi-conservative given forces on asynchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) for which one has the relation t1

∫t

0

∂L d ⎛ ⎞ ⎜ ΔL + q j ∂q dt Δt − LΔt ⎟ dt = 0 j ⎝ ⎠

(20.1.42'')

takes place. Indeed, in this case, the possible displacements Δq j , j = 1, 2,..., s , are independent, and we find again Lagrange’s equations; but, in case of non-holonomous constraints


238

one can use the method of multipliers, finding again Lagrange’s equations with multipliers. The constraints are catastatic, so that ri = 0 , i = 1, 2,..., n , for all particles of the mechanical system; so, a particular configuration of this system is represented by the same point in the space Λs , for all values of the time t . As in the case of the Theorem 20.1.5, the results remain valid for all quasi-conservative given forces which derive from a generalized quasi-potential. This theorem can be used also as a principle: Hölder’s principle, stated in 1896; obviously, this principle is an integral one. If Δt = 0 , then we find again Hamilton’s principle. Using the results of Sect. 20.1.1.4, we can calculate the asynchronous variation of the Lagrangian action in the form t

t1 ⎡ ∂L ⎤1 Δ AL = Δ ∫ L dt = ⎢ Δq j − EΔt ⎥ + t0 ⎣ ∂q j ⎦t 0

where

t1

∫t

0

d ⎛ ∂L ⎞ ⎤ ⎡ ∂L ⎢ ∂q − dt ⎜ ∂q ⎟ ⎥ δq j dt , ⎣ j ⎝ j ⎠⎦ (20.1.43)

E is the generalized mechanical energy E =

∂L q − L . ∂q j j

(20.1.43')

Assuming that we have to do with varied paths with fixed ends and that Δt0 = Δt1 = 0 , there results

Δ AL =

t1

∫t

0

d ⎛ ∂L ⎞ ⎤ ⎡ ∂L ⎢ ∂q − dt ⎜ ∂q ⎟ ⎥ δq j dt . j ⎣ ⎝ j ⎠⎦

(20.1.43'')

We are thus led to Theorem 20.1.8 (A. Voss). Among all possible motions of a mechanical system subjected to ideal constraints, acted upon by quasi-conservative given forces, on asynchronous varied paths with fixed ends (in space and time), only and only the motion (of the representative point on the true curve) which makes stationary the Lagrangian action Δ AL = Δ ∫

t1

t0

L dt = 0

(20.1.44)

takes place. This result is valid in case of holonomic constraints (one obtains Lagrange’s equations), as well as in case of non-holonomic ones (one obtains Lagrange’s equations with multipliers); it remains valid also in case of quasi-conservative forces which derive from a generalized quasi-potential. As above, one can consider this theorem to be a new variational principle: Voss’s principle (1900). It is an extension of Hamilton’s principle in case of asynchronous variations; besides, one can easily obtain this result, taking into account the relation (20.1.30'), which links the asynchronous to the synchronous variation of an action, and making Δt0 = Δt1 = 0 .


239

Along the extremal curves to which lead Lagrange’s equations, the relation t

⎡ ∂L ⎤1 = ⎢ Δq j − EΔt ⎥ , ⎣ ∂q j ⎦t 0

Δ AL

(20.1.43''')

corresponding to the Poincaré-Cartan integral invariant (see Sect. 21.1.2.4) takes place. We mention also the connection between this relation and the relation (19.2.28') concerning Hamilton’s principal function. As well, in case of varied paths with fixed ends, we can write Δ AL + EΔt

t1 t0

= 0.

(20.1.43iv)

Comparing Hölder’s and Voss’s principles, we see that the second one is a variational principle (an advantage), and that the mechanical system is not subjected to catastatic constraints; in compensation, the varied paths must be with fixed ends also for time, a sufficient severe restriction, not implied by Hölder’s principle. On the other hand, in the corresponding demonstration the principle of virtual work was not used, but only the connection with Lagrange’s equations; thus, one must not have L = T + U (as it was supposed, mentioning the action of quasi-conservative given forces), but we can consider mechanical systems which admit a Lagrangian L which verifies the condition (18.2.34'''). Starting form the general integral principle (20.1.35'), which corresponds to asynchronous variations with fixed ends and imposing catastatic constraints ( dT / dt = P ), we can write t1

∫t

0

( ΔW + ΔT + 2T ddt Δt ) dt = 0 .

(20.1.45)

If the given forces are quasi-conservative, then we have ΔW = ΔU − U Δt , so that we obtain ( L = T + U ) t1

∫t

0

( ΔL + 2T ddt Δt − U Δt ) dt = 0 .

(20.1.45')

Assuming that the quasi-conservative forces derive from a simple quasi-potential and observing that, in case of catastatic constraints, we have q j

∂L ∂T = q j = 2T2 = 2T , T = 0 , ∂q j ∂q j

we find again Hölder’s principle. If the given forces are quasi-conservative, then the relation (20.1.35') becomes ( ΔW = ΔU − U Δt , L = T + U and E = T − U ) t1

Δ ∫ L dt + t0

t1

∫t

0

( ddTt − P − U ) Δtdt + ∫

t1 t0

E d( Δt ) = 0 ;


240 integrating by parts, we can write also t1

Δ ∫ L dt + t0

t1

∫t

0

( ddTt − P ) Δtdt − ∫ ( ddEt + U ) Δtdt + E Δt t1

t0

t1 t0

= 0.

In case of catastatic constraints we have dE = dT − dU = dW − dU = −Udt , so that Δ∫

t1 t0

L dt + E Δt tt10 = 0 ;

(20.1.46)

we find thus a variational principle of Hamilton type, in the same conditions Δt0 = Δt1 = 0 . In fact, this is Voss’s principle, stated for L = T + U in the restricted frame of catastatic constraints, case in which the generalized mechanical energy is reduced to the mechanical energy ( E = E ). The condition Δt0 = Δt1 = 0 is very inconvenient. To eliminate it, we can start from the relation (20.1.43iv), which can be written also in the form t1

Δ ∫ ( L + h ) dt + ( E − h ) Δt t0

t1 t0

= 0,

where h is constant with respect to time. If L = 0 , then we obtain Jacobi’s first integral (or of Jacobi type) E = h ; so we can state Theorem 20.1.9 Among all possible motions of a generalized conservative mechanical system, subjected to ideal constraints, on asynchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) for which t1

Δ ∫ ( L + h ) dt = 0 , h = const , t0

(20.1.47)

takes place. Also this result can be used as a principle; in fact, it is a generalization of Hamilton’s principle for which there are not imposed restrictions concerning time variations (at the ends of the time interval). If E = E = h in case of catastatic constraints, we can state the same result for a conservative mechanical system; starting from the relation (20.1.46), one obtains the same principle. As well, one can obtain the same result using Hölder’s principle.

20.1.3.3 Changes of Independent Variable. Conformal Mappings As we have seen, one of the important properties of the variational principles is the possibility to use any system of co-ordinates; the generalized form (20.1.47) of Hamilton’s principle, corresponding to a generalized conservative mechanical system, allows to make a step further, i.e. to introduce a change of independent variable. Let be dt = udθ ,

(20.1.48)


241

where u = u ( q1 , q 2 ,..., qs ) is a given positive function of class C 1 ; we may consider θ as an artificial time, measured in the motion of the representative point in the space Λs as in Sects. 19.2.2.1, 19.2.2.3, and 19.2.2.3. Denoting by L ′ = L ′ ( q1 , q2 ,..., qs , q1′ , q2′ ,..., qs′ ) , q j′ = dq j / dθ , j = 1, 2,..., s , the new form of the function L by introducing the new generalized velocities, we can write the principle (20.1.47) in the form θ1

Δ ∫ Λ dθ = 0 , Λ = u ( L ′ + h ) . θ0

(20.1.47')

The corresponding Euler–Lagrange equations

d ⎛ ∂Λ ⎞ ∂Λ − = 0 , j = 1, 2,..., s , dθ ⎜⎝ ∂q j ⎟⎠ ∂q j

(20.1.49)

hold only for generalized conservative mechanical systems ( L ′ = dL ′ / dθ = 0 and E = h ); one obtains thus the trajectory of the representative point in the phase space, but not the direct connection between the position of the point on the orbit and the time. If we denote by “prime” the change of independent variable in a certain quantity, we can write T2 = θ2T2′ = T2′ / u 2 , T1 = θT1′ = T1′ / u , so that Λ = u (T2 + T1 + T0 + U + h ) =

T2′ + T1′ + u (T0 + U + h ) ; u

(20.1.50)

the corresponding first integral of Jacobi type leads to the condition T2′ − (T0 + U ) = h . u2

(20.1.50')

In particular, let be a mechanical system with two degrees of freedom for which T2 =

1 ( g q 2 + 2g12q1q2 + g22q22 ) , T1 = g1q1 + g2q2 ; 2 11 1

(20.1.51)

by a convenient transformation we can pass to isothermic (isometric) co-ordinates for which (by convention, we can use the same notation for the generalized co-ordinates) T2 =

1 u ( q12 + q22 ) , 2

T1 = α1q1 + α2q2 .

(20.1.51')

By a change of independent variable of the form (20.1.48), we obtain Λ=

1 2 ( q ′ + q2′2 ) + α1q1′ + α2q2′ + β , β = u (T0 + U + h ) ; 2 1

there results the equations of motion

(20.1.51'')


242 q1′′ + γq 2′ =

∂α ∂α ∂β ∂β , q 2′′ − γq1′ = , γ = 1 − 2. ∂q1 ∂q 2 ∂q 2 ∂q1

(20.1.52)

The isoenergetic orbits (of energy h ) are specified by the system (20.1.52), satisfying the relation 1 2 ( q ′ + q2′2 2 1

) = β.

(20.1.52')

We obtain thus the normal form of the equations of motion of a generalized conservative mechanical system with two degrees of freedom; these are the equations of the plane motion of a particle subjected to a field of conservative forces of potential β and to a gyroscopic force γ v . The method of calculation presented above may be used successfully also for Liouville type systems, considered in Sects. 18.3.1.2 and 19.2.2.2, obtaining similar results. Let be a mechanical system with two degrees of freedom, of Lagrangian 1 2

L = ( q12 + q22 ) + U ( q1 , q2 ) .

(20.1.53)

In case of a conformal mapping z = q1 + iq 2 = f ( ζ ) , ζ = ξ1 + iξ2 , where f is a regular function in a convenient domain for ζ , we may write dz = M dζ , M = f ′ ( ζ ) , so that 1 2

L = M 2 ( ξ12 + ξ22 ) + W ( ξ1 , ξ2 ) , W ( ξ1 , ξ2 ) = U ( q1 , q2 ) .

(20.1.53')

Applying the method of change of independent variable, we find Λ=

1 2 ( ξ ′ + ξ2′2 ) + M 2 (W + h ) , 2 1

(20.1.54)

with a condition of energetical nature 1 2 ( ξ ′ + ξ2′2 2 1

) = M 2 (W

+ h ).

(20.1.54')

For instance, the conformal mapping z = ζ 2 corresponds to the study of the Keplerian motion.

20.1.4 Maupertuis’s Principle. Other Variational Principles In what follows, we will present Maupertuis’s principle in various forms, in the general asynchronous case; as well, we will put into evidence the possibility to enounce also other variational principles (for instance, the principle of the least potential action), equivalent to Hamilton’s and Maupertuis’s principles.


243

20.1.4.1 Maupertuis’s Principle We will introduce the kinetical action in the form

AT =

t1

∫t

Td t =

0

1 t1 ∂L q dt , 2 ∫t0 ∂q j j

(20.1.55)

in the hypothesis of a conservative mechanical system and of catastatic constraints. We mention that, often, one uses the quantity A = 2 AT , called action. We admit also that the mechanical energy is constant along any varied path; if E = h on the true curve, then it results E + ΔE = h + Δh on a varied path. Starting from the relation (20.1.42') and using the condition L = 0 (we have T = U = 0 ), we find t1

∫t

0

d ∂L ∂L ⎛ ⎞ ⎜ ΔL + ∂q q j dt Δt ⎟ dt = ∂q Δq j j j ⎝ ⎠

t1

. t0

Applying the operator Δ to the relation (20.4.43'), we can write ⎛ ∂L ⎞ ΔE = Δ ⎜ q j ⎟ − ΔL = Δh , ⎝ ∂q j ⎠

so that d ∂L ∂L ⎡ ⎛ ∂L ⎞ ⎤ ⎢ Δ ⎜ ∂q q j ⎟ − Δh + ∂q q j dt Δt ⎥ dt = ∂q Δq j j j ⎣ ⎝ j ⎠ ⎦

t1

∫t

0

t1

; t0

using the relation (20.1.14), we get Δ AT =

t1 ∂L 1 1 ∂L Δ q dt = Δq j 2 ∫t0 ∂q j j 2 ∂q j

t1 t0

+

1 ( t − t0 ) Δh . 2 1

(20.1.55')

Besides, starting from the relation (20.1.43'') which takes place along the solutions of Lagrange’s equations, and making E = h , we have Δ AL = Δ ∫

t1 t0

Ldt =

∂L Δq j ∂q j

t1

− h ( Δt1 − Δt0 ) ;

t0

by means of the relation (20.1.43'), we obtain Δ∫

t1

t0

∂L ∂L q j dt − Δ [ h ( t1 − t0 ) ] = Δq j ∂q j ∂q j

t1 t0

− h ( Δt1 − Δt0 )


244

and find again the relation (20.1.55'), in the same conditions. Assuming that we have to do with varied paths with fixed ends, for which Δh = 0 , the constants being the same on all these paths (isoenergetic paths), it follows t1

Δ AT = Δ ∫ Tdt = 0 .

(20.1.56)

t0

As a consequence of Lagrange’s equations, in the above mentioned conditions, this relation represents a necessary condition for the motion on the true curve. To put into evidence the sufficiency of this relation, we must show that, starting from the relation (20.1.56), we obtain Lagrange’s equations. To do this, we must solve a problem of extremum with constraints, of the form t1

Δ ∫ Fd t = 0 , F = T + λ ( E − h ) , E = T − U , t0

where λ = λ ( t ) is a Lagrange multiplier. Using the relation (20.1.15) and equating to zero the displacements Δq j , j = 1, 2,..., s , at the ends of the time interval, we may write t1 ∂F ⎛ ⎞ Δ ∫ Fd t = ⎜ F − q j ⎟ Δt t0 ∂q j ⎝ ⎠

t1 t0

+

t1

∫t

0

d ⎛ ∂F ⎞ ⎤ ⎡ ∂F ⎢ ∂q − dt ⎜ ∂q ⎟ ⎥ δq j dt . ⎣ j ⎝ j ⎠⎦

For the sake of simplicity, without loosing the generality, we will suppose that the initial moment at P0 is the same for all varied paths ( Δt0 = 0 ), Δt1 being not fixed; but we fix the co-ordinates q j ( t1 ) , j = 1, 2,..., s , so that we have (we observe that T −U = h ) F −

∂F q = − ( 1 + 2λ )T = 0 ∂q j j

for t = t1 . Using the basic lemma of the variational calculus, we obtain Euler– Lagrange equations in the form d dt

∂T ⎤ ∂T ∂U ⎡ ⎢ ( 1 + λ ) ∂q ⎥ − ( 1 + λ ) ∂q = − λ ∂q , j = 1, 2,..., s , j ⎦ j j ⎣

or in the form ⎡ d ⎛ ∂T ⎞

∂T

∂U ⎤

∂T

∂U

(1 + λ ) ⎢ ⎜ ; − = − λ − ( 1 + 2λ ) ⎟− ∂q j ⎦⎥ ∂q j ∂q j ⎣ dt ⎝ ∂q j ⎠ ∂q j

multiplying by q j and summing for all indices j , we have ⎡ d ⎛ ∂T

⎞

dT

dU ⎤

∂T

dU

(1 + λ ) ⎢ ⎜ q j ⎟ − − = − λ q − ( 1 + 2λ ) , dt ⎦⎥ ∂q j j dt ⎣ dt ⎝ ∂q j ⎠ dt


245

where we took into account that U = T = 0 , corresponding to the catastatic constraints and to the condition L = 0 . Because the kinetic energy is a quadratic form, we can write (1 + λ )

d (T − U ) dU ; = −2T λ − ( 1 + 2λ ) dt dt

taking into account that T − U = h , we remain with −2T λ − ( 1 + 2λ )

dU d = − [ ( 1 + 2λ )T ] = 0 . dt dt

Hence, ( 1 + 2λ )T is constant along the true curve; the condition imposed for t = t1 shows that this constant vanishes, so that λ = −1/ 2 at any moment. With this value of λ , we see that the above Euler–Lagrange equations are just Lagrange’s equations of motion. One can thus state Theorem 20.1.10 (Maupertuis). Among all possible motions of a mechanical system subjected to catastatic ideal constraints, acted upon by conservative given forces, on asynchronous varied isoenergetic paths with fixed ends, only and only the motion (of the representative point on the true curve) which makes stationary the kinetic action (20.1.56) takes place. As in the cases considered in the previous Section, we can consider that this result is a variational principle: the principle of the least action (Maupertuis’s principle). This principle played an important rôle in the development of mechanics, being the first variational principle of mechanics, enounced for the first time by P.L. Moreau de Maupertuis in 1740 in a quite obscure form; this formulation led to a passionate controversy, especially with S. Koenig, a controversy in which we mention also the intervention of François Marie Arouet dit Voltaire (an adversary of Maupertuis). The problem has been taken again by L. Euler (1744) who gave the formulation of the principle in case of central forces; Lagrange (1760) and then Jacobi (1847) have given its general form. Sometimes, the principle of least action is called also the MaupertuisLagrange principle. Obviously, for Δt = 0 we can enounce an analogous synchronous principle. As we have seen, in case of catastatic constraints the relation (20.1.35') is reduced to (20.1.45); comparing with Maupertuis’s principle, written in the developed form t1

∫t

0

( ΔT + T ddt Δt ) dt = 0 ,

and eliminating the term Td ( Δt ) / dt , we can state Theorem 20.1.11 Among all possible motions of a mechanical system, subjected to catastatic ideal constraints, on asynchronous varied paths with fixed ends, only and only the motion (of the representative point on the true curve) which makes stationary the kinetic action (20.1.56) with the condition t1

∫t

0

ΔTdt =

t ∫t ΔLdt 1

0

(20.1.57)


246

takes place. In case of quasi-conservative given forces, deriving from a simple quasi-potential, we have ΔL = ΔU − U Δt , so that the condition (20.1.57) is reduced to

∫t ( ΔE + U Δt ) dt t1

= 0;

(20.1.57')

0

if the forces are conservative, then there result E = h and U = 0 , the condition (20.1.57') being identically verified. We find again Maupertuis’s principle. We observe that the Theorem 20.1.11 represents a generalization (larger conditions) of the Theorem 20.1.10; obviously, we can affirm the same thing for the corresponding variational principles. Eliminating U between L = T + U and T − U = h , it follows L = 2T − h ; from (20.1.38') and (20.1.55) we obtain, in this case,

AL = 2 AT − h ( t1 − t0 ) = A − h ( t1 − t0 ) ,

(20.1.58)

a relation connecting the kinetic and the Lagrangian actions. If t0 = 0 and t1 = t , we find again the relation (19.2.15'), connecting the function S (t ) of the Hamilton-Jacobi theory (to which corresponds AL ) to the function S (t ) , which appears in case of generalized conservative mechanical systems (to which corresponds A = 2 AT , according to the formula (19.2.17)).

20.1.4.2 Other Forms of the Least Action Principle We can express the action

A = 2 AT also in the form

A =

t1 n

mi vi2 dt ∫t i∑ =1 0

for a mechanical system S of n particles Pi , i = 1, 2,..., n ; Maupertuis enounced the least action principle in the synchronous form δ∫

S1 n S0

∑ mi vi dsi

i =1

= 0,

where si is the curvilinear abscissa on the trajectory of the particle Pi ,

(20.1.59)

S0 and S1

being the states of the system S at the time t0 and t1 , respectively. Hence, the elementary action 2Tdt is equal to the elementary work of the momenta of the considered mechanical system S. In case of a single particle P , we obtain δ∫

P1 P0

mv ds = 0 ;

(20.1.59')


247

if on the particle does not act any given force, its motion will be uniform (as it results from the study of the motion in an intrinsic frame of reference), so that v = const , and the relation (20.1.58') is reduced to δ∫

P1 P0

ds = 0 .

(20.1.59'')

The trajectory of the particle P will be, in this case, a “geodesic line” in the constraint conditions imposed to the motion. If the particle moves on a given fixed surface, then the trajectory is a geodesic line on the respective surface, passing through the points P0 and P1 . If there are not constraint relations, then the geodesic is a straight line and we find again the principle of inertia. Using the conservation theorem of mechanical energy, Jacobi eliminates the time in the least action principle. Starting from

2T =

n

∑ mi vi2 =

i =1

n

∑ mi

i =1

( ), dsi dt

2

we find δA = δ ∫

S1 S0

2 T (U + h )dt = δ ∫

= δ∫

S1 S0

2 (U + h )

n

∑

i =1

S1 S0

n

2T

mi dsi2

∑ mi dsi2

i =1

= 0.

(20.1.60)

This synchronous form of the least action principle represents Jacobi’s principle and has a pure geometric character; we observe that the integral A , in which the time disappeared, can be calculated by means of any other parameter. As well, the condition that the motion be isoenergetic on all varied paths becomes irrelevant; this is the great advantage of this formulation. One observes that, using a Lagrangian of the form Λ = 2 T (U + h ) , one obtains s differential equations of Euler–Lagrange type, which are not independent; but they only allow to obtain the trajectory of the representative point in the space Λs . If we impose the condition T − U = h , then the velocity on this trajectory is also determined. In generalized co-ordinates, we have n

∑ mi dsi2

i =1

= gij dqi dq j ,

so that

A =

S1

∫S

2 (U + h ) gij dqi dq j .

(20.1.60')

0

Replacing the mechanical system S by a representative point P in the Lagrangian space Λs , it is convenient to endow this space with the metrics


248

dσ 2 = gij dqi dq j .

(20.1.61)

In the case in which the mechanical system is not subjected to any given force (U = const ), we can neglect a multiplicative constant, remaining the variational condition δ∫

S1 S0

dσ = 0 ;

(20.1.61')

thus, we state Theorem 20.1.12 (Jacobi). Let be given a discrete mechanical system S subjected to holonomic, ideal constraints, which is acted upon by no one given force; the trajectory of the representative point P in the space Λs , endowed with the metrics (20.1.61), is a geodesic line of this space. As in Sect. 7.2.1.6, we can write the differential equations of the geodesic lines in the form g jk qk′′ + [ kl , j ]qk′ql′ = 0 , j = 1, 2,..., s ,

(20.1.62)

where [ kl , j ] is Christoffel’s symbol of the first kind; the differentiation is made with respect to an arbitrary parameter, which can be even the curvilinear abscissa along the trajectory of the representative point. Multiplying by the normalized algebraic complement, summing and using Christoffel’s symbol of the second kind, there result the equations of the geodesic lines in the normal form ⎧⎪ j ⎫⎪ q j′′ + ⎨ ⎬ qk′ ql′ = 0 , j = 1, 2,..., s ; ⎪⎩ k l ⎪⎭

(20.1.62')

we notice the analogy which, obviously, does exists between these equations and the normal form (18.2.47') of Lagrange’s equations, in which we make Q j∗ = 0 , j = 1, 2,..., s . If the given forces are non-zero, hence if the potential U is not constant, then we introduce a manifold Λs of metrics dσ 2 = 2 (U + h ) dσ 2 = gij dqi dq j , gij = 2 (U + h ) gij , i , j = 1, 2,..., s ,

(20.1.63)

so that δA = δ∫

S1 S0

dσ = 0 .

(20.1.63')

We can thus state Theorem 20.1.12' (Jacobi). Let be given a discrete mechanical system S subjected to holonomic, ideal constraints and acted upon by conservative given forces; the trajectory of the representative point P , corresponding to the manifold Λs and endowed with the metrics (20.1.62) is a geodesic line of this manifold.


249

After obtaining the geodesic lines, the law of motion is given by

dt =

dσ dσ = . ( 2 U +h) ( ) 2 U +h

(20.1.64)

The Theorems 20.1.12 and 20.1.12' put in evidence the character of minimum of the considered variational principle, as well as the denomination given to it. In fact, if the functional is not minimized, it remains – at least – stationary.

20.1.4.3 The Analogy with Fermat’s Principle From an optical point of view, a material medium is characterized by the refraction index n =

c , v

(20.1.65)

where c is the velocity of light in vacuum, while v is its velocity in the respective medium. This index is a function of point ( n = n ( x1 , x 2 , x 3 ) ) in a non-homogeneous medium; in an anisotropic medium, it is function also of direction. Let us consider a medium, in general, non-homogeneous; the duration of light propagation is given by (we take t0 = 0 , by convention) t =

s

∫s

0

dσ 1 s = ∫ nd σ , v c s0

where dσ is the arc element, while s is the curvilinear co-ordinate along the path of the light photon. Corresponding to Fermat’s principle, among all paths which join two points of a material medium, the light photon follows the shortest optical path, so that s

δ ∫ ndσ = 0 , s0

(20.1.66)

in a synchronous formulation. We find thus a formal analogy between this optical principle and Maupertuis’s mechanical principle. Using (20.1.60), we can write, for a single particle, δ∫

P P0

2 (U + h )ds = 0 ,

neglecting the factor m . The trajectory of the light photon coincides with the trajectory of the particle if U = n 2 / 2 − h . If we assume that, in the vicinity of the Earth surface, the refraction index decreases linearly with the altitude z , we can write n = n 0 ( 1 − kz / H ) , where H is the height of the atmosphere, n0 is the refraction index at the surface of the Earth, while k is a constant. Neglecting ( z / H )2 with respect to z / H , we obtain


250 U = c − gz , c =

n2 1 2 n0 − h , g = k 0 , H 2

(20.1.67)

hence the potential of a gravitational force in the proximity of the Earth surface, with a conventional gravitational acceleration. Hence, if the refraction index n has a linear variation with the altitude, then the light is propagated along a parabola with a vertical axis. The refraction index will have the form n = n ( q1 , q 2 ,..., qs ) in the space Λs , so that we obtain for the photon a geodesic line in a manifold Λs of metrics dσ = ndσ . But the complete analogy between the two principles has been realized in 1924 by Louis de Broglie in the frame of undulatory mechanics; he replaces the phase velocity v by a group velocity u , so that uv = const . In this case, the principle (20.1.66) takes the form s

δ ∫ udσ = 0 , s0

(20.1.66')

analogous to the form (20.1.59') of Maupertuis’s principle for a particle.

20.1.4.4 Plane Motion in a Field of Conservative Forces In a plane motion of a particle on a curve C, we put in evidence the generalized coordinates s (the curvilinear co-ordinate, starting from the fixed point P0 ) and θ (the angle made by the external normal n to the curve C with the fixed axis Ox (Fig. 20.3)). We assume that s = s ( θ ) is an increasing function; if the curve is closed, then it is also convex.

Fig. 20.3 Plane motion in a field of conservative forces

Let be the integral I =

P1

∫P

2 (U + h )ds ;

0

if we pass from the curve C to the varied path C ′ by a displacement δn = δn (s ) along the external normal, we can write


δI =

P1

∫P

0

δU ds + 2 (U + h )

251 P1

∫P

2 (U + h )δ ds .

0

We have δU = N δn , where N is the component of the field of forces along the external normal, while δ ds = δndθ = δnds / ρ , ρ being the curvature radius at the point P . We obtain thus δI =

P1

∫P

0

W 2 (U + h ) δnds , W = N + . ρ 2 (U + h )

(20.1.68)

Writing the equation of motion along the external normal and taking into account the mechanical energy conservation theorem, there results N = − mv 2 / ρ = −2 (U + h ) / ρ , and we find again Jacobi’s form of the least action principle. Let be a convex and closed simple curve C, and C an analogous curve which contains the first one. We can state Theorem 20.1.13 (Whittaker). If a particle P which is in motion in the interior of a ring formed by two conex and closed simple plane curves C (internal) and C (external) and is acted upon by conservative forces for which the potential U has not singularities in this ring, then it admits at least a periodic orbit if W < 0 on C and W > 0 on C .

20.1.4.5 Other Variational Principles We will introduce, following V. Vâlcovici, a kinetic-potential energy Eα ,β = αT + βU

(20.1.69)

of ( α, β ) kind and a corresponding kinetic-potential action

Aα ,β =

t1

∫t

0

Eα , β d t

(20.1.69')

of the same kind. To establish the conditions in which one can enounce an asynchronous variational principle of the form t1

Δ Aα ,β = Δ ∫ Eα ,β dt = t0

t1

∫t

0

( ΔE

α ,β

+ Eα , β

)

d Δt dt = 0 , dt

we start from the general integral principle (20.1.35'), corresponding to the principle of virtual work, the varied paths having fixed ends; subtracting the relation (20.1.35') from the relation t1

∫t

0

we find the condition

( αΔT + βΔU + ( αT + βU ) ddt Δt ) dt = 0 ,

252


t1

{ α − 1 ΔT + βΔU − ΔW + ( P − ddTt ) Δt + [ α − 2 T + βU ] ddt Δt } dt = 0 ,

∫t

(

0

)

(

)

(20.1.70) which allows the determination of the function Δt . In case of catastatic constraints ( P = dT / dt ) and of quasi-conservative given forces ( ΔW = ΔU − U Δt ), we obtain a much more simple form, i.e. t1

∫t

0

{ α − 1 ΔT + ( β − 1 ) ΔU + U Δt + [ α − 2 T + βU ] ddt Δt } dt = 0 . (

)

(

)

(20.1.70') If the given forces are conservative (U = 0 ), then we can write a mechanical energy conservation theorem ( E = T − U = h ), so that the condition (20.1.70') becomes t1

∫t

0

{ α − 1 Δh + ( α + β − 2 ) ΔU + [ α − 2 h + ( α + β − 2 )U ] ddt Δt } dt = 0 , (

)

(

)

(20.1.70'') where we assumed that along a varied path we have E + ΔE = h + Δh . Supposing further that we have to do with isoenergetic varied paths ( Δh = 0 ), this latter condition has the form t1

( α + β − 2 ) Δ ∫ Udt + ( α − 2 ) h ( Δt1 − Δt0 ) = 0 t0

(20.1.71)

or the form t1

Δ ∫ [ ( α + β − 2 )U + ( α − 2 ) h ] dt = 0 . t0

(20.1.71')

We can thus state Theorem 20.1.14 Among all possible motions of a mechanical system, subjected to catastatic, ideal constraints and acted upon by conservative given forces, on asynchronous varied isoenergetic paths with fixed ends, only and only the motion (of the representative point on the true curve) which makes stationary the action Aα ,β with the condition (20.1.71') takes place. Obviously, in this case too, starting from this theorem, one can obtain a corresponding variational principle. In particular, for α = 2 and β = 0 we find again the Theorem 20.1.10 of Maupertuis. As well, for α = β = 1 and Δt0 = Δt1 we find a particular form of Voss’s principle; but we observe that, if we make α = β = 1 in the condition (20.1.70'), then we arrive at the form considered in Sect. 20.1.3.2. Taking α = 0 and β = 2 , as well as Δt0 = Δt1 , we obtain an asynchronous least potential principle stated first by V. Vâlcovici and afterwards, in general conditions, by P.P. Teodorescu) (the same duration on all varied paths)


253

t1

Δ AU = Δ ∫ U dt = 0 .

(20.1.72)

t0

To can formulate a least mechanical action principle in the form t1

Δ AE = Δ ∫ E dt = 0 ,

(20.1.73)

t0

we must have E ≠ const ; starting from the condition (20.1.70') and making α + β = 0 , we find ( E = T − U , L = T + U ) α∫

t1

t0

( ΔE + E ddt Δt ) dt + ∫ (U Δt − ΔL − 2T ddt Δt ) dt = 0 . t1

t0

The first variation vanishes, so that we can have a least mechanical action principle if the condition t1

∫t

U Δt dt =

0

t1

∫t

0

( ΔL + 2T ddt Δt ) dt

(20.1.73')

is verified; we find a more intricate relation (specifying Δt as a function of t ) as in the previous case.

20.1.4.6 Larmor’s Principle If some ignorable co-ordinates which not appear in the kinetic potential L do exist (see Sect. 18.2.3.6), we can state a variational principle, in the frame of which appears a functional which becomes stationary on a certain class of varied paths. We have thus to do with a holonomic mechanical system with s degrees of freedom, in which the first r generalized co-ordinates are ignorable. Let thus be the functional t ⎛ ∫t ⎜⎝ L 1

0

−

d r ∂L ⎞ dt . qk ∑ dt k =1 ∂qk ⎟⎠

Using the relation (20.1.43'''), we obtain the variation of this functional in the form t

r ⎡ ∂L ⎤1 ⎡ ∂L ⎛ ∂L ⎞ ⎤ 1 − − + Δ q E Δ t Δ q q Δ j ⎜ ∂q ⎟ ⎥ ∑ k k ⎢ ∂q ⎥ ⎢ ⎝ k ⎠ ⎦t 0 ⎣ j ⎦t0 k =1 ⎣ ∂qk t

t

1 r ⎡ s ∂L ⎛ ∂L ⎞ ⎤ = ⎢ ∑ Δq j − EΔt − ∑ qk Δ ⎜ . ⎟ ⎥ ⎝ ∂qk ⎠ ⎦t k =1 ⎣ j =r + 1 ∂q j 0

The varied motion will be subjected to the restrictions (18.2.79'), with the same constant as in the true motion. In this case (we calculate the derivative d / dt under the integral sign) t

r ∂L t1 ⎛ ⎡ s ∂L ⎤1 ⎞ Δ∫ ⎜ L − ∑ qk ⎟ dt = ⎢ ∑ Δq j − EΔt ⎥ . t0 ⎝ ⎠ k = 1 ∂qk ⎣ j = r +1 ∂q j ⎦t 0


254

By means of the relations (18.2.79') and by the normal procedure, we can express the above integrand as a function of the constants C 1 ,C 2 ,...,C r and of the generalized velocities qr +1 , qr + 2 ,..., qs , obtaining thus the function R of Routh, which depends on the palpable co-ordinates qr +1 , qr + 2 ,..., qs . Fixing the palpable co-ordinates at the ends of the time interval (not also the ignorable co-ordinates) and fixing the initial and the final times ( Δt0 = Δt1 = 0 ), there results r t1 t1 ⎛ ∂L ⎞ Δ AR = Δ ∫ Rdt = Δ ∫ ⎜ L − ∑ qk ⎟ dt = 0 . t0 t0 ⎝ q ∂ k ⎠ k =1

(20.1.74)

We may state Theorem 20.1.15 (Larmor). Among all possible motions of a generalized conservative mechanical system, subjected to ideal constraints, acted upon by quasi-conservative given forces, on asynchronous varied paths with fixed ends (in space, for the palpable co-ordinates, and in time), only and only the motion (of the representative point on the true curve) which makes stationary the Routhian action (20.1.74) takes place. This result can be considered as a generalization of the Theorem 20.1.8 and may be used as a variational principle: Larmor’s principle. Starting from this principle, we obtain – in the usual way – Routh’s equations.

20.1.5 Continuous Mechanical Systems After some results with a general character, we consider some particular mechanical systems; we will thus present the basic equations in the motion of strings, of straight bars and of linear elastic solids. We used till now analytical methods of calculus only to study discrete mechanical systems (with a finite number of degrees of freedom); in case of an infinite number of degrees of freedom, the problems impose a specific treatment. For instance, in case of vibrations of an elastic solid, every one of its point participates at this motion; the motion may be described only if all parameters which indicate the position of each point are involved.

20.1.5.1 Lagrangian Formalism To put into evidence the treatment of the problem, we will start from a continuous mechanical system, which will be studied – at the beginning – by discrete methods in a Lagrangian formalism. Let be a linear elastic, isotropic and homogeneous straight bar (see Sect. 12.1.1), subjected to longitudinal small motions; it can be approximated (mathematically modelled) by a discrete system of particles Pi of equal masses m along the axis, separated at equal distances a (in a position of equilibrium) by springs of elastic constants k (Fig. 20.4a); during the motion, the particle Pi has a displacement ui along the bar axis, so that ui = ui ( t ) (Fig. 20.4b). As a matter of fact, we study thus a linear polyatomic molecule (case in which the number of particles is finite). We obtain thus the Lagrangian


L = T +U =

255

1 1 mui2 − ∑ k ( ui + 1 − ui )2 , 2∑ 2 i i

the forces which act on those particles (because of the springs at left and right) being Fi = − k ( ui − ui −1 ) + k ( ui +1 − ui ) . We may also write

L =

∑ aLi

, Li =

i

(

u − ui 1 ⎡m 2 ui − ka i +1 ⎢ a 2⎣a

)

2

⎤; ⎥⎦

there result Lagrange’s equations u −u u −u m u − ka i +1 2 i + ka i 2 i −1 = 0 . a i a a

In a process of passing to limit, m / a tends to the density μ , ( ui +1 − ui ) / a tends to the linear strain ε , Fi tends to the force F , while k tends to the rigidity of axial forces EA , where E is the longitudinal elasticity modulus and A is the area of the cross section ( F = EAε ). In this case, the kinetic potential will be (for a bar of length l)

Fig. 20.4 Discrete model of a linear elastic, isotropic, homogeneous straight bar l

∫ 0 L dx =

( ) ⎤⎥⎦ dx ,

1 ∞ ⎡ 2 ∂u μu − EA ⎢ ∫ −∞ ∂x 2 ⎣

2

(20.1.75)

where u = u ( x ; t ) and x is the abscissa along the bar axis; observing that, by passing to the limit,

( ) − ( ∂∂ux )

EA ⎡ ∂u a ⎣⎢ ∂x

x

x −a

⎤ ⎦⎥

leads to EA ( ∂ 2 u / ∂x 2 ) , we find the equation of motion μu − EA

∂ 2u = 0. ∂x 2

(20.1.75')

So, characteristic aspects of passing from discrete to continuous systems are put in evidence. As one can see, the position co-ordinate x is not a generalized one; it plays


256

the rôle of a continuous index, replacing the numerable index i . As to every index i corresponds a generalized co-ordinate ui , so to every x corresponds a generalized coordinate u ( x ) ; taking into account also the dependence on time, we must have u = u ( x ; t ) . In the above conditions, L is – in fact – a Lagrangian density; in what follows we will use the notation L (as for the Lagrangian), because only the Lagrangian density plays a rôle in describing the motion of the mechanical system. The motion of a three-dimensional continuous mechanical system, which occupies the domain Ω , will be described – in general – by means of the Lagrangian

∫Ω L dΩ , L

=

L ( u , u1 , u2 , u3 , u ; x1 , x 2 , x 3 ; t ) ,

(20.1.76)

where L is the Lagrangian density, while ui = ∂u / ∂x i , i = 1, 2,..., n , u = ∂u / ∂t , x1 , x 2 , x 3 , t being independent variables. Hamilton’s principle in a synchronous case is written in the form t1

δ ∫ dt ∫ t0

Ω

L dΩ = 0 .

(20.1.76')

We notice that, in this case, the parameters x1 , x 2 , x 3 , t have not variations, the variation process having no influence neither on the integration domain Ω , nor on the integration limits with respect to time; thus, δu will vanish on ∂Ω as well as for t0 and t1 . On the basis of the results in Sect. 20.1.1.3, we can write Euler-Ostrogradskiĭ equations d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L − = 0, ⎜ ⎟+ dt ⎝ ∂u ⎠ dx i ⎜⎝ ∂ui ⎟⎠ ∂u

(20.1.77)

where we use the summation convention of dummy indices. Instead of s differential equations of second order for a discrete mechanical system, in case of a continuous one we find a partial differential equation in four variables: x1 , x 2 , x 3 , and t . In general, we may have several types of displacements u α = u α ( x1 , x 2 , x 3 ; t ) , α = 1, 2,...,m ; for instance, in case of vibrations of an elastic solid we have m = 3 . One obtains the equations of motion d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L − = 0 , α = 1, 2,...,m . ⎜ ⎟+ dt ⎝ ∂u α ⎠ dx i ⎜⎝ ∂uiα ⎟⎠ ∂u α

(20.1.78)

Introducing the functional derivative (the variational derivative – an extension of the Euler–Lagrange derivative) δ ∂ d ∂ = − δu α ∂u α dx i ∂uiα

(20.1.79)

and observing that δL / δu α = ∂L / ∂u α , α = 1, 2,...,m , we can write the equations of motion also in the form


257

d ⎛ δL ⎞ δL = 0 , α = 1, 2,...,m , ⎜ ⎟− dt ⎝ δu α ⎠ δu α

(20.1.79')

similar to that of Lagrange’s equations. 20.1.5.2 Hamiltonian Formalism To put in evidence the possibility of using the Hamiltonian formalism, we take again the example considered at the preceding subsection by discretization. We introduce thus the generalized momenta ∂L ∂L =a i ∂ui ∂ui

pi =

and build up the Hamiltonian H =

∑ pi ui

∂L

∑ a ∂uii ui

−L =

i

−L =

i

∂L

∑ a ⎛⎜⎝ ∂uii ui i

− Li ⎞⎟ . ⎠

Using the same process of passing to limit and introducing the Lagrangian density L , we obtain the Hamiltonian H =

∞

⎛ ∂L u − L ⎞ dx ; ⎟ ∂u ⎠

∫ −∞ ⎜⎝

we are thus led to use further the notion of density. In the motion of a three-dimensional continuous mechanical system which occupies the domain Ω and the position of which is specified by m generalized co-ordinates u α = u α ( x1 , x 2 , x 3 ; t ) , α = 1, 2,...,m , we introduce the generalized momenta density πα =

∂L , ∂u α

(20.1.80)

hence, the Hamiltonian density

H =

m

∑ πα u α − L

,

(20.1.81)

α =1

of the form H = H ( u α , u1α , u2α , u3α , u α , π α ; x1 , x 2 , x 3 ; t ) . Hamilton’s function is thus given by H =

so that

∫Ω H

dΩ =

⎛

m

α α ∫Ω ⎜⎝ α∑=1 π u

⎞ − L ⎟ dΩ , ⎠

(20.1.81')


258 dH =

⎡

∂H

⎛ α ∫Ω ⎢⎣ α∑=1 ⎜⎝ ∂u α du m

+

∂H ∂H dπ α + α duiα ⎞⎟ + ∂π α ∂ui ⎠

H dt ⎤⎥ dΩ . ⎦

Integrating by parts and using the same procedure as in Sect. 20.1.1.3, we obtain dH =

⎧

∂H

∂H d ⎛ ∂H ⎞ α ⎤ ⎫ dπ α + du + H dt ⎬ dΩ ∂π α dx i ⎝⎜ ∂uiα ⎠⎟ i ⎦⎥ ⎭ ∂ H ⎡ m ⎛ ∂H ⎤ = ∫ ⎢ ∑ ⎜ α du α + α dπ α ⎞⎟ + H dt ⎥ dΩ , Ω⎣ ∂π ⎠ ⎦ α = 1 ⎝ ∂u

⎡ α ∫Ω ⎨⎩ α∑=1 ⎢⎣ ∂u α du m

+

where we used the notation (20.1.79) and observed that δ H / δπ α = ∂H / ∂π α . If we differentiate, starting from the second form (20.1.81'), we can write (we use the functional derivative) dH =

δL δL ⎡ m ⎛ α α ⎤ α α α α ⎞ ∫Ω ⎢⎣ α∑=1 ⎜⎝ u dπ +π du − δu α du − δu α du ⎟⎠ − Ldt ⎥⎦ dΩ ;

the notation (20.1.80) and the equations (20.1.79'), written in the form δL δL α , α =π α , α =π δu δu

(20.1.82)

lead to dH =

⎡

m

α α ∫Ω ⎢⎣ α∑=1 ( − π du

⎤ + u α dπ α ) − Ldt ⎥ dΩ . ⎦

Comparing the two expressions of dH , we find the equations of motion u α =

δH δH , π α = − α , δu δπ α

(20.1.83)

analogue to Hamilton’s equations, as well as the relation

H = − L .

(20.1.83')

In a developed form, we have u α =

∂H d ⎛ ∂H ⎞ ∂H , π α = − α + , ∂π α dx i ⎜⎝ ∂uiα ⎟⎠ ∂u

(20.1.83'')

obtaining a form which has no more a symmetric character. The canonical form of Hamilton’s synchronous principle will be δ∫

t1 t0

⎡ m α α ⎤ α α α α α ∫Ω ⎢⎣ α∑=1 π u − H ( u , u1 , u2 , u3 , π ; x1 , x 2 , x 3 ; t ) ⎥⎦ dΩdt = 0 .

(20.1.84)


259

We may calculate dH = dt

⎡

m

∂H

⎛ α ∫Ω ⎢⎣ α∑=1 ⎜⎝ ∂u α u

+

∂H α ⎞ ⎤ π ⎟ + H ⎥ dΩ ; ∂π α ⎠ ⎦

with the aid of the equations of motion (20.1.83) we observe that, along the corresponding integral curves, we obtain dH = dt

∫Ω H dΩ .

(20.1.85)

Hence, the Hamiltonian H is conserved in time if the Hamiltonian density H does not depend explicitly on time ( H = 0 ). In general, let be the functional F =

∫Ω F dΩ ,

(20.1.86)

where F = F ( u α , u1α , u2α , u3α , u α , π α ; x1 , x 2 , x 3 ; t ) . We can write dF = dt

⎡ m ⎛ ∂F α ∂F α ⎞ ⎤ ∫Ω ⎢⎣ α∑=1 ⎜⎝ ∂u α u + ∂πα π ⎟⎠ + F ⎥⎦ dΩ ;

taking into account the equations of motion (20.1.83), there results dF = dt

⎡

m

δF δ H δF δ H ⎞ ⎤ ⎟ + F ⎥ dΩ . α − δπ δπ α δu α ⎠ ⎦

⎛ ∫Ω ⎢⎣ α∑=1 ⎜⎝ δu α

(20.1.87)

We observe thus that we obtain a quantity similar to the Poisson bracket; using an analogous notation, we may write dF = dt

∫Ω ⎡⎣ ( F , H

) + F ⎤⎦ dΩ .

(20.1.87')

We may thus affirm that F is conserved in time if simultaneously ( F , H ) = 0 and

F = 0 ; one can also say that F is a constant of motion. The conservation theorems for integral quantities can be obtained as in the previous chapter, and one can develop analogous methods of calculation. But we must observe that, besides the theorems concerning “macroscopic” constants of integration (for integral quantities), one can state theorems for “microscopic” such constants (for densities); the latter results are of a special interest in mechanics of continuous deformable media. We remark that all the above considerations may be used also for a field description with the aid of variational principles (even in the absence of a mechanical system). 20.1.5.3 Motion of Linear Elastic Solids To determine the density L of the Lagrangian in case of a linear elastic solid, we must calculate the densities T and V of the kinetic energy, and of the potential energy, respectively, so that


260 T =

∫Ω TdΩ , V

=

∫ Ω Vd Ω ;

(20.1.88)

to do this, we need some preliminary considerations. Let f0 ( x10 , x 20 , x 30 ; t ) be a function of class C 1 , expressed by means of material coordinates; using spatial co-ordinates, we obtain the function in the form f ( x1 , x 2 , x 3 ; t ) (see Sect. 12.1.1.1). The partial derivative of first order with respect to one of the variables, e.g. the variable x10 , is written in the form

∂u1 ⎞ ∂f ∂f ∂x k ∂f ⎛ ∂f ∂u2 ∂f ∂u3 , = = + ⎜ 1 + 0 ⎟ + ∂x 0 0 0 ∂ x ∂ x ∂ x 3 ∂x10 ∂x1 ∂x1 ⎠ 1 ⎝ 2 ∂x 1 k ∂x 1

(20.1.89)

where we took into account (12.1.3''), u j , j = 1, 2, 3 , being the components of the displacement vector; neglecting the displacement gradient (the tensor αij = ∂u j / ∂x10 ,

i , j = 1, 2, 3 ) with respect to unity, one has ∂f0 ∂f . = 0 ∂x1 ∂x1

(20.1.89')

Hence, in this case, we obtain the same results, in the same form, either we use material or spatial co-ordinates. The symmetric part of the tensor αij is the strain tensor ( εij = u( i , j ) = ( ui , j + u j ,i ) / 2 = α( i , j ) ); if i = j one obtains the linear strains, while if i ≠ j , γij = 2 εij , i , j = 1, 2, 3 , represent the angular strains. The hypothesis of neglecting

the displacement gradient with respect to unity includes thus the case of infinitesimal deformations, case in which the relations between strains and displacements are linear. The antisymmetric part of the tensor αij is the local rotation of rigid body tensor ( ωij = u[ j ,i ] = ( u j ,i − ui , j ) / 2 = α[ i , j ] ) is also neglected with respect to unity (as the displacement gradient). We may thus speak about a linear theory from a geometric point of view (case in which relations of the form (20.1.89') hold). Taking into account the notions introduced in Sect. 12.1.1.1, from relations (12.1.2'') and (12.1.3'') we obtain the velocity and the acceleration (in spatial description) v =

du ∂u ∂u dv d2 u ∂v ∂v vj + , a= vj + = = 2 = dt ∂x j ∂t x dt ∂ ∂t dt j

(20.1.90)

for a point of the solid; the vector r0 (corresponding to the initial moment) is constant. Neglecting the non-linear terms, we remain with the displacement velocity (neglecting the spatial derivative, the total derivative is reduced to the time one) v =

∂u = u ∂t

(20.1.90')


261

and with the displacement acceleration a =

∂v ∂2 u . = v = 2 = u ∂t ∂t

(20.1.90'')

In this case, the kinetic energy density will be given by 1 2

T = μui ui ,

(20.1.91)

where μ = μ ( r ; t ) is the density. The work effected by the external given forces is stored by the elastic solid and given back by unloading and bringing to the initial form as a work of internal forces (or internal work); it is called also deformation work or strain energy. We denote by w the unit deformation work (the deformation work density or the elastic potential), that is the work corresponding to the unit volume in the vicinity of a point of the body, and which the state of stress (the totality of stresses around a point) yields by the state of strain (the totality of strains around a point) of the solid at the very same point. We call elementary deformation work (elementary strain energy) the work corresponding to a volume element dΩ of the domain Ω occupied by the solid; we have dWint = wdΩ .

(20.1.92)

The internal work is then given by Wint =

∫Ω wdΩ ,

(20.1.92')

1 σ ε 2 ij ij

(20.1.93)

where w =

is the elastic potential. We have introduced the stress tensor σij , i , j = 1, 2, 3 (if i = j we obtain the normal stresses, while if i ≠ j we obtain the tangential stresses). If we take into account the linear constitutive law of R. Hooke

(

)

1 σij = λ ⎡ εll δij − 2 − ε ⎤ , i , j = 1, 2, 3 , ⎢⎣ ν ij ⎥⎦

(20.1.94)

in case of a homogeneous, isotropic body, where λ is Lamé’s elastic constant and ν is the transverse contraction coefficient of Poisson (Poisson’s ratio), we may write also w =

(

)

1 ⎡ 1 λ ( ε )2 − 2 − ε ε ν ij ij 2 ⎢⎣ ll

⎤. ⎥⎦

(20.1.93')


262

Finally, the potential energy density is given by

V = w − F ⋅u,

(20.1.95)

where F is the volumic force (the external given force on the unit volume). We find thus the kinetic potential density of Lagrange

(

)

L = μu j u j − λ ⎡⎢ (u j , j )2 − 1 − (ui , j + u j ,i )(ui , j + u j ,i ) ⎤ + F ⋅ u ; ⎥⎦ 2 2 ⎣ 2 2ν 1

1

1

1

(20.1.96) Euler–Ostrogradskiĭ equations will be of the form (we remind observations in Sect. 20.1.1.3) d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L + − = 0 , j = 1, 2, 3 . dt ⎜⎝ ∂u j ⎟⎠ dx i ⎜⎝ ∂u j ,i ⎟⎠ ∂u j

(20.1.96')

Finally,

(

)

∂ 1 ( μu j ) − λ ⎡ ul ,lj − 1 − (ui , j + u j ,i ),i ⎤ + Fj = 0 , j = 1, 2, 3 , ⎢ ⎥⎦ ∂t 2ν ⎣

so that (1 − 2 ν ) Δu j + ui ,ij +

2ν ⎡ ∂ F − ( μu j ) ⎤ = 0 , j = 1, 2, 3 . λ ⎢⎣ j ∂t ⎦⎥

(20.1.97)

We obtain thus the equations of motion of the homogeneous, isotropic linear elastic solid, subjected to infinitesimal deformations; these equations, expressed in displacements, are G. Lamé’s equations. If μ = const , we can write (1 − 2 ν ) Δu j + ui ,ij +

2ν [ Fj − μuj λ

] = 0,

j = 1, 2, 3 .

(20.1.97')

In vectorial form, we have (1 − 2 ν ) Δu + grad div u +

2ν ) = 0 , (F − u λ

(20.1.97'')

where the unknown function is the displacement vector u = u ( r ; t ) .

20.1.5.4 Motion of Strings Let be a string of length l, extended along the axis Ox 3 by a static tension T0 , without any external given force; in this case, the kinetic energy is given by T =

μ l 1 l 2 u μdx 3 = ∫ u 32 dx 3 , 2 ∫0 3 2 0

(20.1.98)


263

where μ = const is the linear density of the string (see Sect. 12.2.1.3). The potential energy is written in the form (we assume that both states of stress and strain are one-dimensional) V =

A l EA l 2 EA l 2 σ33 ε33 dx 3 = ε33 dx 3 = u dx . ∫ ∫ 2 0 2 0 2 ∫ 0 3,3 3

(20.1.98')

μ 2 EA 2 u − u . 2 3 2 3,3

(20.1.98'')

In this case,

L =

The Euler–Ostrogradskiĭ equation

d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L + − =0 ⎜ ⎟ dt ⎝ ∂u 3 ⎠ dx 3 ⎜⎝ ∂u3,3 ⎟⎠ ∂u 3 leads to the equation of longitudinal proper vibration of strings

EAu3,33 − μu3 = 0 ,

(20.1.99)

corresponding to the equations (12.2.20') and (20.1.75'). In case of displacements normal to the axis, let be this one along the Ox 1 –axis, the kinetic energy is given by T =

μ l 1 l 2 u μdx 3 = ∫ u12 dx 3 , 2 ∫0 1 2 0

(20.1.100)

the linear density being constant. The potential energy is calculated in the form (ds is along the deformed form of the string; see also Fig. 12.8) V =

l

l

∫0T0 (ds − dx 3 ) = T0 ∫0 (

l 2 1 2 1 + u1,3 dx 3 , − 1)dx 3 = T0 ∫ u1,3 0 2

(20.1.100')

where we have expanded the integrand into a series after Newton’s binomial and we have neglected terms of superior order (corresponding to the linearization hypothesis). We obtain

L =

μ 2 T0 2 u − u . 2 1 2 1,3

The Euler–Ostrogradskiĭ equation

d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L + − =0 ⎜ ⎟ dt ⎝ ∂u1 ⎠ dx 3 ⎜⎝ ∂u1,3 ⎟⎠ ∂u1

(20.1.100'')


264 leads to

μu1 − T0 u1,33 = 0 ,

(20.1.101)

hence to the equation of proper transverse vibrations of strings, corresponding to the equations (12.2.22').

20.1.5.5 Motion of Bars In case of a straight bar one must take into account the transverse contraction, so that the transverse displacements of the bar Ox 3 –axis will be u1 = − νx1u3,3 , u2 = − ν x 2 u3,3 ,

the

corresponding

displacement

velocities

being

given

by

u1 = − νx1u 3,3 , u 2 = − ν x 2 u 3,3 . We obtain the kinetic energy T =

μ 2 μ l 1 2 u1 + u 22 + u 32 ) dΩ = ∫ ( u 32 + ν 2 iO2 u 3,3 ( ) dx 3 , ∫ 2 ΩA 2 0

(20.1.102)

where iO is the gyration radius of the cross section with respect to the pole O, given by AiO2 =

∫∫ A ( x1

2

+ x 22 ) dA ,

(20.1.102')

while μ is the linear density, constant on the cross section. The potential energy is given by a formula of the form (20.1.98'), so that

L =

μ 2 1 2 2 u 3 + ν 2iO2 u 3,3 − EAu3,3 . ( ) 2 2

(20.1.102'')

The Euler–Ostrogradskiĭ equation (in the form of the equation (7.2.11')) d2 ⎛ ∂L ⎞ d ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L − − + =0 dx 3 dt ⎜⎝ ∂u 3,3 ⎟⎠ dt ⎜⎝ ∂u 3 ⎟⎠ dx 3 ⎜⎝ ∂u 3,3 ⎟⎠ ∂u 3

leads to the A.E.H. Love’s equation EAu3,33 − μ ( u3 − ν 2 iO2 u3,33

) = 0.

(20.1.103)

Neglecting the term which multiplies ν 2 , we find the longitudinal proper vibrations equation of the straight bars in the classical form (20.1.99), in fact the equation (12.2.42). In case of the transverse vibrations of the straight bar, the kinetic energy due to the motion of translation and to the rotation of the cross section is T =

2 l ∂ l 2 1 l 2 1μ 1 l 1 I 2 ∫ ⎡ (u1,3 ) ⎤ dx 3 = μ ∫ u12 dx 3 + μi22 ∫ u1,3 μ ∫ u1 dx 3 + dx 3 , ⎢ ⎥ 0 0 0 0 2 2A 2 2 ⎣ ∂t ⎦ (20.1.104)


265

where I 2 is the moment of inertia with respect to the neutral axis of the bar ( Ox 2 -axis), i2 being the corresponding gyration radius. Observing that ε33 = − x1u1,33 , σ33 = E ε33 , we can write

V =

l 1 1 2 E u 2 dx x 2 dx dx = EI 2 u1,33 , 2 ∫ 0 1,33 3 ∫A 1 1 2 2

(20.1.104')

so that 1 2

1 2

2 2 L = μ ( u12 + i22 u1,3 . ) − EI 2u1,33

(20.1.104'')

As in the preceding case, the Euler–Ostrogradskiĭ equation d2 ⎛ ∂L ⎞ d2 ⎛ ∂L ⎞ d ⎛ ∂L ⎞ d + − − ⎜ ⎟ dx 3 dt ⎝ ∂u1,3 ⎠ dx 32 ⎜⎝ ∂u1,33 ⎟⎠ dt ⎜⎝ ∂u1 ⎟⎠ dx 3

⎛ ∂L ⎞ ∂L ⎜ ∂u ⎟ + ∂u = 0 1 ⎝ 1,3 ⎠

leads to the proper transverse vibrations equation of the straight bar in the form EI 2 u1,3333 + μ ( u1 − i22 u1,33

) = 0,

(20.1.105)

corresponding to the equation (12.2.47).

20.2 Canonical Transformations In general, applying directly the Hamiltonian formalism (instead of the Lagrangian one), the difficulty to solve a problem of mechanics does not diminish sensibly; the differential equations are practically equivalent. The advantage of using a Hamiltonian formalism consists – particularly – in a profound understanding of the formal structure of mechanics; there appears a greater liberty to choose canonical co-ordinates (generalized co-ordinates and momenta). The corresponding formulations, just because of their abstraction, allow to elaborate new theories concerning matter, nature, Universe; they constitute a starting point for quantum as well as for statistical mechanics. In Sect. 18.2.3.1 we have seen that point transformations let invariant Lagrange’s equations; a study of the transformations which let invariant Hamilton’s equations is thus necessary, so that the use of the Hamiltonian formalism is once more justified.

20.2.1 General Considerations. Conditions of Canonicity In what follows, canonical transformations are introduced; especially, accent is put on the statement of canonicity conditions. We mention also various applications, including those in the perturbation theory. A particular attention is given to infinitesimal canonical transformations.

266


20.2.1.1 Preliminary Considerations. Examples Let be a transformation of canonical variables of the form Q j = Q j ( q , p ; t ) , Pj = Pj ( q , p ; t ) , j = 1, 2,..., s ,

(20.2.1)

where Q j and Pj are functions of class C 2 , having the Jacobian ∂ (Q , P ) ⎤ ≠ 0, J = det ⎡ ⎢⎣ ∂ ( q , p ) ⎥⎦

(20.2.1')

Q ≡ {Q1 ,Q2 ,...,Qs } , P ≡ {P1 , P2 ,..., Ps } , q ≡ {q1 , q 2 ,..., qs } , with p ≡ { p1 , p2 , ..., ps } ; the time t plays the rôle of a parameter in this transformation. Hamilton’s function becomes H ( q , p ;t ) = H ( q (Q , P ;t ) , p (Q , P ; t ) ;t ) = H ( Q , P ; t ) . The transformation (20.2.1), (20.2.1') is called canonical transformation if Hamilton’s canonical system (19.1.14) keeps its form, hence if

∂H ∂H , Pj = − , j = 1, 2,..., s . Q = ∂ Pj ∂Q j

(20.2.2)

The theorem of implicit functions and the condition (20.2.1') ensure the existence of the inverse transformation qk = qk (Q , P ; t ) , pk = pk ( Q , P ; t ) , k = 1, 2,..., s ,

(20.2.1'')

these functions being also of class C 2 . The canonical co-ordinates q , p and Q , P correspond to the same representative point in the space Γ 2 s . The transformations (20.2.1), (20.2.1'), and (20.2.1'') are called complete canonical transformations if they do not depend explicitly on time. Together with A. Sommerfeld, these transformations are called contact transformations too (Sommerfeld, A., 1962). Other authors give this denomination to the transformations (20.2.1) to which we add the function T = T ( q , p ;t ) ,

(20.2.3)

of the same class C 2 (hence, for which t is no more a parameter), so that ∂ ( Q , P ;T ) ⎤ det ⎡ ≠ 0. ⎢⎣ ∂ ( q , p ; t ) ⎥⎦

(20.2.3')

But it is not necessary to consider such transformations in the following study; they can be useful in a relativistic modelling of mechanical phenomena. Starting from canonical equations of a complicated form, we can obtain simpler equations of motion by means of canonical transformations; practically, the Hamiltonian H can have a simpler structure than Hamilton’s function H . In particular,


267

if we find the canonical transformation for which H = const (independent of Q and P ) along the trajectory of the representative point, then the solution of the canonical system (20.2.2) will be Q j = Aj , Pj = B j , j = 1, 2,..., s ,

(20.2.4)

where Aj and B j are constants; the functions (20.2.1) are, in this case, first integrals of the canonical system (19.1.14). Let be the transformation Q j = αq j , Pj = δ p j , j = 1, 2,..., s ,

(20.2.5)

where α and δ are non-vanishing constants; the equations (20.2.2) are easily verified, together with the equations (19.1.14), in case of the new Hamiltonian

H = αδH .

(20.2.5')

Hence, this transformation is a complete canonical transformation. As well, the transformation Q j = β p j , Pj = γq j , j = 1, 2,..., s ,

(20.2.6)

where β and γ are non-zero constants, and for which H = − βγ H ,

(20.2.6')

is also a complete canonical transformation. Analogically, we can show that the transformation Q j = p j tan t , Pj = q j cot t , j = 1, 2,..., s ,

(20.2.7)

with H = −H +

2 QP , sin 2t j j

(20.2.7')

is a canonical one. The direct proof of this affirmation is rather complicated. For instance, we have 1 p j + p j tan t , cos2 t ∂H ∂H 2 ∂H 1 tan t + = − + Q = − pj ; sin 2t j ∂ Pj ∂ Pj ∂Q j cos2 t Q j =

taking into account (19.1.14), the first equation (20.2.2) is verified. As well, it is not completely clear how one can obtain such transformations. For this goal, we will give – in what follows – various conditions of canonicity and will put in evidence the possibility to construct canonical transformations.


268

20.2.1.2 Complete Canonical Transformations Let be a complete canonical transformation Q j = Q j (q , p ) , Pj = Pj (q , p ) , j = 1, 2,.., s ,

(20.2.8)

where Q j , Pj are functions of class C 2 which verify the condition (20.2.1'). We consider the differential form of the first degree (the one-form) ψ = pk dqk − Pj dQ j .

(20.2.9)

The canonical form (20.1.41) of Hamilton’s principle leads to t1

t1

t0

t0

δ ∫ [ pk dqk − H ( q , p ; t ) dt ] = δ ∫

[ Pj dQ j

− H ( Q , P ; t ) dt ] .

To obtain the canonical equations (20.2.2), corresponding to Livens’s Theorem 20.1.6, as a consequence of the complete canonical transformation (20.2.1), it is sufficient that the one-form (20.2.9) be an exact differential, hence dψ = 0 ; as a matter of fact, this represents a sufficient condition of complete canonicity for the transformation (20.2.8). On the basis of Poincaré’s lemma, we have d(dψ ) = 0 , the operator d being Cartan’s external differential operator (see App., Subsec. 1.2.2). Corresponding to the reciprocal of this lemma, if dψ = 0 , then there exists a scalar function V so that ψ = dV ,

(20.2.9')

with V = V ( q , p ;Q , P ) , which cannot depend explicitly on time; in fact, the function V can depend only on two from the four sets of co-ordinates q , p ,Q and P . Corresponding to the form (19.1.45), we can write Pfaff’s forms of Hamilton ω = pk dqk − H ( q , p ; t ) dt , Ω = Pj dQ j − H ( Q , P ; t ) dt .

(20.2.10)

We have proved in Sect. 19.1.1.9 that the two forms have the same bilinear covariant, hence they lead both to the same canonical equations (the same associate differential system), if and only if Ω = c ω − dV ,

(20.2.10')

where c is a constant, while V = V ( q , p ;Q , P ) is a function of class C 2 ; we make the same observation as above for the arguments of the function V . The necessity of introducing the constant c will be clearly put in evidence in Sect. 21.1.2.3, in the study of integral invariants. We notice that c ω − Ω = cpk dqk − Pj dQ j − ( cH − H ) dt = dV ;

because V = ∂V / ∂t = 0 , it follows


269 (20.2.10'')

H = cH .

We introduce thus the differential form ψc = cpk dqk − Pj dQ j = dVc , c ≠ 0 , c = const .

(20.2.11)

The constant c is called the valency of the canonical transformation. If c = 1 , then the transformation is univalent; in this case H = H . Usually, one has to do with univalent canonical transformations. We observe that the condition (20.2.9') (with c = 1 ) is always sufficient; in some particular cases, it is also necessary (the transformations are univalent). In the particular case in which V = 0 , the complete canonical transformation is called, together with E. Mathieu, homogeneous, and the relation pk dqk = Pk dQk

(20.2.9'')

takes place; in this case, if the variable pk is multiplied by a factor, then the variable Pk is multiplied by the same factor. Taking into account (20.2.9), (20.2.9'), we may write ∂Q j ∂V ⎛ ∂Q j ⎞ ∂V pk dqk − Pj ⎜ dqk + dpk ⎟ = dqk + dpk , ∂ pk ∂ pk ⎝ ∂qk ⎠ ∂qk

where we have considered V = V (q , p ) ; there results

pk − Pl

∂Ql ∂Ql ∂V ∂V = = , − Pl , k = 1, 2,..., s . ∂qk ∂qk ∂ pk ∂ pk

Eliminating the function V between these relations or (it is the same thing) writing that the mixed derivatives of second order of this function −

−

∂ Pl ∂Ql ∂ 2Ql ∂ P ∂Ql ∂ 2Ql ∂ 2V ∂ 2V − Pl = , − l − Pl = , ∂q j ∂qk ∂q j ∂qk ∂q j ∂qk ∂qk ∂q j ∂qk ∂q j ∂qk ∂q j

∂ Pl ∂Ql ∂ 2Ql ∂ P ∂Ql ∂ 2Ql ∂ 2V ∂ 2V , − l , − Pl = − Pl = ∂ p j ∂ pk ∂ p j ∂ pk ∂ p j ∂ pk ∂ pk ∂ p j ∂ pk ∂ p j ∂ pk ∂ p j

δjk −

∂ Pl ∂Ql ∂ 2Ql ∂ P ∂Ql ∂ 2Ql ∂ 2V ∂ 2V − Pl = , − l − Pl = ∂ p j ∂qk ∂ p j ∂qk ∂ p j ∂qk ∂qk ∂ p j ∂qk ∂ p j ∂qk ∂ p j

do not depend on the order of differentiation, we obtain the conditions of complete canonicity of the considered transformation in the form

[q j , qk ] = 0 , [ p j , pk ] = 0 , [q j , pk ] = δjk ,

j , k = 1, 2,..., s ,

(20.2.12)


270

where δjk is Kronecker’s symbol; we can write 2 [ s (s − 1)/ 2 ] + s 2 = s (2s − 1) such conditions. We have introduced the Lagrange brackets by relations of the form (19.1.50), where q j , p j , j = 1, 2,..., s , play the rôle of integration constants in the integrals (20.2.8) of the canonical system (20.2.2). The conditions (20.2.12) can be written also in the form

∂Ql ⎞ ∂Ql ⎞ ∂ ⎛ ∂ ⎛ pk − Pl p j − Pl = , ⎜ ⎟ ⎜ ∂q j ⎝ ∂qk ⎠ ∂qk ⎝ ∂q j ⎟⎠ ∂ ⎛ ∂Ql ⎞ ∂ ⎛ ∂Ql ⎞ P P = , ∂ p j ⎜⎝ l ∂ pk ⎟⎠ ∂ pk ⎜⎝ l ∂ p j ⎟⎠ ∂Ql ⎞ ∂Ql ⎞ ∂ ⎛ ∂ ⎛ ; p − Pl = −P ∂ p j ⎜⎝ k ∂qk ⎟⎠ ∂qk ⎜⎝ l ∂ p j ⎟⎠ there exists thus a function V so that ∂V / ∂qk and ∂V / ∂ pk be given by the relations from which we started. We obtain then (20.2.9) and (20.2.9'). Hence, the conditions (20.2.12) are equivalent to (20.2.9), (20.2.9'), being thus sufficient conditions of complete canonicity. If c ≠ 1 , then the conditions (20.2.12) take the form

[q j , qk ] = 0 , [ p j , pk ] = 0 , [q j , pk ] = cδjk ,

j , k = 1, 2,..., s ,

(20.2.12')

being necessary and sufficient conditions of complete canonicity. Obviously, we can write also the sufficient conditions of complete canonicity

[Q j ,Qk ] = 0 , [ Pj , Pk ] = 0 , [Q j , Pk ] = δjk ,

j , k = 1, 2,..., s ,

(20.2.12'')

corresponding to the inverse complete canonical transformations. As well, if c ≠ 1 , then we may write the necessary and sufficient conditions of complete canonicity 1

[Q j ,Qk ] = 0 , [ Pj , Pk ] = 0 , [Q j , Pk ] = c δjk ,

j , k = 1, 2,..., s .

(20.2.12''')

Using the Poisson brackets (defined by the relation (19.1.54)), we see that one can write Q j =

∂Q j ∂Q j ∂Q j ∂ H ∂Q j ∂ H q + p = − = ( Q j , H ) , Pj = ( Pj , H ) , ∂qk k ∂ pk k ∂qk ∂ pk ∂ pk ∂qk

along the integral curves of the canonical system (19.1.14); as well, we have dQ j =

∂Q j ∂Q j ∂ Pj ∂ Pj dqk + dp , dPj = dq + dp . ∂qk ∂ pk k ∂qk k ∂ pk k

Taking into account the Lagrange brackets (20.2.12), we obtain


271

∂Ql ∂ H ⎞ ⎛ ∂ Pl ∂ Pl ⎛ ∂Q ∂ H ⎞ Ql dPl − Pl dQl = ⎜ l − ⎟ ⎜ ∂q dqk + ∂ p dpk ⎟ ∂ ∂ ∂ ∂ q p p q ⎠ j j j ⎠⎝ k k ⎝ j ∂ Pl ∂ H ⎞ ⎛ ∂Ql ∂Ql ⎛ ∂P ∂H ⎞ −⎜ l − ⎟ ⎜ ∂q dqk + ∂ p dpk ⎟ q p p q ∂ ∂ ∂ ∂ ⎠ j j j ⎠⎝ k k ⎝ j ∂H ∂H ⎞ ∂H ∂H ⎛ ⎛ = ⎜ [ qk , p j ] + [ q j , qk ] ⎟ dqk + ⎜ [ q j , pk ] ∂ p + [ pk , p j ] ∂q ∂ ∂ q p j j j j ⎝ ⎠ ⎝ ∂H ∂H = dq + dp = dH − H dt ; ∂qk k ∂ pk k

⎞ ⎟ dpk ⎠

if H = H , then we are led to the canonical system (20.2.2) (the expression dH − H dt is invariant by a complete canonical transformation). Thus, we verify once more that (20.2.9') constitutes a sufficient condition for the considered transformations be complete canonical, Hamilton’s function preserving its form also in the new canonical co-ordinates. Let us represent the Jacobian (20.2.1') as the determinant of a compound matrix ⎡ ⎡ ∂Q j ⎢ ⎢ ∂q ⎣ k J = det ⎢ ⎢ ⎡ ∂ Pj ⎢ ⎢ ∂q ⎣⎣ k

⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k ⎤ ⎡ ∂ Pj ⎥ ⎢ ⎦ ⎣ ∂ pk

⎤⎤ ⎥⎥ ⎦⎥ , j , k = 1, 2,..., s , ⎤⎥ ⎥⎥ ⎦⎦

the index j indicating the row and the index k the column in the four submatrices with s × s elements. We will interchange the first row of submatrices with the second one, and then the first column of submatrices with the second one; the determinant of the new matrix will be as well J (with the same sign). We multiply then the submatrices of the first row, as well as the submatrices of the first column, successively, by −1 ; transposing this matrix, we obtain the determinant

J∗

T ⎡ ⎡ ∂ Pj ⎤ T ⎡ ⎡ ∂ Pj ⎤ ⎡ ∂ Pj ⎤ ⎤ − ⎢ ⎢ ∂p ⎥ ⎢ ⎢ ∂ p ⎥ ⎢ ∂q ⎥ ⎥ ⎣ k ⎦ ⎣ ⎣ k ⎦ k ⎦ ⎢ ⎥ = det = det ⎢ ⎢ ∂ Q ∂ Q ⎢⎡ ∂P T j ⎤ ⎡ j ⎤ ⎥ ⎢ ⎡⎢ − j ⎤⎥ ⎢ ⎢ − ∂ p ⎥ ⎢ ∂q ⎥ ⎥ ⎣⎣ ⎣ k ⎦ ⎦ k ⎦ ⎣⎢ ⎣ ∂qk ⎦

T ⎡ ∂Q j ⎤ − ⎢ ∂p ⎥ ⎣ k ⎦ T ⎡ ∂Q j ⎤ ⎢ ⎥ ⎣ ∂qk ⎦

⎤ ⎥ ⎥. ⎥ ⎥ ⎦⎥

We notice that J ∗ = J , so that

⎡ [ [ q j , pk ] ] J 2 = JJ ∗ = det ⎢ ⎣⎢ [ [ p j , pk ] ]

[ [ qk , q j ] ] ⎤ ⎥ [ [ qk , p j ] ] ⎦⎥

= 1,

(20.2.13)

where we took into account the conditions of complete canonicity (20.2.12); hence J = ±1 . In case of a transformation of valency c , we have J = ±c s . Because the Jacobian J does not vanish, the condition (20.2.1') is always fulfilled in case of a complete canonical transformation.


272

Let F = F ( q , p ) be a function of class C 1 . We construct the differential form of first degree ϕ = − ( Pj , F ) dQ j + ( Q j , F ) dPj ;

(20.2.14)

developing the Poisson brackets ( Q j , F ) , ( Pj , F ) , j = 1, 2,..., s , we obtain

∂F ∂F ⎞ ∂F ∂F ⎞ ⎛ ⎛ ϕ = ⎜ [ qk , p j ] + [ q j , qk ] ⎟ dqk + ⎜ [ pk , p j ] ∂q + [ q j , pk ] ∂ p ⎟ dpk . ∂ ∂ q p j j ⎠ j j ⎠ ⎝ ⎝ If the relations (20.2.12) take place, then the transformation ( Q , P ) → ( q , p ) is complete canonical, the differential form being given by ω = dF , where F is an arbitrary function of class C 1 . Taking into account (20.2.14), we have

(Q j , F ) =

∂F ∂F , ( Pj , F ) = − , j = 1, 2,..., s . ∂ Pj ∂Q j

(20.2.14')

In particular, if F = H , then we find Hamilton’s equations, written by means of the Poisson brackets (equations of the form (19.1.60')) Q j = ( Q j , H ) , Pj = ( Pj , H ) , j = 1, 2,..., s .

(20.2.14'')

If we equate F to Qk and Pk , successively, we obtain

(Q j ,Qk ) = 0 , ( Pj , Pk ) = 0 , (Q j , Pk ) = δjk ,

j , k = 1, 2,..., s ;

(20.2.15)

these relations constitute sufficient conditions of complete canonicity, equivalent to the conditions (20.2.12). Analogically, the conditions ( c ≠ 1 )

(Q j ,Qk ) = 0 , ( Pj , Pk ) = 0 , (Q j , Pk ) = cδjk ,

j , k = 1, 2,..., s ,

(20.2.15')

represent necessary and sufficient conditions of complete canonicity. Obviously, we may write also the sufficient conditions of complete canonicity

( q j , qk ) = 0 , ( p j , pk ) = 0 , ( q j , pk ) = δjk ,

j , k = 1, 2,..., s ,

(20.2.15'')

corresponding to the inverse complete canonical transformations. As well, if c ≠ 1 , then we can write the necessary and sufficient conditions of complete canonicity 1

( q j , qk ) = 0 , ( p j , pk ) = 0 , ( q j , pk ) = c δjk , Let be the Pfaffian

j , k = 1, 2,..., s .

(20.2.15''')


273

∂ pk ∂q ∂ pk ⎛ ∂qk ∂q ∂q ∂ p ∂p dq − k dp = dQl + k dPl ⎟⎞ − k ⎜⎛ k dQl + k dPl ⎟⎞ ∂ Pj k ∂ Pj k ∂ Pj ⎜⎝ ∂Ql ∂ Pl ∂ P ∂ Q ∂ Pl ⎠ ⎠ j ⎝ l ∂Q j ∂Q j = [Ql , Pj ] dQl + [ Pl , Pj ] dPj = dQ j = dqk + dp , ∂qk ∂ pk k

where we took into account the sufficient conditions of complete canonicity (20.2.12''); analogically ∂ Pj ∂ Pj ∂ pk ∂q dqk − k dpk = − dPj = − dqk − dp . ∂Q j ∂Q j ∂qk ∂ pk k

By identification, we obtain the relations ∂Q j ∂ Pj ∂ Pj ∂ pk ∂Q j ∂q ∂p ∂qk = = − k , = − k , = , , j , k = 1, 2,..., s . ∂qk ∂ Pj ∂ pk ∂ Pj ∂qk ∂Q j ∂ pk ∂Q j

(20.2.16) In this case, we may write

(Q j ,Qk ) =

∂Q j ∂Qk ∂Q j ∂Qk ∂ p ∂ql ∂q ∂ pl − = − l + l , ∂ql ∂ pl ∂ pl ∂ql ∂ Pj ∂ Pk ∂ Pj ∂ Pk

as well as other analogous relations; we find thus the relations between Poisson brackets for the inverse complete canonical transformations (20.2.8) and Lagrange brackets for the corresponding direct transformations

(Q j ,Qk ) = [ Pj , Pk ] , ( Pj , Pk ) = [Q j ,Qk ] , (Q j , Pk ) = [Q j , Pk ] .

(20.2.16')

If c ≠ 1 , then the relations (20.2.16) are of the form ∂Q j ∂ Pj ∂ Pj ∂ p ∂Q j ∂q ∂p ∂q =c k , = −c k , = −c k , = c k , j , k = 1, 2,..., s , ∂qk ∂ Pj ∂ pk ∂ Pj ∂qk ∂Q j ∂ pk ∂Q j

(20.2.16'') while the relations (20.2.16') become

(Q j ,Qk ) = c 2 [ Pj , Pk ] , ( Pj , Pk ) = c 2 [Q j ,Qk ] , (Q j , Pk ) = c 2 [Q j , Pk ] , j , k = 1, 2,..., s .

(20.2.16''')

Analogically, one can establish the relations 1 1 [ p j , pk ] , ( p j , pk ) = 2 [ q j , qk ] , c2 c 1 ( q j , pk ) = 2 [ q j , pk ] , j , k = 1, 2,..., s ; c

( q j , qk ) =

(20.2.16iv)


274 if c = 1 , then we have

( q j , qk ) = [ p j , pk ] , ( p j , pk ) = [ q j , qk ] , ( q j , pk ) = [ q j , pk ] ,

j , k = 1, 2,..., s .

(20.2.16v) Thus, starting from (20.2.12'') and (20.2.12), we find again the sufficient conditions of complete canonicity (20.2.15) and (20.2.15''), respectively; as well, the conditions (20.2.12''') and (20.2.12') allow us to write the necessary and sufficient conditions of complete canonicity (20.2.15') and (20.2.15'''), respectively.

20.2.1.3 Canonical Transformations In case of an arbitrary transformation (20.2.1), (20.2.1'), in which the time appears explicitly, we consider the one-form ψ = pk d qk − Pj dQ j − ( H − H ) d t .

(20.2.17)

In this case too, a sufficient condition of canonicity imposes the differential form (20.2.17) to be an exact differential, hence ψ = dW ,

(20.2.17')

where W = W ( q , p ,Q , P ; t ) is a function of class C 2 ; taking into account the canonical form (20.1.41) of Hamilton’s principle, one can easily prove this affirmation. If W = 0 , then the transformation is homogeneous. Introducing Pfaff’s forms of Hamilton (20.2.10), the necessary and sufficient condition of canonicity will be Ω = c ω − dW ,

(20.2.18)

where c is a non-vanishing constant, while W = W ( q , p ,Q , P ; t ) is a function of class C 2 or c ω − Ω = cpk dqk − Pj dQ j − ( cH − H ) dt = dW .

(20.2.18')

As a matter of fact, in this case we introduce the differential form ψc = cpk dqk − Pj dQ j − ( cH − H ) dt = dWc .

(20.2.18'')

For t = t ∗ , arbitrary but fixed, the condition (20.2.18') becomes Pj dQ j = cpk dqk − dW ( q , p ,Q , P ; t ∗ ) ;

(20.2.19)

this is a necessary and sufficient condition for the transformation Q j = Q j ( q , p ; t ∗ ) , Pj = Pj ( q , p ; t ∗ ) , j = 1, 2,..., s , be complete canonical. For the transformation

(20.2.1) be canonical, it is necessary to be canonical for any fixed t (the time is considered as a parameter), with the same valency c .


275

Supposing that the transformation (20.2.1) is canonical for any t , it follows that we have (20.2.19) for any t . After transformation, let us define the Hamiltonian H by H = cH + W + PjQ j ;

(20.2.19')

we obtain thus

(

)

Pj Q j dt = −W dt − c H − H dt .

Summing member by member with (20.2.19) (where the differentials dQ j and dW are not extended to the time t , which is fixed), we find again the condition (20.2.18'). Hence, the condition (20.2.18) is a necessary and sufficient condition of canonicity. On the basis of the above affirmation, it results that also the conditions (20.2.12) are sufficient conditions of canonicity, while the conditions (20.2.12') are not only sufficient but also necessary for the canonicity of the transformation. One can then make analogous affirmations for all conditions established at the previous subsection by means of Lagrange’s or Poisson’s brackets. As well, the Jacobian (20.2.1') is non-zero in the general case of canonical transformations too. Let qk = qk ( q 0 , p 0 ; t ) , pk = pk ( q 0 , p 0 ; t ) , k = 1, 2,..., s ,

(20.2.20)

be the solution of the canonical equations (19.1.14), with the initial conditions q 0 = q ( t0 ) and p 0 = p ( t0 ) . According to Lagrange’s Theorem 19.1.4, the Lagrange brackets [ q j0 , qk0 ] , [ p j0 , pk0 ] , [ q j0 , pk0 ] do not depend on time; for t = t0 we have

[q j0 , qk0 ] = 0 , [ p j0 , pk0 ] = 0 , [q j0 , pk0 ] = δjk , Hence, the transformation

( q 0 , p0 ) → ( q , p )

j , k = 1, 2,..., s .

given by (20.2.20) is a univalent

canonical transformation. More general, we can state that the values of the canonical co-ordinates at the moment t + τ can be expressed as functions of the values of the same co-ordinates at the moment t in the form q j ( t + τ ) = q j ( q ( t ) , p ( t ) ; τ ) , p j ( t + τ ) = p j ( q ( t ) , p ( t ) ; τ ) , j = 1, 2,..., s ;

,

(20.2.20')

hence, the variation of the canonical co-ordinates q and p during the motion can be considered as a canonical transformation. We can state that the general integral of a dynamical system, on one hand, and the canonical transformations of this system, on the other hand, are two problems covering reciprocally themselves from the point of view of their contents.

20.2.1.4 Point Transformations A transformation of the form Q j = Q j ( q ) , Pj = Pj ( q , p ) , j = 1, 2,..., s ,

(20.2.21)


276 with the functional determinant

∂ (Q , P ) ⎤ ⎡ ∂Q j J = det ⎡ = det ⎢ ⎢⎣ ∂ ( q , p ) ⎥⎦ ⎣ ∂qk

⎤ ⎡ ∂ Pj ⎤ ⎥ det ⎢ ∂ p ⎥ ≠ 0 , ⎦ ⎣ k ⎦

(20.2.21')

is called a point transformation; in this case, we also have ⎡ ∂Q j ⎤ ⎡ ∂ Pj ⎤ ≠ 0 , det ⎢ det ⎢ ⎥ ⎥ ≠ 0. ⎣ ∂qk ⎦ ⎣ ∂ pk ⎦

(20.2.21'')

The condition (20.2.9') of complete canonicity can be written in the form pk dqk = Pj dQ j = Pj

∂Q j dqk , ∂qk

in case of a homogeneous transformation; hence, it is sufficient to have pk = Pj

∂Q j , k = 1, 2,..., s , ∂qk

for the point transformation (20.2.21), (20.2.21') be completely canonical and homogeneous. Multiplying both members by ∂qk / ∂Ql , k , l = 1, 2,..., s , and summing, we may state that the point transformation Q j = Q j ( q ) , Pj = pk

∂qk , j = 1, 2,..., s , ∂Q j

(20.2.22)

linear with respect to the generalized momenta, is complete canonical and homogeneous. We notice that, in the second group of relations, one introduces Q j = Q j ( q ) , j = 1, 2,..., s , to have the transformations written in the form (20.2.21). The condition (20.2.21') is, in this case, automatically fulfilled, as we have seen in Sect. 20.2.1.2. We can write the relations of inverse transformation qk = qk ( Q ) , pk = Pj

∂Q j , k = 1, 2,..., s , ∂qk

(20.2.22')

where we introduce qk = qk ( Q ) in the second group of relations, to have the transformation written in the form pk = pk ( Q , P ) , k = 1, 2,..., s . Let us introduce also a more general transformation, of the form Q j = Q j (q ; t ) , Pj = Pj (q , p ; t ) , j = 1, 2,..., s ,

(20.2.23)

where the time appears explicitly, and which verifies the condition (20.2.21') (implicitly the conditions (20.2.21'')). The canonicity condition is


277

⎛ ∂Q j ⎞ dqk + Q dt ⎟ + ( H − H ) dt ; pk dqk = Pj dQ j + ( H − H ) dt = Pj ⎜ ∂ q ⎝ k ⎠

hence, it follows pk = Pj

∂Q j , k = 1, 2,..., s . ∂qk

As in the previous case, we can state that the generalized point transformation Q j = Q j ( q ; t ) , Pj = pk

∂qk , j = 1, 2,..., s , ∂Q j

(20.2.24)

linear with respect to the generalized momenta, is canonical and homogeneous; we make Q j = Q j ( q ; t ) in the second group of relations, to have transformations of the form (20.2.23). We notice that the Hamiltonian is transformed in the form H = H + Pj Q j .

(20.2.24')

The relations of inverse transformation are qk = qk ( Q ; t ) , pk = Pj

∂Q j , k = 1, 2,..., s , ∂qk

(20.2.25)

where we introduce qk = qk ( Q ; t ) , k = 1, 2,..., s , in the second group of relations; the Hamiltonian is given by H = H + pk qk .

(20.2.25')

20.2.1.5 Free Canonical Transformations. Generating Functions of Canonical Transformations Let be a transformation (20.2.1), (20.2.1') for which the supplementary condition ⎡ ∂Q j ⎤ det ⎢ ⎥ ≠0 ⎣ ∂ pk ⎦

(20.2.26)

takes place; such a transformation is called a free transformation. In this case, on the basis of the theorem of implicit functions, the first group of relations (20.2.1) allows us to calculate pk = pk ( Q , q ; t ) , and then Pj = Pj ( Q , q ; t ) , j , k = 1, 2,..., s ; as well, the function W can be expressed in the form W ( q , p ; t ) = S ( Q , q ; t ) . Thus, the sufficient condition (20.2.17), (20.2.17') takes the form pk dqk − Pj dQ j − ( H − H ) dt =

∂S ∂S dQ j + dq + Sdt ; ∂Q j ∂qk k


278 we obtain

pk =

∂S ∂S , Pj = − , j , k = 1, 2,..., s , ∂Q j ∂qk H = H + S .

(20.2.27) (20.2.28)

These relations define the canonical transformations (sufficient conditions). Indeed, differentiating the first relation (20.2.27) with respect to time, we may write pk =

∂ 2S ∂ 2S ∂ 2S ; ql + Ql + ∂ql ∂qk ∂Ql ∂qk ∂t ∂qk

inverting the differentiation order and using the relations (20.2.27), (20.2.28), we obtain pk =

∂ pl ∂P ∂ q − l Q + (H − H ) . ∂qk l ∂qk l ∂qk

We have H ( Q , P ; t ) = H ( Q , P ( Q , q ; t ) ; t ) and H ( q , p ; t ) = H ( q , p ( Q , q ; t ) ; t ) , so that ∂ (H − H ∂qk

)

=

∂ H ∂ Pl ∂ H ∂ H ∂ pl ; − − ∂ Pl ∂qk ∂qk ∂ pl ∂qk

we find the relation pk +

∂ pl ∂H = ∂qk ∂qk

⎛ q − ∂ H ⎞ − ∂ Pl ⎜ l ∂ p ⎟ ∂q ⎝ l ⎠ k

∂H ⎞ ⎛ ⎜ Ql − ∂ P ⎟ . ⎝ l ⎠

Analogically, the second relation (20.2.27) leads to ∂P ∂H Pj + = − l ∂Q j ∂Q j

∂ H ⎞ ∂ pk ⎛ ⎜ Ql − ∂ P ⎟ − ∂Q ⎝ j l ⎠

⎛ q − ∂ H ⎞ . ⎜ k ∂p ⎟ ⎝ k ⎠

Assuming that Hamilton’s equations (20.2.2) take place and taking into account (20.2.26), the second group of relations leads to qk = ∂ H / ∂ pk ; replacing in the first group of relations, we obtain the second group of Hamilton’s equations (19.1.14). Hence, the representation (20.2.27) is sufficient to obtain a free canonical transformation. If the Hessian of the function S is non-zero ⎡ ∂ 2S ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂Q j ∂qk ⎦

(20.2.29)

then we obtain the generalized co-ordinates Q from the first subsystem (20.2.27); by replacing in the second subsystem (20.2.27), we are led to the complete determination


279

of the canonical co-ordinates Q , P as functions of the co-ordinates q , p and of the time t , hence to the transformation (20.2.1). As well, starting from the condition (20.2.29) we obtain the generalized co-ordinates q from the second subsystem (20.2.27); by replacing in the first subsystem (20.2.27), we get the inverse canonical transformation (20.2.1''). The relation (20.2.28) puts into evidence the connection between Hamilton’s functions corresponding to the two canonical systems. Hence, we may state Theorem 20.2.1 (Jacobi). A function S ( Q , q ; t ) of class C 2 which verifies the condition (20.2.29) generates a free canonical transformation by means of the relations (20.2.27). Indeed, starting from the first subsystem (20.2.27), we obtain Q j = Q j ( q , p ; t ) ,

j = 1, 2,..., s ; this subsystem will be identically satisfied if we replace the generalized co-ordinates Q j thus expressed. Differentiating with respect to pl , we obtain (we observe that the function S depends on pl only by means of the generalized coordinates Q j ) ∂ pk ∂ 2 S ∂Q j = = δk l . ∂Q j ∂qk ∂ pl ∂ pl

The Hessian (20.2.29) does not vanish, hence the corresponding matrix is non-singular; therefore, the matrix [ ∂Q j / ∂ pl ] , which is the inverse of the previous one, is also nonsingular, the condition (20.2.26) being thus fulfilled. The function S is called the generating function of free canonical transformations. If the Jacobian (20.2.26) is identically zero, then there exists at least one relation connecting the generalized co-ordinates q ,Q and the time t . Let us assume that the matrix [ ∂Q j / ∂ pk ] is of rank s − 1 ; in this case, there is only one relation of the above mentioned form, let be Ω ( Q , q ; t ) = 0 . The relations (20.2.27), (20.2.28) are replaced by pk =

∂S ∂Ω ∂S ∂Ω , Pj = − , j , k = 1, 2,..., s , +λ −λ ∂Q j ∂Q j ∂qk ∂qk H = H + S + λ Ω ,

(20.2.27') (20.2.28')

where λ is a non-determinate Lagrange’s multiplier. Analogically, if the matrix [ ∂Q j / ∂ pk ] is of rank s − r , then there exist r relations connecting the generalized co-ordinates q ,Q and the time t , while the formulae of type (20.2.27'), (20.2.28') contain r Lagrange’s multipliers. For example, a point transformation of the form (20.2.21), which is a homogeneous complete canonical transformation, includes s relations linking the generalized coordinates q to the generalized co-ordinates Q ; as a matter of fact, any homogeneous complete canonical transformation includes at least such a relation. Indeed, the condition of complete canonicity pk dqk = Pj dQ j implies s homogeneous equations


280 Pj

∂Q j = 0 , k = 1, 2,..., s ; ∂ pk

hence, to have non-zero solutions Pj , det [ ∂Q j / ∂ pk

]

must vanish, so that

Ω (Q , q ) = 0 . Let us put the problem to determine the generating function S , which leads to a special form of the Hamiltonian H . For instance, if H = 0 , then we have S + H = 0 , where H = H ( q , p ; t ) = H ( q1 , q 2 ,..., qs , ∂ S / ∂q1 , ∂S / ∂q 2 ,..., ∂ S / ∂qs ; t ) ; we find thus again the Hamilton-Jacobi equation (19.2.9), the function S being the corresponding unknown function. We have seen in Sec. 20.2.1.1 that, in this case, 2s first integrals of the initial canonical system are determined; the two subsystems (20.2.27) correspond to the two sequences (19.2.10), which give the solution of Hamilton’s system in the Hamilton–Jacobi theorem. Hence, the problem to determine the canonical transformation for which H = 0 is equivalent to the integration of Hamilton’s canonical system. If we use the necessary and sufficient condition (20.2.18'), then the generating function S = S ( Q , q ; t ) leads to the representation

cpk =

∂S ∂S , Pj = − , j , k = 1, 2,..., s , ∂qk ∂Q j

(20.2.27'')

with

H = cH + S , c ≠ 0 , c = const ,

(20.2.28'')

which, in general, is not univalent. In case of a complete canonical transformation, the generating function S = S ( Q , q ) does not depend explicitly on time ( S = 0 ) and the Hamiltonian is transformed by the relation (20.2.10''). We notice that, in such a transformation, the function H does not change essentially; to can obtain a canonical transformation having a simpler Hamiltonian, we must use a generating function S which contains the time explicitly. For instance, let be the affine transformation Q j = αq j + β p j , Pj = γq j + δ p j , αδ − βγ ≠ 0 ,

(20.2.30)

where α, β , γ , δ are constants; one sees easily that, if c = αδ − βγ , then the necessary and sufficient condition (20.2.11) is fulfilled. In this case, 1 ( αγq j q j + 2 βγq j p j + βδ p j p j ) 2 1 1 1 = − ( αq j + β p j ) ( γq j + δ p j ) + cq j p j = − ( Q j Pj − cq j p j ) . 2 2 2 V = −

We notice that det [ ∂Q j / ∂ pk

] = β s ; hence, if

(20.2.30')

β ≠ 0 , then the transformation is free

(as one can directly see from the first relation (20.2.30)). If β = γ = 0 , then we find


281

again the transformation (20.2.5), which is not free; as well, if α = δ = 0 , then we obtain the transformation (20.2.6), which is free. We mention that the transformation (20.2.7) is also a free transformation too. For a natural mechanical system, the generalized co-ordinates q1 , q 2 ,..., qs define its position and, together with the generalized momenta p1 , p2 ,..., ps , define its state (positions and velocities of its points). But this specificity of canonical co-ordinates does not remain valid, in general, in case of a canonical transformation; indeed, the quantities Q1 ,Q2 ,...,Qs do no more define the position of the mechanical system, but define its state only together with the quantities P1 , P2 ,...., Ps . The quantities Q1 ,Q2 ,...,Qs define the position of the mechanical system only in the case of a point transformation. However, the non-point canonical transformations are important in the theory of Hamiltonian systems, because they can lead to a Hamiltonian having a simpler structure. Let be the canonical transformation generated by the function S (Q , q ) = Q j q j ;

(20.2.31)

Q j = p j , Pj = −q j , j = 1, 2,..., s .

(20.2.31')

we obtain

The canonical co-ordinates invert their rôles, the above affirmations being thus justified. At the same time, the denomination of conjugate co-ordinates given to qk and pk is also justified. In Sect. 19.2.1.6 we have introduced Hamilton’s principal function S , which satisfies the Hamilton–Jacobi equation; indeed, taking into account the deterministic character of mechanics (we have det [ ∂q j / ∂ pk0 ] ≠ 0 ) and using the final equations of motion (20.2.20), Hamilton obtained pk0 = pk0 ( q , q 0 ; t ) , k = 1, 2,..., s . In this case,

the function

S = S ( q ,q 0 ;t )

will be a generating function of canonical

transformations, so that the transformation (20.2.20) can be considered as a univalent free canonical transformation.

20.2.1.6 Other Functions Generating Canonical Transformations We notice that d (Q j Pj

) = Pj dQ j

+ Q j dPj ; the sufficient condition of canonicity

(20.2.17), (20.2.17') has the form pk dqk + Q j dPj − ( H − H ) dt = dW1 , W1 = W + Q j Pj

in this case. Assuming the supplementary condition ⎡ ∂ Pj ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂ pk ⎦

(20.2.32)


282

the second group of relations (20.2.1) leads to pk = pk ( P , q ; t ) and then to Q j = Q j ( P , q ; t ) , j , k = 1, 2,..., s ; as well, the function W1 can be expressed in the form W1 ( Q , P , q , p ; t ) = S1 ( P , q ; t ) . The condition of canonicity allows us to write pk =

∂ S1 ∂ S1 , Qj = , j , k = 1, 2,..., s , ∂qk ∂ Pj

(20.2.33)

as well as H = H + S1 .

(20.2.33')

⎡ ∂ 2 S1 ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂ Pj ∂qk ⎦

(20.2.32')

If the Hessian

then we obtain the canonical transformation (20.2.1) from the first subsystem (20.2.33). Hence, a function S1 ( P , q ; t ) of class C 2 , which verifies the condition (20.2.32'), determines a canonical transformation (20.2.33), (20.2.33'). As well, the relation d ( qk pk ) = pk dqk + qk dpk leads to the sufficient condition of canonicity −qk dpk − Pj dQ j − ( H − H ) dt = dW2 , W2 = W − qk pk .

If the supplementary condition ⎡ ∂Q j ⎤ det ⎢ ⎥ ≠0 ⎣ ∂qk ⎦

(20.2.34)

is fulfilled, then the first group of relations (20.2.1) leads to qk = qk ( Q , p ; t ) , so that Pj = Pj ( Q , p ; t ) , j , k = 1, 2,..., s ; analogically, the function W2 can be expressed in the form W2 ( Q , P , q , p ; t ) = S 2 ( Q , p ; t ) . The sufficient condition of canonicity leads to the representation qk = −

∂S2 ∂S , Pj = − 2 , j , k = 1, 2,..., s , ∂ pk ∂q j

(20.2.35)

as well as to H = H + S2 .

(20.2.35')

⎡ ∂ 2S2 ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂Q j ∂ pk ⎦

(20.2.34')

If the Hessian


283

then we obtain the generalized co-ordinates Q j from the first subsystem (20.2.35); replacing then in the second subsystem (20.2.35), the canonical transformation (20.2.1) is completely specified. By way of consequence, a function S 2 ( Q , p ; t ) of class C 2 , which verifies the condition (20.2.34'), leads to a canonical transformation (20.2.35), (20.2.35'). Analogically, using the relations d (Q j Pj

) = Pj dQ j

+ Q j dPj , d ( qk pk ) = pk dqk + qk dpk ,

the sufficient condition of canonicity −qk dpk + Q j dPj − ( H − H ) dt = dW3 , W3 = W − qk pk + Q j Pj

is resulting. If ⎡ ∂ Pj ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂qk ⎦

(20.2.36)

then the second group of relations (20.2.1) allows us to write qk = qk ( P , p ; t ) and then Q j = Q j ( P , p ; t ) , j , k = 1, 2,..., s , wherefrom W3 ( Q , P , q , p ; t ) = S 3 ( P , p ; t ) . By means of the sufficient canonicity condition, we obtain the representation qk = −

∂S 3 ∂S 3 , Qj = , j , k = 1, 2,..., s , ∂ Pj ∂ pk

(20.2.37)

as well as H = H + S 3 .

(20.2.37')

⎡ ∂ 2S3 ⎤ det ⎢ ⎥ ≠ 0. ⎣ ∂ Pj ∂ pk ⎦

(20.2.36')

We assume that

In this case, from the first subsystem (20.2.37) there result the generalized momenta Pj ; replacing in the second subsystem (20.2.37), we obtain also the generalized coordinates Q j , the canonical transformation (20.2.1) being thus specified. Wherefrom, a function S 3 ( P , p ; t ) of class C 2 , which verifies the condition (20.2.36'), determines the canonical transformation (20.2.37), (20.2.37'). Using the necessary and sufficient condition of canonicity (20.2.18'), we obtain representations corresponding to canonical transformations of non-unitary valency c . If the functions which occur do not depend explicitly on time (for instance, S1 = S1 ( P , q ) , S 2 = S 2 ( Q , p ) , S 3 = S 3 ( P , p ) ), then one obtains representations corresponding to complete canonical transformations.


284

Starting from the free transformation (20.2.27), (20.2.28) with H = 0 , we find again the Hamilton–Jacobi equation (19.2.9); one obtains the same result starting from the transformation (20.2.33), (20.2.33'). As well, starting from the transformation (20.2.35), (20.2.35') or from the transformation (20.2.37), (20.2.37'), we find again the HamiltonJacobi type equation (19.2.68). As an application, let us consider the generating function S1 ( P , q ; t ) = f j ( q ; t ) Pj ,

(20.2.38)

⎡ ∂ 2 S1 ⎤ ⎡ ∂ fj ⎤ = det ⎢ det ⎢ ⎥ ⎥ ≠ 0. ⎣ ∂qk ⎦ ⎣ ∂ Pj ∂qk ⎦

(20.2.38')

for which

The formulae (20.2.33) allow us to write Q j = f j ( q ; t ) , pk =

∂ fj P , j , k = 1, 2,..., s , ∂qk j

(20.2.38'')

obtaining thus a point transformation; if we take into account (20.2.38'), then we may calculate Pj = Pj ( q , p ; t ) from the second relation (20.2.38''). We also notice that H = H + fj Pj ,

(20.2.38''')

finding thus again the results of Sect. 20. 2.1.4. An interesting particular case is S1 = q j Pj ,

(20.2.39)

to which corresponds the identical transformation Q j = q j , Pj = p j , j = 1, 2,..., s .

(20.2.39')

As well, let q j0 , p j0 be a stationary point for an autonomous system (which does not depend explicitly on t ) of Hamiltonian H , so that q j = q j0 , p j = p j0 , j = 1, 2,..., s , is a solution for the canonical equations. The generating function S1 = ( q j − q j0

)( Pj

− p j0

)

(20.2.40)

leads to the transformation Q j = q j0 + q j , Pj = p j0 + p j , j = 1, 2,..., s ,

(20.2.40')

which measures the deviation with respect to the equilibrium solution. More general, may be q j0 = q j0 ( t ) and p j0 = p j0 ( t ) , j = 1, 2,..., s , do represent a known solution


285

of the canonical equations of an autonomous system; it can be a first application in the perturbation theory, which will be considered in the next subsection.

20.2.1.7 Applications to the Perturbation Theory Let (20.2.20) be the solution of the canonical equations (19.1.14) of a mechanical system of Hamiltonian H and initial conditions q 0 = q ( t0 ) and p 0 = p ( t0 ) ; we try now to obtain the motion of a "perturbed" system of Hamiltonian H + H 1 , hence the solution of the canonical system qk =

∂ ∂ ( H + H 1 ) , pk = − ( H + H 1 ) , k = 1, 2,..., s . ∂ pk ∂qk

(20.2.41)

If the initial conditions are considered as new variables in (20.2.20), then these formulae determine a univalent free canonical transformation (as we have seen in Sects. 20.2.1.3 and 20.2.1.5), which transforms the canonical system (19.1.14) into a canonical system of Hamiltonian H = 0 , so that q j0 = 0 and p j0 = 0 , j = 1, 2,..., s ; as well, (20.2.20) transforms the canonical system (20.2.41) into a canonical system of Hamiltonian H = H 1 (because H − ( H + H1 ) = 0 − H = S , where S is the generating function), from which q j0 =

∂H1 ∂H , p j0 = − 01 , j = 1, 2,..., s . 0 ∂ pj ∂q j

(20.2.42)

So, the variables q 0 and p 0 have the following property: they are constant for the nonperturbed motion (equal to the initial conditions), while for the perturbed motion they are functions of time and initial values; by convention, we represent these variables in the form qk[ 0 ] = qk[ 0 ] ( q 0 , p 0 ; t ) , pk[ 0 ] = pk[ 0 ] ( q 0 , p 0 ; t ) , k = 1, 2,..., s , being – in fact – the general solution of the system (20.2.20), in which the Hamiltonian is the "perturbed energy" H 1 . By replacing this function in formulae (20.2.20), we find the solution corresponding to the perturbed motion.

Fig. 20.5. Perturbed motion of a particle.

Using the theory of the canonical transformations, the integration of the canonical system (20.2.41) was replaced by the integration of the canonical system (19.1.14), (20.2.42); the general solution of the system (20.2.41) will thus be


286

q j = q j ( q [ 0 ] ( q 0 , p 0 ; t ) , p[ 0 ] ( q 0 , p 0 ; t ) ) , p j = p j ( q [ 0 ] ( q 0 , p 0 ; t ) , p[ 0 ] ( q 0 , p 0 ; t ) ) , j = 1, 2,..., s .

(20.2.41')

We notice that the “energy perturbation” of the mechanical system is equivalent to the perturbation of the “initial conditions”. Let us consider a fixed representative point P0 in the hyperplane t = t0 and let us build up in this hyperplane a non-perturbed trajectory P0 P , given by (20.2.20); as well, the trajectory P0Q0 in the same hyperplane corresponds to the displacement of the initial position P0 (at the moment t0 ) till the initial position Q0 (at the moment t ), given by (20.2.41') (Fig. 20.5). From the representative point Q0 , let us set up the nonperturbed corresponding trajectory Q0Q . The representative point Q will correspond to the position of the system in the perturbed motion at the moment t , while P0Q will correspond to the trajectory in the perturbed motion. The perturbation is thus put in evidence by the “displacement” PQ . Thus, the perturbed motion can be considered as a “compound motion” in the phase space; the representative point is in motion along the non-perturbed trajectory, but this trajectory is displaced (in general, deformed) because of the perturbations of the initial conditions.

20.2.1.8 Infinitesimal Canonical Transformations Let be the infinitesimal transformation Q j = q j + δq j = q j + ε f j ( q , p ; t ) , Pj = p j + δ p j = p j + εg j ( q , p ; t ) , j = 1, 2,..., s ,

(20.2.43)

where ε is a small parameter. One can obtain such a transformation starting from the canonical transformation Q j = Q j ( q , p ; t ; λ ) , Pj = Pj ( q , p ; t ; λ ) , where λ is a parameter which for λ = λ0 leads to the identical transformation (20.2.40'); if we put λ = λ0 + ε , and expand into a series after the powers of ε , neglecting

O ( ε2 ) , then

we find the canonical transformation (20.2.43), which is completely determined if we know the functions f j and g j , j = 1, 2,..., s . Hamilton’s function is transformed in the form H = H + δH = H + εH 1 ,

(20.2.43')

while the generating function will be of the form S1 + δS1 = q j Pj + εK ,

(20.2.43'')

where K = K ( q , P ; t ) ; we notice that K ( q , P ; t ) = K ( q , p ; t ) + O ( ε ) . Applying the formulae (20.2.33), one has Qj = q j + ε

∂K ∂K , Pj = p j − ε , j = 1, 2,..., s , ∂ pj ∂q j

(20.2.44)


287

as well as

H = H + εK ;

(20.2.44')

hence δq j = ε f j = ε

∂K ∂K , δ p j = εg j = − ε , δH = εH 1 = εK . ∂ pj ∂q j

(20.2.44'')

Thus, for each function K = K ( q , p ; t ) of class C 1 one obtains an infinitesimal canonical transformation; as a matter of fact, taking into account the condition (20.2.32'), the function K must be of class C 2 . If K = K ( q , p ) , non-depending explicitly on time, then H = H , and the transformation is an infinitesimal complete canonical transformation. In this case, let F ( q , p ) be a function of class C 1 ; for an infinitesimal complete canonical transformation given by (20.2.44) we have a variation of the form δF =

∂F ∂F ∂F ∂K ⎞ ⎛ ∂F ∂K − = ε (F,K ) , δq j + δ pj = ε ⎜ ∂q j ∂ pj ∂ p j ∂q j ⎟⎠ ⎝ ∂q j ∂ p j

(20.2.45)

where a Poisson bracket has been put in evidence. The operator D=

∂K ∂ ∂K ∂ − ∂ p j ∂q j ∂q j ∂ p j

(20.2.45')

is the operator of the infinitesimal canonical transformation; in this case δF = εDF .

(20.2.45'')

We notice that the infinitesimal canonical transformation (20.2.44) transforms the function F in itself if (F,K ) = 0 .

(20.2.45''')

In particular, for F = qk and F = pk respectively, we get δqk = ε ( qk , K ) = ε Dqk , δ pk = ε ( pk , K ) = ε Dpk , k = 1, 2,..., s ;

(20.2.46)

we can thus express the variations of the canonical co-ordinates, hence the infinitesimal complete canonical transformation corresponding to a generating function K ( q , p ) of class C 1 by means of a Poisson bracket, using the relations (20.2.46). By these relations one can show that the infinitesimal generator of the spatial translations along a fixed direction is the projection of the linear momentum of the mechanical system on the same direction; as well, the infinitesimal generator of the spatial rotations about a fixed axis is the projection of the angular momentum of the mechanical system on the very same axis.


288 If F = H in (20.2.45), then we have

δH = ε ( H , K ) ;

(20.2.46')

comparing with the third relation (20.2.44''), we obtain ( ( H , K ) = − ( K , H ) ) ( K , H ) + K = 0 .

(20.2.47)

Taking into account Theorem 19.1.5, we may state that the generating function K is a first integral of the canonical system (19.1.14). As well, we may state that the first integrals of the motion are generators of the infinitesimal canonical transformations. Particularly, for K = H we see that we must have H = 0 , hence H = H ( q , p ) ; wherefrom, an infinitesimal canonical transformation for which the Hamiltonian is a generating function is an infinitesimal complete canonical transformation. If, in this case, we put ε = δt , introducing a time variation, then we find again Hamilton’s equations written by means of the Poisson brackets in the form δqk = ( qk , H ) δt , δ pk = ( pk , H ) δt , k = 1, 2,..., s .

(20.2.48)

To pass from the co-ordinates q , p at the moment t to the co-ordinates q + δq , p + δp at the moment t + δt , it is sufficient to effect an infinitesimal complete canonical transformation of the co-ordinates q , p , of small parameter δt and of generating function H ( q , p ) . Especially, starting from the initial values of the generalized co-ordinates, we attain their values at a neighbouring moment, obtaining the incipient motion. One may thus state that the motion of the representative point on its trajectory is obtained by a succession of infinitesimal complete canonical transformations. Hence, the motion of a mechanical system corresponds to the evolution or to the continuous development of a canonical transformation; we can say that the Hamiltonian is the generator of this motion in time. If DF = 0 , then F ( q , p ) is an invariant of the canonical transformation, hence a first integral of the equations of motion. We can consider the equations (20.2.48) as being the canonical equations written in finite differences; we are thus led to an iterative method to integrate them (obtaining a finite number of points of the trajectory of the representative point). Let qk ( t ) , pk ( t ) and qk ( t ) + δqk , pk ( t ) + δ pk , k = 1, 2,..., s , be two neighbouring solutions of the system of canonical equations (19.1.14), corresponding to two neighbouring initial states. In this case, the functions δqk (t ) and δ pk (t ) , of class C 1 , verify the equations of the variations d ∂ 2H ∂ 2H δq j = δqk + δp , dt ∂ p j ∂qk ∂ p j ∂ pk k d ∂ 2H ∂ 2H δ pj = − δqk − δ p , j = 1, 2,..., s . dt ∂q j ∂qk ∂q j ∂ pk k

(20.2.49)

The knowledge of a first integral f ( q , p ; t ) = const of the canonical system leads to the obtaining of a particular solution for these equations. We will thus show that


δq j = ε

289

∂f ∂f , δ pj = − ε , j = 1, 2,..., s , ∂ pj ∂q j

(20.2.49')

are the solutions of the equations (20.2.49). Indeed, introducing in these equations, we have

d ∂ 2H ∂ f ⎞ ∂H ∂ f ⎞ ⎛ ∂ 2H ∂ f ⎡ ∂ ⎛ ∂H ∂ f δq j = ε ⎜ − ⎟ = ε ⎢ ∂ p ⎜ ∂q ∂ p − ∂ p ∂q ⎟ dt ∂ p ∂ q ∂ p ∂ p ∂ p ∂ q j k k k ⎠ k k k ⎠ ⎝ j k ⎣ j ⎝ k ∂f ∂H ∂ 2 f ⎞ ⎤ ∂ ⎛ ∂f ⎛ ∂H ∂ 2 f ⎞ −⎜ − ⎟ ⎥ = − ε ∂ p ⎜ ∂ p pk + ∂q qk ⎟ ∂ ∂ ∂ ∂ ∂ ∂ q p p p p q ⎠ j j j ⎝ k k k ⎠⎦ k k ⎝ k

(

)

∂2 f ∂ df ⎛ ∂2 f ⎞ ⎡ d ⎛ ∂f ⎞ ∂ ⎛ ∂f ⎞⎤ +ε ⎜ − f + ε ⎢ ⎜ p k + qk ⎟ = − ε ⎟− ⎜ ⎟⎥ d ∂ ∂ ∂ ∂ ∂ p p p q p t j j j k k ⎝ ⎠ ⎣ dt ⎝ ∂ p j ⎠ ∂t ⎝ ∂ p j ⎠ ⎦

= −ε

∂ df d ⎛ ∂f ⎞ d ⎛ ∂f ⎞ ε +ε ⎜ = , j = 1, 2,..., s , ⎟ ∂ p j dt dt ⎝ ∂ p j ⎠ dt ⎜⎝ ∂ p j ⎟⎠

where we have supposed that f is a function of class C 2 and have taken into consideration that df / dt = 0 ; analogically, d d ⎛ ∂f ⎞ , j = 1, 2,..., s , δp = − ⎜ ε dt j dt ⎝ ∂q j ⎟⎠

the above affirmation being thus proved. Hence, the infinitesimal transformation (20.2.49') transforms each trajectory in a neighbouring one and the whole family of paths in itself.

20.2.2 Structure of Canonical Transformations. Properties In what follows, we will consider some structure properties of the canonical transformations; we mention especially the group properties and the rôle played by the Symplectic group. The invariance of Lagrange and Poisson brackets and of the volume element to such transformations, as well as the important rôle played by the involution functions are put into evidence.

20.2.2.1 Structure of a Canonical Transformation. The Generating Function In Sects. 20.2.1.5 and 20.2.1.6 we have shown how one can obtain canonical (eventually, complete canonical) transformations, starting from the generating functions S , S1 , S 2 or S 3 . These generating functions characterize the structure of the respective canonical transformations; for instance, if there exists a generating function S , then we have to do with a free transformation. In general, let be 4s quantities qk , pk ,Q j , Pj , j , k = 1, 2,..., s , linked by the canonical transformations (20.2.1), for which the condition (20.2.1') is verified; one can choose 2s independent quantities in the form


290

q1 , q 2 ,..., qr , pr +1 , pr + 2 ,..., ps ,Q1 ,Q2 ,...,Qm , Pm +1 , Pm + 2 ,..., Ps , 0 ≤ r , m ≤ s . (20.2.50)

Indeed, we may state Theorem 20.2.2 (Carathéodory’s lemma). Let be 2s dependent functions Q1 ,Q2 ,...,Qs , P1 , P2 ,..., Ps , which depend on the independent quantities q1 , q 2 ,..., qs , p1 , p2 ,..., ps ; from the 4s quantities Q , P , q , p one may always choose 2s independent quantities so that between them do not be any pair of conjugate quantities qk , pk or Q j , Pj , j , k = 1, 2,..., s . We must mention that the proof of this lemma has been given by Carathéodory for the particular case in which the relations connecting the dependent variables to the independent ones correspond to a canonical transformation; in fact, the lemma has a general character. If we conveniently index again the generalized co-ordinates q and Q , and the generalized momenta p and P , then the 2s independent quantities can be represented in the form (20.2.50). We use the relations s

∑

k =r +1 s

∑

j =m + 1

pk dqk = d

Pj dQ j = d

s

∑

qk pk −

∑

Q j Pj −

k =r +1 s

j = m +1

s

∑

k =r +1 s

∑

qk dpk ,

j =m + 1

Q j dPj ;

in this case, the necessary and sufficient condition of canonicity (20.2.18') is written in the form s ⎛ r ⎞ ⎛ m c ⎜ ∑ pk dqk − ∑ qk dpk ⎟ − ⎜ ∑ Pj dQ j − ⎝ k =1 ⎠ ⎝ j =1 k =r +1

⎞ Q j dPj ⎟ − ( cH − H ) dt = dU , j = m +1 ⎠ (20.2.51) s

∑

where U =W −c

s

∑

k = r +1

qk pk +

s

∑

j =m + 1

Q j Pj ;

(20.2.51')

we may assume that the function U depends on the variables (20.2.50) and on time. In this case, ∂U ∂U , cql = − , k = 1, 2,..., r , l = r + 1, r + 2,..., s , ∂qk ∂ pl ∂U ∂U , Qj = , i = 1, 2,..., m , j = m + 1, m + 2,..., s , Pi = − ∂Qi ∂ Pj

cpk =

(20.2.52) (20.2.52')


291

the relations thus obtained corresponding to the considered canonical transformations of valency c . The function U is the generating function of the canonical transformation. Let U be a function of class C 2 for which we suppose that ⎡ ∂ 2U ⎤ det ⎢ ⎥ ≠ 0, ⎣ ∂ Rj ∂rk ⎦

(20.2.53)

where rk , Rj , j , k = 1, 2,..., s , represent the variables (20.2.56) in the mentioned order; we can solve the equations (20.2.52) with respect to Pi , i = 1, 2,..., m , and Q j , j = m + 1, m + 2,..., s . Introducing in the relations (20.2.52'), we obtain the searched relations of the canonical transformations (20.2.1).

20.2.2.2 Group Properties. The Symplectic Group We introduce the 2s × 2s matrix associated to the Jacobian (20.2.1') in the form ⎡ ⎡ ∂Q j ∂ ( Q , P ) ⎤ ⎢ ⎢⎣ ∂qk = ⎢ J = ⎡ ⎢⎣ ∂ ( q , p ) ⎥⎦ ⎢ ⎡ ∂ Pj ⎢ ⎢ ∂q ⎣⎣ k

⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k ⎤ ⎡ ∂ Pj ⎥ ⎢ ⎦ ⎣ ∂ pk

⎤⎤ ⎥⎥ ⎦⎥ , ⎤⎥ ⎥⎥ ⎦⎦

(20.2.54)

where j , k = 1, 2,..., s ; we consider also the matrix ⎡ 0 −E⎤ I= ⎢ ⎥, ⎣E 0 ⎦

(20.2.55)

where E is the s × s unit matrix. The matrix I is called the canonical matrix. We notice that 2 ⎡ 0 −E⎤ ⎡ 0 −E⎤ ⎡ −E I2 = ⎢ = ⎥⎢ ⎥ ⎢ ⎣ E 0 ⎦ ⎣ E 0 ⎦ ⎣⎢ 0

0 ⎤ ⎡ E2 = − ⎥ ⎢ − E2 ⎦⎥ ⎣⎢ 0

0 ⎤ ⎡E 0 ⎤ −1 = − ⎥ ⎢ 0 E ⎥ = − E = − II , E2 ⎦⎥ ⎣ ⎦

the last unit matrix having 2s rows and 2s columns; one obtains I −1 = − I , so that this matrix is non-singular. Introducing the transpose matrix J T , let us calculate ⎡ ⎡ ∂Q j ⎢ ⎢ ∂q ⎣ k T ⎢ J I= ⎢ ⎡ ∂Q j ⎢⎢ ⎣⎢ ⎣ ∂ pk

then

⎤ ⎥ ⎦

T

⎤ ⎥ ⎦

T

⎡ ∂ Pj ⎢ ⎣ ∂qk ⎡ ∂ Pj ⎢ ⎣ ∂ pk

T ⎡ ⎡ ∂ Pj ⎤ ⎤ ⎥ ⎥ ⎡ 0 − E ⎤ ⎢ ⎢ ∂q ⎦ ⎥ ⎢⎣ k ⎥ = ⎢ T ⎥ ⎢E 0 ⎦ ⎢ ⎡ ∂ Pj ⎤ ⎥⎣ ⎥ ⎥ ⎢ ⎦ ⎦ ⎣⎢ ⎣ ∂ pk

⎤ ⎥ ⎦

T

⎤ ⎥ ⎦

T

⎡ ∂Q j −⎢ ⎣ ∂qk ⎡ ∂Q j −⎢ ⎣ ∂ pk

T ⎤ ⎤ ⎥ ⎥ ⎦ ⎥ ; T ⎤ ⎥⎥ ⎥ ⎦ ⎦⎥


292 ⎡ ⎡ ∂ Pj ⎢ ⎢ ∂q ⎣ k J T IJ = ⎢ ⎢ ⎡ ∂ Pj ⎢⎢ ⎢⎣ ⎣ ∂ pk

T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂q ⎦ ⎣ k T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂q ⎦ ⎣ k

⎤ ⎡ ∂Q j ⎥ − ⎢ ∂q ⎦ ⎣ k ⎤ ⎡ ∂Q j ⎥ − ⎢ ∂p ⎦ ⎣ k

⎡ [ [ q j , qk = ⎢ ⎢⎣ [ [ q j , pk

]] ]]

T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂q ⎦ ⎣ k T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂q ⎦ ⎣ k

− [ [ q j , pk

⎤ ⎥ ⎦ ⎤ ⎥ ⎦

]] ⎤ ⎥ [ [ p j , pk ] ] ⎥⎦

⎡ ∂ Pj ⎢ ∂q ⎣ k ⎡ ∂ Pj ⎢ ∂p ⎣ k

T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k T ⎤ ⎡ ∂Q j ⎥ ⎢ ∂p ⎦ ⎣ k

⎤ ⎡ ∂Q j ⎥ − ⎢ ∂q ⎦ ⎣ k ⎤ ⎡ ∂Q j ⎥ − ⎢ ∂p ⎦ ⎣ k

T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂p ⎦ ⎣ k T ⎤ ⎡ ∂ Pj ⎥ ⎢ ∂p ⎦ ⎣ k

⎤⎤ ⎥⎥ ⎦⎥ ⎤ ⎥⎥ ⎥ ⎦ ⎦⎥

⎡ 0 −c E ⎤ = ⎢ = cI , 0 ⎥⎦ ⎣cE

where we took into account the necessary and sufficient conditions of canonicity (20.2.12'). We obtain thus a necessary and sufficient condition of canonicity, equivalent to the conditions (20.2.12'), and we can state Theorem 20.2.3 The necessary and sufficient condition for the transformation (20.2.1) in the space Γ 2s be canonical consists in the existence of a scalar constant c ≠ 0 , such that J T IJ = c I .

(20.2.56)

This condition is written in the form J T IJ = I

(20.2.57)

in case of a univalent canonical transformation. A matrix J which satisfies the condition (20.2.57) is called a Symplectic matrix. Because detI = ±1 and the determinant of a product of matrices is equal to the product of the determinants of the corresponding matrices, we obtain detJ = ±1 , hence the Symplectic matrices are nonsingular; as a matter of fact, we have given also another proof in Sect. 20.2.1.2. A matrix J which satisfies the relation (20.2.56) will be called a generalized Symplectic matrix of valency c ≠ 0 ; starting from this relation, one can show that detJ = ±c s , obtaining again a known result. Hence, the generalized Symplectic matrices are also non-singular. Let J1 and J 2 be two 2s × 2s generalized Symplectic matrices; the matrix J 2 J1 is a matrix which has the same number of rows and columns. We have J1T IJ1 = c1 I , J 2T IJ 2 = c2 I , wherefrom J1T ( c2−1 J 2T IJ 2 ) J1 = c1 I ; but J1T J 2T = ( J 2 J1 )T , so that

( J 2 J1 )T I ( J 2 J1 ) = c2c1 I , the matrix J 2 J1 being also a Symplectic matrix. The associativity property of the Symplectic matrices follows from the associativity property of the matrix product. Let be the unit matrix E , so that EJ = JE = J , ET = E and ET IE = I ; one notices that this matrix is Symplectic, playing the rôle of neutral element in the set of these matrices (the corresponding canonical transformation is univalent). Multiplying the relation (20.2.56) at left by the matrix I −1 and at right by the matrix J −1 and taking into account that II −1 = JJ −1 = E , we find the remarkable relation I -1 JI = c J −1 ;

(20.2.58)


293

hence, the inverse matrix J −1 does exist. We notice that detJ ≠ 0 , so that

( J −1 )T

= ( JT

)−1 ; in this case, taking into account the relation (20.2.56) (we assume

that the matrix J is a generalized Symplectic matrix), we may write

( J −1 )T

= IJ −1 = ( J T

)

−1

IJ −1 = c −1 ( J T

−1

) ( J T IJ ) J −1

= c −1 I ,

the matrix J −1 being Symplectic too. We can thus state that the set of Symplectic matrices forms a group: the Symplectic group, defined in a real linear space, of even dimension 2s , and denoted by Sp ( 2s , \ ) . The essential property of this group consists in making invariant certain antisymmetric bilinear form defined on \2s . There results that the set of canonical transformations forms a group too. Analogously, the set of generalized Symplectic matrices forms the generalized Symplectic group. We notice also that, if J is a generalized Symplectic matrix, besides its inverse J −1 , the transpose matrix J T is also a generalized Symplectic matrix; indeed, starting from the relation

( J −1 )T IJ −1

= c −1 I and calculating the inverse at right and at left, we

have ⎡ ( J −1 ⎣

)T IJ −1 ⎤⎦

−1

= ( J −1

= − ( JT

)−1 I −1 ⎡⎣ ( J −1 )T ⎤⎦

)T IJ T

−1

= − JI ⎡⎣ ( J −1

)−1 ⎤⎦

T

= −c I −1 = c I .

Hence, we may write the necessary and sufficient condition of canonicity (20.2.56) also in the form JIJ T = c I ;

(20.2.56')

as well, in case of univalent canonical transformations, the condition (20.2.57) becomes JIJ T = I .

(20.2.57')

20.2.2.3 Invariance Properties The univalent canonical transformations admit three basic invariants: the volume element in the phase space, the Poisson bracket and the Lagrange bracket. For instance, let us define the univalent canonical transformation by means of the generating function S1 = S1 ( P , q ; t ) . Before the transformation, the volume element is given by ∂ (q, p ) ⎤ dq1 dq2 ...dqs dp1 dp2 ...dps = det ⎡⎢ dq1 dq2 ...dqs dP1 dP2 ...dPs ⎣ ∂ ( q , P ) ⎥⎦

and after the transformation by ∂ (Q , P ) ⎤ dQ1 dQ2 ...dQs dP1 dP2 ...dPs = det ⎡⎢ dq1 dq2 ...dqs dP1 dP2 ...dPs , ⎣ ∂ ( q , P ) ⎥⎦


294 with

0 ⎤ ⎡ E 2 ∂ (q, p ) ⎤ ⎡ ⎢ ⎥ = det ⎡ ∂ pk ⎤ = det ⎡ ∂ S1 = det ⎢ det ∂ ∂ p p ⎢ ∂P ⎥ ⎢ ∂ P ∂q ⎤ ⎡ ⎤⎥ ⎢⎡ ⎣ ∂ ( q , P ) ⎥⎦ ⎣ j ⎦ ⎣ j k ⎣⎢ ⎢⎣ ∂q ⎥⎦ ⎢⎣ ∂ P ⎥⎦ ⎦⎥ ⎡ ⎡ ∂Q ⎤ ⎡ ∂Q ⎤ ⎤ ∂ (Q , P ) ⎤ ⎡ ∂ 2 S1 ⎡ ∂Q j ⎤ ⎡ = det ⎢ ⎢⎣ ∂q ⎥⎦ ⎢⎣ ∂ P ⎥⎦ ⎥ = det ⎢ = det det ⎢ ∂q ∂ P ⎥ ⎢⎣ ∂ ( q , P ) ⎥⎦ ⎢ ⎥ ⎣ ∂qk ⎦ ⎣ k j E ⎦⎥ ⎣⎢ 0

⎤ ⎥, ⎦ ⎤ ⎥, ⎦

where we took into account the representation (20.2.33). The two determinants at the right are equal because the corresponding matrices are one the transpose of the other (or, using Schwartz’s theorem, if S1 is a function of class C 2 ); it follows that dQ1 dQ2 ...dQs dP1 dP2 ...dPs = dq1 dq2 ...dqs dp1 dp2 ...dps .

(20.2.59)

We can thus state Theorem 20.2.4 The volume element in the phase space is invariant with respect to a univalent canonical transformation. We obtain the same result, observing that ∂ (Q , P ) ⎤ dQ1 dQ2 ...dQs dP1 dP2 ...dPs = det ⎡⎢ dq1 dq2 ...dqs dp1 dp2 ...dps ; ⎣ ∂ ( q , p ) ⎥⎦

in Sect. 20.2.1.2 we have seen that J = ±1 , while for the identical transformation (20.2.39') we have J = 1 , so that we find again the relation (20.2.59). In case of a canonical transformation of valency c we obtain dQ1 dQ2 ...dQs dP1 dP2 ...dPs = cs dq1 dq2 ...dqs dp1 dp2 ...dps .

(20.2.59')

Concerning the Poisson bracket of the functions ϕ = ϕ ( q , p ; t ) , ψ = ψ ( q , p ; t ) we can write

∂ϕ ∂ψ ∂ϕ ∂ψ ⎛ ∂Φ ∂Q j ∂Φ ∂ Pj ⎞ ⎛ ∂Ψ − =⎜ + ∂ql ∂ pl ∂ pl ∂ql ∂ Pj ∂ql ⎟⎠ ⎜⎝ ∂Qk ⎝ ∂Q j ∂ql ∂Φ ∂ Pj ⎞ ⎛ ∂Ψ ∂Qk ∂Ψ ∂ Pk ⎞ ∂Φ ⎛ ∂Φ ∂Q j −⎜ + + = ⎜ ⎟ ⎟ ∂ Pj ∂ pl ⎠ ⎝ ∂Qk ∂ql ∂ Pk ∂ql ⎠ ∂Q j ⎝ ∂Q j ∂ pl

∂Qk ∂Ψ ∂ Pk ⎞ + ∂ pl ∂ Pk ∂ pl ⎟⎠

( ϕ, ψ ) =

+

∂Φ ∂Ψ (Q j , Pk ∂Q j ∂ Pk

∂Φ ∂Ψ ( Pj ,Qk j ∂Qk

) + ∂P

∂Ψ (Q j ,Qk ∂Qk

∂Φ ∂Ψ ( Pj , Pk ) , j ∂ Pk

) + ∂P

where Φ (Q , P ; t ) = ϕ ( q (Q , P ;t ) , p (Q , P ;t ) ;t ) , Ψ (Q , P ;t ) = ψ ( q (Q , P ;t ) , p (Q , P ;t ) ; t ) ;

)


295

taking into account the sufficient conditions of canonicity (20.2.15), it follows

( ϕ, ψ ) = ( Φ, Ψ ) ,

(20.2.60)

so that we can state Theorem 20.2.5 The Poisson bracket is invariant with respect to a univalent canonical transformation. In case of a canonical transformation of valency c , the necessary and sufficient conditions of canonicity (20.2.15') lead to

( ϕ, ψ ) = c ( Φ, Ψ ) .

(20.2.60')

Let us consider also the Lagrange bracket with respect to the parameters u and v in the form ∂ql ∂ pl ∂q ∂ pl ∂q ∂ Pj ⎞ ⎛ ∂ pl ∂Qk ∂ pl ∂ Pk ⎞ ⎛ ∂q ∂Q j − l =⎜ l + l ⎟ ⎜ ∂Q ∂v + ∂ P ∂v ⎟ Q u P u ∂u ∂v ∂v ∂ u ∂ ∂ ∂ ∂ j j k ⎠ ⎝ ⎠⎝ k ∂q ∂ Pk ⎞ ⎛ ∂ pl ∂Q j ∂ pl ∂ Pj ⎞ ∂Q j ∂Qk ⎛ ∂q ∂Qk −⎜ l + l ⎟ ⎜ ∂Q ∂u + ∂ P ∂u ⎟ = ∂u ∂v [Q j ,Qk ] Q v P v ∂ ∂ ∂ ∂ j j k ⎝ k ⎠⎝ ⎠ ∂Q j ∂ Pk ∂ Pj ∂Qk ∂ Pj ∂ Pk Q j , Pk ] + Pj ,Qk ] + + [ [ [ Pj , Pk ] , ∂u ∂ v ∂u ∂v ∂u ∂v

[ u , v ]q , p =

where Qj ( t ;u , v ) = Qj ( q ( t ;u , v ) , p ( t ;u , v ) ;t ) , Pj ( t ; u , v ) = Pj ( q ( t ; u , v ) , p ( t ; u , v ) ; t ) , j = 1, 2,..., s ;

the sufficient conditions of canonicity (20.2.12'') lead to

[ u , v ]q , p = [ u , v ]Q ,P ,

(20.2.61)

and we may state Theorem 20.2.6 The Lagrange bracket is invariant with respect to a univalent canonical transformation. If the transformation is of valency c , then the necessary and sufficient conditions of canonicity (20.2.12''') allow us to write

[ u , v ]q , p =

1 [ u , v ]Q ,P . c

(20.2.61')

20.2.2.4 Functions in involution. Lie’s Theorem In Sect. 19.1.2.3 we have introduced the canonical conjugate first integrals (the Poisson bracket of which is constant) and the first integrals in involution (the Poisson bracket of which is zero). In general, a set of functions of q , p and t , of class C 1 , for which the Poisson bracket of each pair is equal to zero, is a set of functions in


296

involution; obviously, the generalized co-ordinates Q j = Q j ( q , p ; t )

and the

generalized momenta Pj = Pj ( q , p ; t ) , j = 1, 2,..., s , have, everyone, this property. In general, s arbitrary functions of q , p and t , of class C 1 , cannot be the generalized co-ordinates Q 1 ,Q2 ,...,Qs of a canonical transformation; the following question arises: is it sufficient that these functions be in involution to have this property ? More general, we can formulate the question: Let be given s functions Q j = Q j ( q , p ; t ) in involution; can one find other s functions Pj = Pj ( q , p ; t ) , j = 1, 2,..., s , so that the transformation ( q , p ) → ( Q , P ) be canonical ? Obviously, the answer is affirmative in case of a point transformation. In general, let be the functions in involution ϕi = ϕi ( q , p ; t ) , i = 1, 2,..., s , of class C 2 , so that det [ ∂ϕi / ∂ pk ] ≠ 0 ; because there is not an identical relation connecting the generalized co-ordinates q and Q and the time t , we have to solve the equations Qi = ϕi ( q , p ; t ) with respect to p , obtaining pl = ψl ( q ,Q ; t ) , i , l = 1, 2,..., s . We can write the identity ϕi ( q , ψ ; t ) − Qi = 0 , i = 1, 2,..., s , with respect to the variables q and Q , where we have denoted ψ ≡ { ψ1 , ψ2 ,..., ψs } . Calculating the partial derivatives with respect to ql , there results ∂ϕi ∂ϕ ∂ψl + i = 0 , i , k = 1, 2,..., s ; ∂qk ∂ pl ∂qk

multiplying by ∂ϕ j / ∂ pk and summing for all the values of k , we find ∂ϕi ∂ϕj ∂ϕ ∂ϕj ∂ψl + i = 0 , i , j = 1, 2,..., s . ∂qk ∂ pk ∂ pl ∂ pk ∂qk

Taking the antisymmetric part with respect to the indices i and j and observing that ( ϕi , ϕj ) = 0 , i , j = 1, 2,..., s , we may write (we invert the dummy indices k and l in the first sum) ∂ϕj ∂ϕi ∂ψl ∂ψ ∂ϕi ∂ϕj ∂ψl ∂ϕi ∂ϕj ⎛ ∂ψk − = − l ⎞⎟ = 0 , i , j = 1, 2,..., s ; ⎜ ∂ pl ∂ pk ∂qk ∂ pl ∂ pk ∂qk ∂ pk ∂ pl ⎝ ∂ql ∂qk ⎠

we obtain thus a system of s 2 homogeneous linear equations to determine s 2 unknowns ∂ψk / ∂ql − ∂ψl / ∂qk , the determinant of which is equal to

( det [ ∂ϕi / ∂ pk ] )2 s ≠ 0 . It results ∂ψk ∂ψl = , k , l = 1, 2,..., s ; ∂ql ∂qk

(20.2.62)

hence, there exists a function S = S ( q ,Q ; t ) so that pk = ψk ( q ,Q ; t ) =

∂S , k = 1, 2,..., s , ∂qk

(20.2.62')


297

[ ∂ 2 S / ∂Q j ∂qk ]

(if not, an identical relation connecting

with a non-singular matrix

q , p and t must exist). If we take also

Pj = −

∂S , j = 1, 2,..., s , ∂q j

(20.2.62'')

we find a free canonical transformation, the answer to the above question being, in general, affirmative. Let us consider the case in which the s functions ϕi , i = 1, 2,..., s , are first integrals of the system of canonical equations. The canonical equations in new canonical co-ordinates derive from Hamilton’s function H = H + S = H ( Q , P ; t ) ; but the canonical co-ordinates Q are constant in time along the integral curves of the canonical system in q , p and t , while the function H cannot depend on the generalized momenta P . Much more, we can choose the canonical transformation so as to have H = 0 , the generalized momenta P being also constant in time along the trajectory of the representative point of the mechanical system. Hence, the knowledge of s first integrals in involution gives the possibility to determine other s first integrals, so that the 2s distinct first integrals do specify, completely, the solution of the problem. Indeed, if we replace Q by ϕ ≡ {ϕ1 , ϕ2 ,..., ϕs } in (20.2.62'), then we obtain an identity in q , p and t ; differentiating successively with respect to q j , p j and t , it follows ∂ψk ∂ψ ∂ϕl ∂ψk ∂ϕl + k = 0, = δ jk , j , k = 1, 2,..., s , ∂q j ∂Ql ∂q j ∂Ql ∂ p j

(20.2.63)

∂ψ ψk + k ϕ l = 0 , k = 1, 2,..., s . ∂Ql

(20.2.63')

Because ϕl is a first integral (it verifies a relation of the form (19.1.61)), we can write ∂ψ ∂ψk ∂ψ ⎛ ∂ϕ ∂ H ∂ H ∂ϕl ⎞ − ; ψk = − k ϕ l = (ϕ ,H ) = k ⎜ l ∂Ql ∂Ql l ∂Ql ⎝ ∂q j ∂ p j ∂q j ∂ p j ⎟⎠

taking into account (20.2.63), we obtain ∂ H ∂ψk ∂H , k = 1, 2,..., s . − ψk = − ∂ p j ∂q j ∂qk

Replacing p by ψ in H ( q , p ; t ) , we obtain G ( Q , q ; t ) , hence ∂G ∂H ∂ H ∂ϕj = + , k = 1, 2,..., s , ∂qk ∂qk ∂ p j ∂qk


298 so that

∂G ψk + = 0 , k = 1, 2,..., s . ∂qk

(20.2.64)

From (20.2.62) and (20.2.64) it follows that

ψk =

∂S , G = − S , ∂qk

(20.2.65)

with S = S ( Q , q ; t ) ; if this function corresponds to the canonical transformation (20.2.62'), (20.2.62''), then the new function of Hamilton is H = G + S , expressed by

Q , P and t . Taking into account (20.2.65), we have H = 0 . We obtain thus Pj = Pj ( q , p ; t ) , j = 1, 2,..., s , that is s new first integrals of the canonical system in q , p and t ; these first integrals are in involution, because the Poisson brackets of all pairs which can be formed vanish. We notice that this result corresponds to Liouville’s Theorem 19.1.9. The above considerations can be presented also in a slightly different form. Starting from ϕk ( q , p ; t ) = ak , ak = const , we may calculate pk = ψk ( q , a ; t ) , k = 1, 2,..., s , with a ≡ {a1 , a2 ,..., as } . Replacing the generalized momenta p by ψ in H ( q , p ; t ) , we obtain the function G ( q , a ; t ) . In this case, ψk dqk − Gdt = dS , with S = S ( q , a ; t ) , is an exact differential; other s relations ∂ S / ∂ak = bk , k = 1, 2,..., s , give the generalized co-ordinates. We build up the function S by means of the s first integrals in involution, and then find again the Hamilton–Jacobi equation (19.2.9). In the particular case in which ϕ k = 0 , k = 1, 2,..., s , and H = 0 , with ϕs = H ( q , p ) , as = h being the energy constant, we have pk = ψk ( q , a ) = ∂ S / ∂qk , k = 1, 2,..., s , with S = S ( q , a ) ; we find again the relations (19.2.15'''), obtaining thus the solution of the problem. A system of functions u1 , u2 ,..., um of class C 2 in a domain D of variables q1 , q 2 ,..., qs , p1 , p2 ,..., ps , t , so that ( u j , uk

) = 0,

j , k = 1, 2,..., s , is called a system

in involution. We may state Theorem 20.2.7 (S. Lie). If v = v (u ) , w = w (u ) are functions of class C 2 , where u ≡ {u1 , u2 ,..., um } is a system in involution defined on the domain D of variables q , p and t , then ( v,w ) = 0 .

(20.2.66)

A direct calculus proves the theorem. We have seen in Sect. 20.2.1.8 that a function F is transformed in itself by an infinitesimal canonical transformation if ( F , K ) = 0 (the formula (20.2.45''')). Let be such a transformation with K = u j ; because we have to do with a system in involution,


299

any function uk is transformed in itself, hence also the function v . We have thus

( uj ,v ) = 0 ,

j = 1, 2,..., s . Let us consider now an infinitesimal canonical

transformation with K = v ; any of the functions u j is transformed in itself, hence also the function w . The relation (20.2.66) takes place, the Theorem 20.1.7 being once more justified. This theorem can be stated also in a modified form, sometimes useful. Thus, if the equations v = 0 and w = 0 are consequences of the equations u j = 0 , j = 1, 2,..., s , then ( v , w ) = 0 . In fact, this form of the theorem is known as Lie’s theorem. As an application, let us consider the functions ϕ1 , ϕ2 ,..., ϕs in involution and the equations Q j − ϕj ( q , p ; t ) = 0 , j = 1, 2,..., s ; solving with respect to the generalized momenta, we find p j − ψj ( Q , q ; t ) = 0 , j = 1, 2,..., s . Equating the generalized coordinates to constants, the last equations are consequences of the first ones. But the first members of these equations form a system in involution; we can thus say the same thing about the first members of the second system of equations, so that

( pj

∂ ( p j − ψj ) ∂ ( pk − ψk ) ∂ ( p j − ψj ) ∂ ( pk − ψk ) − ∂qi ∂ pi ∂ pi ∂qi ∂ψj ∂ψj ∂ψ ∂ψ δ + k δ ji = − = − + k = 0. ∂qi k i ∂qi ∂qk ∂q j

− ψj , pk − ψk

)=

We find again the relation (20.2.62) as a consequence of Lie’s theorem.

20.2.2.5 First Integrals Linear with Respect to the Generalized Momenta As it is known, in case of a dynamical system described by Lagrangian co-ordinates, one of them (for example, qs ) being cyclic, the corresponding generalized momentum remains constant during the motion (see Chap.19, Subsec. 1.1.8). One can state an inverse property, i.e. Theorem 20.2.8 Any autonomous system which has a first integral linear with respect to the generalized momenta can be described by means of some generalized coordinates conveniently chosen, so that one of them be cyclic. Indeed, let be the first integral ψj p j , where ψj = ψj (q ) , j = 1, 2,..., s . We consider the system of equations dq1 dq dqs = 2 = ... = , ψ1 ψ2 ψs

(20.2.67)

which defines a family of curves in the space Λs ; the solutions are defined as intersections of s − 1 independent integrals of the partial derivatives equation ψj

∂z = 0. ∂q j

(20.2.67')

We denote these integrals by ϕj ( q ) = const , j = 1, 2,..., s − 1 ; let be now the point transformation


300

Q j = ϕ j , j = 1, 2,..., s ,

(20.2.68)

where ϕ1 , ϕ2 ,..., ϕs −1 are the above obtained functions. We choose then Qs = ϕs , so that we have dq1 dq dqs = 2 = ... = = dQs ψ1 ψ2 ψs

(20.2.69)

for the previously defined displacements. To obtain Qs , we integrate, e.g., dQs =

dq1 , ψ1

where Ψ1 = ψ1 ( q1 ,Q1 ,Q2 ,...,Qs −1 ) . Let us suppose that the transformation (20.2.68) is reversible, so that we may obtain qk = Fk ( Q1 ,Q2 ,...,Qs ) , k = 1, 2,..., s ; for a displacement along one of the paths defined by (20.2.67) (for which Q j = const , j = 1, 2,..., s − 1 ) we obtain ∂ Fk / ∂Qs = ψk , k = 1, 2,..., s . We consider now the point transformation (see Sect. 20.2.1.4) qk = Fk , Pk = p j

∂ Fj , k = 1, 2,..., s ; ∂Qk

we have

Ps = p j

∂ Fj = p j ψj = const , ∂Qs

in the new variables, so that Qs is a cyclic co-ordinate, the Theorem 20.2.8 being thus proved.

20.3 Symmetry Transformations. Noether’s Theorem. Conservation Laws In 1918, Emmy Noether proved her famous theorem relating the physical properties of a mechanical (in general, physical) system and the conservation laws, when the equations of motion emerge from a variational principle. Thus, she established a general method to derive conservation laws in classical mechanics; but this method can be applied equally well in case of relativistic mechanics, quantum mechanics, electromagnetism etc. This study is, obviously, on the line of the Erlangen programme. The importance of the results thus obtained is significant because, in general, the conservation theorems are not deduced systematically, but only in particular cases. By introducing symmetry transformations, certain properties leading to Noether’s theory are put into evidence; one obtains thus integration constants (first integrals). Some considerations concerning Lie groups are made, and then various applications to discrete mechanical systems, resulting conservation laws, are given.


301

20.3.1 Symmetry Transformations. Noether’s Theorem In what follows, we will present the symmetry transformations which allow to state Noether’s theorem; an important particular case is that of the Lagrangian which does not depend explicitly on time. We mention also the rôle played by the reciprocal of Noether’s theorem. 20.3.1.1 Equations of Motion Let be a physical system (for instance, the mechanical system considered in Sect. 20.1.5.1), the states of which are determined by means of the independent variables x ≡ { x i , i = 1, 2,..., n } and the functions of state (dependent variables) u α , α = 1, 2,...,m ; the derivatives of the functions of state are denoted by uiα = ∂u α / ∂ xi , i = 1, 2,..., n . The time t has not a privileged position and can be anyone of the variables xi . We assume that the equations of motion derive from the functional

∫Ω L ( x i , u

A =

α

, uiα ) dΩ ,

(20.3.1)

where dΩ = dx1 dx 2 ...dxn is the volume element of the domain Ω , and the Lagrangian density L (corresponding to the unit volume) is of class C 2 . To obtain the equations of motion, we introduce the variations of the independent variables xi′ = xi + δxi = xi + εηi ( x ) , i = 1, 2,..., n ,

(20.3.2)

where ε is a small parameter, while ηi and δx i are arbitrary functions and infinitesimal functions of class C 2 , respectively. The relations (20.3.2) define an infinitesimal transformation of the domain Ω into Ω ′ . Consequently, the state functions and their derivatives are transformed according to the relations (analogously, we take δu α = εξ α ( x ) , where ξ α are arbitrary functions of class C 2 ) δu α ( x ) = u ′α ( x ′) − u α ( x ) , δuiα = ui′α ( x ′) − uiα ( x ) , i = 1, 2,..., n , α = 1, 2,...,m ,

(20.3.3)

where we used similar notations, while the variation of the functional (20.3.1) is given by δA =

∫Ω ′ L ( xi′ , u ′

α

, ui′α ) dΩ ′ −

∫Ω L ( x i , u

α

, uiα ) dΩ ,

(20.3.1')

where dΩ ′ = dx1′dx 2′ ...dx n′ . The Jacobian of the transformation (20.3.2) is ⎡ ∂ ( x j + δx j ) ⎤ ∂ ⎡ ∂ x j′ ⎤ ⎡ J = det ⎢ ⎥ = det ⎢ δ jk + ∂ x δx j ⎥ = det ⎢ ∂ ∂ x x ⎣ k k ⎣ k ⎦ ⎣ ⎦

⎤, ⎥⎦


302 hence J =1+

∂ δx + O ( ε 2 ) , ∂x j j

(20.3.4)

where we have used Einstein’s summation convention for dummy Latin indices from 1 to n. The volume elements associated with the domains Ω and Ω ′ are thus related by dΩ ′ = JdΩ , and for the inverse transformation we will have J −1 = 1 −

∂ δx + O ( ε 2 ) . ∂xi i

(20.3.4')

With these definitions and using Einstein’s summation convention for dummy Greek indices from 1 to m, we may write the Taylor series expansion

L ( x i′ , u ′α , ui′α ) = L ( x i + δx i , u α + δu α , uiα + δuiα ) ∂L ∂L ∂L α α 2 = L ( x i , u α , uiα ) + δx i + α δu + α δui + O ( ε ) . ∂xi

∂u

∂ui

Hence, taking into account (20.3.4), the variation (20.3.1') of the functional A can be written in the form δA =

⎛

∫Ω ⎜⎝ L

∂ ∂L ∂L ∂L δx + δx + δu α + δu α ∂ x i i ∂ x i i ∂u α ∂uiα i

⎞ dΩ . ⎟ ⎠

(20.3.1'')

As, in general, δuiα ≠ ∂ ( δu α ) / ∂ x i , we introduce the variations δ∗u α ( x ′ ) = u ′α ( x ′ ) − u α ( x ′ ) , δ∗uiα ( x ′ ) = ui′α ( x ′ ) − uiα ( x ′ ) ,

(20.3.5)

to integrate by parts the last term in (20.3.1''). Thus, we have the Taylor series

δ∗u α ( x ′ ) = δ∗u α ( x + δx ) = δ∗u α (x ) +

∂ ( δ∗u α ) δx i + O ( ε2 ) , ∂xi

where δ∗u α and δx i are variations of order ε; by neglecting O ( ε2 ) , we have δ∗u α ( x ′ ) = δ∗u α (x ) . Therefore, from (20.3.3) we obtain δu α ( x ) = [ u ′α ( x ′ ) − u α ( x ′ ) ] + [ u α ( x ′ ) − u α ( x ) ] , δuiα ( x ) = [ ui′α ( x ′ ) − uiα ( x ′ ) ] + [ uiα ( x ′ ) − uiα ( x ) ] ,

while the notations (20.3.5) and new expansions into Taylor series lead to δu α (x ) = δ∗u α (x ) + uiα δx i , δuiα (x ) = δ∗uiu (x ) + uijα δx j ,

(20.3.5')


303

where the mixed derivatives of second order uijα = ∂ 2u α / ∂ x i ∂ x j , i , j = 1, 2,..., n , have been also introduced. The second relation (20.3.5) may be written also in the form δ∗uiα ( x ′ ) =

∂ ∂ δ∗u α ( x ′ ) , [ u ′α ( x ′ ) − u α ( x ′ ) ] = ∂ xi′ ∂ xi′

(20.3.5'')

where we took into consideration the first relation (20.3.5); analogically,

δ∗uiα (x ) =

∂ ∂ δ∗u α (x ) , δuiα (x ) = δ∗u α (x ) + uijα δx j . ∂ xi ∂xi

(20.3.5''')

Introducing the operator of total differentiation (L depends on xi explicitly or by the agency of the functions u α and uiα ) d ∂ ∂ ∂ , i = 1, 2,..., n , = + uiα + u jiα dxi ∂ xi ∂u α ∂u jα

(20.3.6)

we observe that

d ∂ ∂L ∂L ∂L δx + δx + uiα δx + uijα δx ; ( L δx i ) = L dx i ∂xi i ∂ xi i ∂u α i ∂uiα j taking into account also the relations (20.3.5'), the variation (20.3.1'') of the functional

A becomes δA =

∂L

⎡ d

∫Ω ⎢⎣ d xi ( L δxi ) + ∂u α δ∗u

α

+

∂L δ∗uiα ∂uiα

⎤ dΩ . ⎥⎦

(20.3.7)

The last term in δA can be written in the form ∂L d ⎛ ∂L d ⎛ ∂L ⎞ α α ⎞ α ⎜ ⎜ α ⎟ δ∗u , α δ∗ui = α δ∗u ⎟ − ∂ui dx i ⎝ ∂ui ⎠ dx i ⎝ ∂ui ⎠

so that δA =

⎡ d ⎛

∫Ω ⎢⎣ dxi ⎝⎜ L δxi

+

∂L δ∗u α ⎞⎟ + [ L ]α δ∗u α ∂uiα ⎠

⎤ dΩ , ⎦⎥

(20.3.7')

where

[ L ]α =

∂L d ⎛ ∂L ⎞ , α = 1, 2,...,m , − ∂u α dx i ⎜⎝ ∂uiα ⎟⎠

(20.3.8)

is the Euler–Lagrange derivative (corresponding to the functional derivative (20.1.79)). Finally, by means of the equations (20.3.5') we obtain the variation of the action A in the form


304 δA =

⎡ d ⎛


∂L ∂L α u δx ⎞ + [ L ]α ( δu α − u αj δx j ) ⎤⎥ dΩ . δu α − ∂uiα ∂uiα j j ⎠⎟ ⎦

+

(20.3.7'')

To obtain the equations of motion, according to the conditions of Hamilton’s principle, we assume that the variations δx i vanish identically inside and on the boundary of the domain Ω (i.e. δx i ≡ 0 in Ω and on ∂Ω, and the integration domain remains unchanged), and the variations of the functions of state u α are identically zero on the boundary of Ω ( δu α ∂Ω ≡ 0 ). By way of consequence, applying the fluxdivergence formula (A.2.67'), we show that the first term in (20.3.7'') vanishes and we obtain δA =

∫Ω [ L ]

α

δu α dΩ ;

(20.3.9)

equating to zero the variation δA and using the second basic lemma of the variational calculus, we obtain the equation of motion in the differential form

[ L ]α = 0 , α = 1, 2,...,m .

(20.3.10)

The function L which leads to a certain type of equations of motion is not uniquely determined. Indeed, these equations are invariant if we replace the function

L by λL,

where λ is an arbitrary non-zero constant (scale transformation) or by L + df j / dx j , where f j ( xi , u α ) , j = 1, 2,..., n , are arbitrary functions of class C 2 (divergence transformation). Indeed, observing that df j ∂ fj ∂ fj α ∂ fi ∂ ⎛ df j ⎞ = + , = α uj , dx j ∂uiα ⎜⎝ dx j ⎟⎠ ∂u α ∂x j ∂u ∂ 2 fj ∂ 2 fj ∂ ⎛ df j ⎞ ∂ ⎛ ∂ fj ⎞ ∂ ⎛ ∂ fj = + uα = + ⎜ ⎟ β ⎜ dx ⎟ β β α j β ∂ x j ⎝ ∂u ⎠ ∂u α ⎜⎝ ∂u β ∂u ⎝ j ⎠ ∂u ∂ x j ∂u ∂u

it is easy to show that

[ df j / dx j ]α

d ⎛ ∂ fj ⎞ ⎞ α ⎟ u j = dx ⎜ β ⎟ , j ⎝ ∂u ⎠ ⎠

= 0 . In particular, when the only independent

variable is the time t we obtain the gauge transformations discussed in Sect. 18.2.3.2. 20.3.1.2 Symmetry Transformations. Noether’s Theorem By symmetry transformations we denote the transformations of independent and dependent variables under which the form of the equations of motion remains invariant. Thus, the general form of a transformation of independent variables is xi′ = ϕi ( x ) , i = 1, 2,..., n ,

and it changes the functions of state into

(20.3.11)


305

u ′α ( x ′) = Φ α ( x , u ( x ) ) , α = 1, 2,...,m ,

(20.3.11')

where u ≡ {u 1 , u 2 ,..., u m } . The functional (20.3.1) is invariant with respect to the transformation (20.3.11) if the relation

L ′ ( x i′ , u ′α , ui′α ) dΩ ′ = L ( x i , u α , uiα ) dΩ

(20.3.12)

holds. On the other hand, the equations of motion (20.3.10) are invariant if

L ′ ( x i′ , u ′α , ui′α ) = L ( x i′ , u ′α , ui′α ) +

df j . dx j′

(20.3.12')

Thus, the transformations (20.3.11) are symmetry transformations for the considered mechanical system if and only if both conditions (20.3.12) and (20.3.12') are satisfied. In the case of the infinitesimal transformations (20.3.2), (20.3.3), these conditions become

L ′ ( x i + δxi , u α + δu α , uiα + δuiα ) dΩ ′ = L ( x i , u α , uiα ) dΩ , L ′ ( x i + δxi , u α + δu α , uiα + δuiα ) dΩ ′ = L ( x i + δx i , u α + δu α , uiα + δuiα ) dΩ +

d ( δ f j ) dΩ ; dx j

in the last term we have introduced the variation δ f j (a quantity of order ε) and we have used the relation δ f j ( x ′ ) = δ f j (x ) (we neglect O ( ε2

)

with respect to

observing that dx j′ and dΩ ′ differ from dxi and dΩ, respectively, by

O ( ε ) ),

O ( ε ) (which

can be neglected, because δ f j is of the order of ε). Eliminating

L ′ ( x i + δx i , u α + δu α , uiα + δuiα ) dΩ ′ , we obtain

L ( xi + δxi , u α + δu α , uiα + δuiα ) = ⎡⎢ L ( xi , u α , uiα ) − ( δ fj dx j ⎣ d

) ⎤⎥ J −1 ; ⎦

finally, substituting (20.3.4') and neglecting O ( ε2 ) , we obtain (expansion in a Taylor series) ⎛ δx ∂ + δu α ∂ + δu α ∂ + ∂ δx ⎞ L = − d δ f ( x , u α ) . ⎜ i ∂x i ∂u α ∂uiα ∂ x i i ⎟⎠ dx j j i ⎝ i

(20.3.13)

Hence, an infinitesimal transformation of the form (20.3.2), (20.3.3) is a symmetry transformation if, for a given L, there exist functions f j ( x i , u α ) of class C 2 , so that the equation (20.3.13) be satisfied. It can be shown that, for a given L, the symmetry transformations form a group, which represents the symmetry group of the considered mechanical system.


306

If, in particular, together with E.L. Hill, the left member of the equation (20.3.13) is identically zero, and if the Jacobian of the transformation is equal to unity, then we say that the density L has an invariant form. From (20.3.1') and (20.3.13) we have (we integrate on the arbitrary domain Ω and on the domain Ω ′ , respectively, δA +

d

∫Ω dxi ( δ fi ) dΩ = 0 ,

(20.3.13')

so that (20.3.7'') can be written in the form ⎡ d ⎛


+

∂L ∂L α u δx + δ fi ⎟⎞ + [ L ]α ( δu α − u αj δx j ) ⎤⎥ dΩ = 0 . δu α − ∂uiα ∂uiα j j ⎠ ⎦ (20.3.14)

This integral has to vanish on any domain Ω; if we assume that the integrand is a continuous function, then it must vanish at any point x ∈ Ω . Consequently, when the equations of motion (20.3.10) are satisfied, then we obtain an equation of conservation of the form d dx i

⎛ L δx + ∂L δu α − ∂ L u α δx + δ f ⎞ = 0 . ⎜ i i ⎟ ∂uiα ∂uiα j j ⎝ ⎠

(20.3.15)

It follows that an infinitesimal symmetry transformation of the mechanical system is associated with a certain equation of conservation; as the symmetry transformations form a group, the relation (20.3.15) establishes the connection between the symmetry group of a mechanical (physical) system and a certain conservation law. The general connection between the symmetry properties of a mechanical system and the conservation laws is given by Theorem 20.3.1 (E. Noether). If the Lagrangian of a physical (mechanical) system is invariant with respect to a continuous group of transformations with m parameters, then there exist m quantities which are conserved during the evolution of the system. 20.3.1.3 Case of a Generalized Conservative System Let be a mechanical system described by a Lagrangian L = L ( q , q ) , q ≡ {q1 , q 2 ,..., qs } , q ≡ {q1 , q2 ,..., qs } , which does not depend explicitly on time ( L = 0 ), for instance a mechanical system subjected to catastatic constraints and conservative given forces; see Sect. 18.2.3.4). We also assume that neither the Hamiltonian H = H ( q , p ) , p ≡ { p1 , p2 ,..., ps } (which is a consequence of the considered constraints), nor the integrals of motion of the system do not depend explicitly on time. For an infinitesimal transformation of the generalized co-ordinates


qk′ = qk + δqk , k = 1, 2,..., s ,

307 (20.3.16)

we have δqk =

d δq . dt k

(20.3.17)

We obtain thus the relation between the Lagrangian L ( q , q ) and the Hamiltonian H ( q , p ) by means of the generalized momenta given by (18.2.80); to pass from H to L we use Hamilton’s first equation (19.1.14). If C ( q , p ) is an integral of motion in the Hamiltonian formalism, then in the Lagrangian formalism this integral of motion will take the form D ( q , q ) = C ( q , p ) ; also, along the trajectory of the representative point, this constant of motion will satisfy the relations d d C (q, p ) = D ( q , q ) = 0 . dt dt

(20.3.18)

The variation of the Lagrangian, induced by the transformation (20.3.16), can be written in the form (we use the summation convention of dummy indices from 1 to s; see Sect. 20.1.1.2) δL =

d ⎛ ∂L ⎞ δq − [ L ]k δqk , dt ⎜⎝ ∂qk k ⎟⎠

(20.3.19)

which, along the trajectory of the representative point in Λs , becomes δL =

d ⎛ ∂L ⎞ δq . dt ⎜⎝ ∂qk k ⎟⎠

(20.3.19')

We consider three cases when the relation (20.3.19') leads to conserved quantities (integrals of motion), i.e.: (i) If δL = 0 , then the corresponding integral of motion is D ( q , q ) δτ =

∂L δq = pk δqk , ∂qk k

(20.3.20)

where pk are the generalized momenta in the Hamiltonian formulation and δτ is a constant factor. (ii) If there exists a function f (q ) such that δL =

d δ f (q ) , dt

(20.3.21)

then we obtain the integral of motion ∂f ⎞ ⎛ ∂L δq . D ( q , q ) δτ = ⎜ − ∂ ∂ q qk ⎟⎠ k ⎝ k

(20.3.21')


308

In this case, we may assume that the Lagrangian of the system has the form L ′ = L − df / dt , so that δL ′ = 0 . Let us notice that the two Lagrangians (L and L ′ ) are related by a divergence transformation; consequently, they are equivalent in the sense that the form of the equations is preserved. (iii) If δL =

d g ( q , q ) , dt

(20.3.22)

then the associated integral of motion is

D ( q , q ) δτ =

∂L δq − g ( q , q ) . ∂qk k

(20.3.22')

In this case, there is no one Lagrangian L ′ to be equivalent to L and to satisfy the relation δL ′ = 0 . It is evident that the first and the second case are particular cases of the third case. Actually, there is no other case (i.e. δL ≠ dg ( q , q ) / dt ) which could lead to an integral of motion associated with the infinitesimal transformation (20.3.16). Note that not all the three cases considered above correspond to symmetry transformations. Indeed, the only independent variable is the time t, while the Lagrangian L does not explicitly depend on t. Hence, the defining relation (20.3.13) takes the form ∂ ∂ ⎞ d ⎛ δL ≡ ⎜ δqk + δqk L = − δ f ( q ;t ) ; ∂qk ∂qk ⎟⎠ dt ⎝

(20.3.23)

it follows that only the first case and the second one correspond to symmetry transformations. We may state Theorem 20.3.1' (E. Noether). To every infinitesimal transformation of the form (20.3.16), inducing a variation of the Lagrangian of the form (20.3.22), it corresponds a conserved quantity defined by (20.3.22'). If the infinitesimal transformation (20.3.16) is a symmetry transformation (i.e., g ( q , q ) = δ f ( q ) ), then there exists an invariant Lagrangian

L′=L −

d f (q ) . dt

(20.3.24)

20.3.1.4 The Reciprocal of Noether’s Theorem The Hessian of a given Lagrangian with respect to the generalized velocities has the elements

H jk = Hkj =

∂ pj ∂ pk ∂ 2L , j , k = 1, 2,..., s , = = ∂q j ∂qk ∂qk ∂q j

(20.3.25)


309

and det ⎡⎣ H jk ⎤⎦ ≠ 0 is the necessary and sufficient condition for the integrability of Lagrange’s equations. The inverse of the matrix ⎡⎣ H jk ⎤⎦ has the elements

H jk−1 = Hkj−1 =

∂q j ∂q = k , j , k = 1, 2,..., s . ∂ pk ∂ pj

(20.3.25')

The relation (20.3.22') represents the condition which must be satisfied by the most general integral of motion of a generalized conservative mechanical system whose evolution is described by Lagrange’s equations. An equivalent form of this condition is given by (20.3.18); thus we have ∂D ∂D d D ( q , q ) = q + q = 0 . dt ∂qk k ∂qk k

(20.3.18')

In our case L = L ( q , q ) , H = H ( q , p ) and qk = qk ( q , p ) , k = 1, 2,..., s , so that (we use the relations (19.1.13), (19.1.14)) ∂qk ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂H ⎞ ∂ ⎛ ∂L ⎞ ∂ 2 L ∂ql = = = − =− ; ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ∂q j ∂q j ⎝ ∂ pk ⎠ ∂ pk ⎝ ∂q j ⎠ ∂ pk ⎝ ∂q j ⎠ ∂ql ∂q j ∂ pk

we obtain the generalized accelerations in the form (we use the relations (20.3.25')) qk =

∂q ∂q ∂q ∂ L ∂ 2 L ∂ql d qk = k q j + k p j = − q + k ∂q j ∂ pj ∂ql ∂q j ∂ pk j ∂ p j ∂q j dt

∂ 2L ⎛ ∂L ⎞ ∂q =⎜ − ql ⎟ k , k = 1, 2,..., s . q q q ∂ ∂ ∂ j l ⎝ j ⎠ ∂ pj

(20.3.26)

In terms of the elements of the inverse Hessian (20.3.25'), and by means of (20.3.26), the condition (20.3.18') becomes

qk

∂D ∂L ⎞ −1 ∂ D ⎛ ∂ 2L =⎜ ql − , H ∂qk ∂qk ⎟⎠ kj ∂q j ⎝ ∂qk ∂ql

(20.3.27)

where D ( q , q ) is an integral of motion determined by (20.3.22'). The condition (20.3.27) can also be expressed by means of the Poisson bracket (note that ∂C ( q , p ) / ∂t = 0 ) (C ( q , p ) , H ) = 0 ,

(20.3.28)

this being a necessary and sufficient condition for C ( q , p ) be a first integral of the canonical system. For K = C and ε = δτ in (20.2.46) and by means of (20.3.25'), we may rewrite the variations of the generalized co-ordinates in the form


310 δqk = ( qk ,C ) δτ =

∂C ∂ D ∂q j ∂D δτ = δτ = H −1 δτ , k = 1, 2,..., s ; ∂ pk ∂q j ∂ pk ∂q j jk

(20.3.27')

the conditions (20.3.27) and (20.3.27') are equivalent to (20.3.22'). It follows that an integral of motion D ( q , q ) is associated with an infinitesimal transformation defined by (20.3.16) and (20.3.27'), such that the condition (20.3.22') be satisfied, and the function g ( q , q ) is determined by ⎛ ∂L ∂D ⎞ g ( q , q ) = ⎜ H jk−1 − D ⎟ δτ . ⎝ ∂qk ∂q j ⎠

(20.3.27'')

The variation of the Lagrangian, induced by this infinitesimal transformation, is given by (20.3.22). The condition (20.3.27) is a necessary condition which has to be satisfied by the most general integral of motion given by Noether’s theorem. Because this condition is a homogeneous partial differential equation of first order, in 2s independent variables ( q , q ) , it has 2s − 1 functional independent solutions D ( q , q ) . Hence, the equation (20.3.27) yields all the constants of motion D ( q , q ) , which do not explicitly depend on time. As well, the equation (20.3.27) is also a sufficient condition for the integrals of motion; we may rewrite it in the form d ⎡ d ⎛ ∂L ⎞ ∂L ⎤ ∂ D D ( q , q ) = ⎢ ⎜ H jk−1 , ⎟− dt ⎣ dt ⎝ ∂qk ⎠ ∂qk ⎦⎥ ∂q j

(20.3.29)

as one can easily prove. We are thus led to d D ( q , q ) / dt = 0 along the integral curves of the system. In case of integrals of motion corresponding to symmetry transformations, we have g ( q , q ) = δ f ( q ) = ( ∂ f / ∂qk ) δqk , hence g ( q , q ) =

∂ f ∂D H −1 δτ ; ∂qk ∂q j jk

(20.3.30)

by means of the relations (20.3.22), (20.3.27'') we obtain D ( q , q ) = E ( q , p ′ ) = pk′

∂E , ∂ pk′

(20.3.31)

where we denoted p ′ ≡ { p1′ , p2′ ,..., ps′ } , with pk′ =

∂L ∂f ∂L ′ − = , k = 1, 2,..., s . ∂qk ∂qk ∂qk

(20.3.31')

Thus, the integral of motion D ( q , q ) has to be a homogeneous function of the generalized momenta (20.3.31'). This condition is equivalent to the condition which has to be satisfied by the generating function of infinitesimal, homogeneous, canonical


311

transformations; the relation (20.3.27') shows how to obtain the related infinitesimal canonical transformations, starting from a given integral of motion. Also, the conditions (20.3.31), (20.3.31') or the equivalent condition ∂ f ⎞ ∂D ⎛ ∂L D ( q , q ) = ⎜ − H −1 ∂ q ∂ qk ⎟⎠ ∂q j jk ⎝ k

(20.3.31'')

represent a sufficient condition for the case (ii) discussed above. All these results can be summarized in the form of Theorem 20.3.1'' (E. Noether; reciprocal). Suppose that D ( q , q ) is an integral of motion. To this integral of motion we associate an infinitesimal variation (20.3.27') of the generalized co-ordinates, which induces a variation

δL =

d ⎡ ⎛ ∂L ∂ D ⎞ ⎤ H −1 − D ⎟ δτ ⎥ dt ⎢⎣ ⎜⎝ ∂qk ∂q j jk ⎠ ⎦

(20.3.32)

of the Lagrangian. If and only if there is a Lagrangian L ′ equivalent to the Lagrangian L, as defined in (20.3.24), and D ( q , q ) is a homogeneous function of the first degree in the generalized momenta (20.3.31'), then the integral of motion D ( q , q ) can be correlated with an invariance property of the Lagrangian ( δL ′ = 0 ).

20.3.2 Lie Groups In what follows we present some notions concerning Lie groups, useful in the study of conservation laws; we consider first the one-parameter Lie groups and then the Lie groups with m parameters. 20.3.2.1 One-Parameter Lie Groups We consider the transformations xi′ = fi ( x ;a ) , x ≡ {x1 , x 2 ,..., x 2 s } , i = 1, 2,..., 2s ,

(20.3.33)

which depend on one parameter a, and the identity transformation corresponding to the zero value of the parameter ( a = 0 ), that is x i = fi ( x ;0 ) , i = 1, 2,..., 2s .

(20.3.33')

We suppose that there exists a value a of the parameter a, so that xi = fi ( x ′;a ) , x ′ ≡ { x1′ , x 2′ ,..., x 2′ s } , i = 1, 2,..., 2s ;

(20.3.33'')

hence there is an inverse transformation of the transformation (20.3.33). Let xi′′ = fi ( x ′;b ) , i = 1, 2,..., 2s ,

(20.3.34)

be some other transformation of the form (20.3.33). These transformations constitute a finite one-parameter topological group if there is a value c of the parameter such that


312

xi′′ = fi ( x ;c ) , i = 1, 2,..., 2s .

(20.3.34')

We notice that the parameter c is a function of the parameters a and b c = ϕ ( a ,b ) , a = ϕ ( a , 0 ) .

(20.3.35)

The group of transformations (20.3.33) is a Lie group if the functions fi are analytic functions in the parameter a and if the function ϕ is analytic in both arguments. In our case, 1 is the minimum number of independent parameters necessary to characterize the elements of the Lie group; this parameter is called essential, while 1 is the order or the dimension of the group. In terms of the general method for infinitesimal transformations, we have

xi′′ = xi′ + dxi′ = fi ( x ′; δa ) = x i′ + c = a + da = ϕ ( a , δa ) = a +

∂ fi ∂a

a =0

δa = xi′ + ui ( x ′ ) δa ,

∂ϕ (a , b ) δa = a + μ (a )δa , ∂b b = 0

(20.3.36) (20.3.36')

where c is related to the transformation xi′ + dx i′ = fi ( x ;a + da ) .

(20.3.36'')

Indeed, the infinitesimal displacement of the point x can be obtained in two ways: as a result of successive applications of two transformations or directly. As well, the relation (20.3.36') is a consequence of the property of associativity. Noting that δa = λ (a )da , λ (a ) = 1/ μ (a ) , we can write the system of differential equations of the group in the form dxi′ = ui ( x ′)λ (a ) , i = 1, 2,..., 2s , da

(20.3.37)

where the functions ui ( x ) form a velocity field. By introducing the parameter α=

a

∫ 0 λ(a)da ,

(20.3.37')

we obtain dx i′ = ui ( x ′) , i = 1, 2,..., 2s . dα

(20.3.37'')

The new parameter α is an additive quantity with respect to the composition of infinitesimal transformations; for an infinitesimal parameter β we have fi ( f ( x ; α ) ; β ) = fi ( x ; α + β ) , f ≡ { f1 , f2 ,..., f2 s } , i = 1, 2,..., 2s .

(20.3.38)


313

Let be a function F ( x ) of class C 2 . By an infinitesimal transformation of the point x (corresponding to an infinitesimal displacement), the function F has a variation of the form dF ( x ) =

2s

∂F

∑ ∂xi dxi

i =1

=

∂F

2s

∑ ui (x ) ∂ xi δa = XF (x )δa

i =1

=

∂F δa ; ∂a

(20.3.39)

the operator X =

∂

2s

∑ ui (x ) ∂ xi

i =1

(20.3.40)

is called the generator of the transformations group. From (20.3.39) it follows that we can express this generator also in the form X =

∂ . ∂a

(20.3.40')

Hence, the solution of the system of differential equations (20.3.37'') can be expanded in a power series xi′ = xi + α Xxi +

α2 2 X xi + ... , i = 1, 2,..., 2s , 2!

(20.3.41)

which, for infinitesimal transformations, is reduced to the simple form xi′ = xi + α Xxi = xi + αui ( x ) , i = 1, 2,..., 2s .

(20.3.42)

From (20.3.41) it follows x i′ = fi ( x ; α ) = e αX x i , i = 1, 2,..., 2s ,

(20.3.43)

and we are led to the equations ∂ fi = Xfi , i = 1, 2,..., 2s . ∂α

(20.3.43')

We are now in a position to establish the connection between the canonical transformations and the one-parameter Lie group, formed by the transformations (20.3.33). This will allow us to make use of Noether’s theorem, to find the corresponding conserved quantities. First of all, it is worthwhile to note that a special form of (20.3.43') is obtained if we set fi ( x ; α ) = xi ( α ) , i = 1, 2,..., 2s , and α = t , such that they become the canonical equations dxi = Xt xi , i = 1, 2,..., 2s , dt

(20.3.44)


314

where we have attached the label t to the generator to emphasize the associated parameter. Now we assume that the transformations (20.3.33) constitute a Lie group and note that, in the phase space Γ 2s , these transformations can be written in a form similar to (20.2.1), that is qi′ ≡ Qi = fi ( q , p ; α ) , pi′ ≡ Pi = qi ( q , p ; α ) , i = 1, 2,..., s .

(20.3.45)

We sought the conditions that have to be verified by these transformations (which form a Lie group) such that they be also canonical transformations. A sufficient condition consisting in the existence of a function V = V ( Q , P ) , whose exact differential is a differential one-form (20.2.9). Thus, if we denote λi =

∂ fi ∂q , μi = i , i = 1, 2,..., s , ∂α ∂α

(20.3.46)

we obtain dQi dα

α=0

= λi ( Q , P ) ,

dPi dα

α =0

= μi ( Q , P ) , i = 1, 2,..., s ,

from (20.3.45), where the dependence of λi and μi on Q and P was put in evidence. Also, the generator (20.3.40) becomes X = λi

∂ ∂ + μi , ∂qi ∂ pi

(20.3.46')

where we have used the summation convention for dummy indices from 1 to s. By means of the power series (20.3.41) and assuming the series expansion V (Q , P ) =

∞

∑ αkVk (Q , P ) ,

k =0

we obtain the sufficient condition

α2 2 α2 ⎛ ⎞ ⎛ ⎞ X μi + ... ⎟ ⎜ dqi + αdλi + pi dqi − ⎜ pi + αμi + dX 2 λi + ... ⎟ 2! 2! ⎠ ⎝ ⎠ ⎝ =

∞

∑ αk dVk (Q , P ) ,

k =0

which has to be identically satisfied. Hence, equating the coefficients of equal power of α yields the relations dV0 = 0 , dV1 = − ( μi dqi + pi dλi ) ,...


315

All these relations are simultaneously satisfied if we assume that the function V has the form V = αV1 +

α2 α3 2 XV1 + X V1 + ... . 2! 3!

(20.3.47)

To find the functions λi and μi , we introduce the function U ( q , p ) = −V1 − pi λi ,

(20.3.47')

such that dU =

∂U ∂U dq + dp = μi dqi − λi dpi . ∂qi i ∂pi i

The last equality is identically satisfied in the variables qi and pi if we choose λi = −

∂U ∂U , μi = , i = 1, 2,..., s . ∂ pi ∂qi

Thus, we obtain the condition which has to be satisfied such that the transformations (20.3.45) be canonical transformations, the function U being arbitrary. Consequently, in terms of the Poisson bracket, the generator (20.3.46') takes the form X =

∂U ∂ ∂U ∂ − ≡ (U , ) . ∂qi ∂ pi ∂ pi ∂qi

(20.3.46'')

Finally, by applying this operator to the functions ∂U / ∂qk and ∂U / ∂ pk , we obtain the whole group of canonical transformations, that is Qi = qi − α

∂U α2 ∂U + − ..., X ∂ pi 2! ∂ pi

∂U α2 ∂U + + ..., i = 1, 2,..., s . Pi = pi + α X ∂qi 2! ∂qi

(20.3.48)

Note that we are led to the same result by integrating the system of partial differential equations ∂ fi = (U , fi ) , i = 1, 2,..., s , ∂α

(20.3.49)

or by integrating the system of differential equations (20.3.46), written in the form dQi ∂U dPi ∂U , , i = 1, 2,..., s , = =− dα ∂ Pi dα ∂Qi

(20.3.50)


316

where U ( Q , P ) = −U ( q , p ) . All these three methods used to determine the Lie group of canonical transformations are equivalent. The function U is the generating function of canonical transformations, while the operator X = (U , ) is the generator of the Lie group. It is also easy to see that, for α = t and U = H ( Q , P ) , the system of differential equations (20.3.50) is identical with Hamilton’s equations. We denote by x the set of canonical variables {q , p } . Then, a function f ( x ) is invariant with respect to the Lie group of canonical transformations if the equation f (x ′) =

∞

αk

∑ k ! Xk f ( x ) = f ( x )

(20.3.51)

k =0

is identically satisfied. This can be accomplished if and only if Xf ( x ) = 0 , ∀x ∈ \2 s . Explicitly, by using the expression (20.3.46'') for the generator X, we may rewrite this condition in terms of canonical variables

∂U ∂ f ∂U ∂ f − ≡ (U , f ) = 0 , ∂qi ∂ pi ∂ pi ∂qi

(20.3.52)

where we have introduced the Poisson bracket. It follows that the conserved mechanical quantities are invariants of the group of canonical transformations and also solutions of the equation (20.3.52). If we find m independent invariants of this kind, then we may choose them as generalized co-ordinates (cyclic co-ordinates in this case), and there remain to integrate only 2s − m equations of motion. 20.3.2.2 Lie Groups with m Parameters Let be xi′ = fi ( x ;a ) , a ≡ {a1 , a2 ,..., am } , i = 1, 2,..., s ,

(20.3.53)

a set of general transformations, which depend on m independent parameters, real and continuous, and are such that for a = 0 (i.e. a1 = a2 = ... = am = 0 ) we obtain the identical transformation. We assume that a are essential parameters, and therefore the transformations (20.3.53) form a Lie group with m parameters. In this case, dxi′ =

∂ fi ( x ;a ) δak = ∂ak k =1 a =0 m

∑

∂ fi ( x ;a ) da j , i = 1, 2,..., s ; ∂a j j =1 m

∑

denoting uik ( x ) =

we may write

∂ fi ( x ;a ) , i = 1, 2,..., s , k = 1, 2,..., m , ∂ak a =0

(20.3.54)

Variational Principles. Canonical Transformations m

∑ uik (x )δak , i

dxi =

317

= 1, 2,..., s .

(20.3.54')

k =1

As in case of one-parameter Lie groups, the infinitesimal displacement of the point x will be a differential displacement, and there are two possibilities to obtain it: as a result of two successive transformations xi′ = fi ( x ;a ) and xi′ + dx i′ = fi ( x ; δa ) or directly, xi′ + dx i′ = fi ( x ;a + da ) ; taking into account the associativity property, the considered parameters are connected by relations of the form a j + da j = ϕ ( a ; δa ) = a j +

m

∑ μkj (a )δak

, j = 1, 2,..., m ,

(20.3.55)

k =1

where μkj (a ) =

∂ϕ j (a , b ) , j , k = 1, 2,..., m , ∂bk b =0

b ≡ {b1 , b2 ,..., bm } being parameters in i = 1, 2,..., s . The relations (20.3.55) lead to δak =

m

∑ λjk (a )da j , k j =1

the

(20.3.55')

transformation

xi′′ = fi ( x ′, b ) ,

= 1, 2,..., m ,

(20.3.56)

where m

∑ λlj μkj j =1

= δkl , k , l = 1, 2,..., m ,

(20.3.56')

δkl being Kronecker’s symbol. From the relations (20.3.54') and (20.3.56) we obtain the differential equations of the group ∂ xi = ∂a j

m

∑ uki (x )λjk (a ) , i

= 1, 2,..., s , j = 1, 2,..., m .

(20.3.57)

k =1

If the functions uki are linear independent (what we assume), then we say that the parameters a j are essential. Let F ( x ) be a function of class C 2 . This function has a variation of the form dF ( x ) =

s

∂F

∑ ∂xi dxi

i =1

=

s

m

∂F

∑ ∑ uij (x ) ∂xi δa j

i = 1 j =1

≡

m

∑ X j F (x )δa j j =1

=

m

∂F

∑ ∂a j δa j ; j =1

(20.3.58) by an infinitesimal transformation of the point x (corresponding to an infinitesimal displacement); the operators


318 Xj =

∂

s

∑ uij (x ) ∂xi

, j = 1, 2,..., m ,

(20.3.59)

i =1

are called the generators of the transformation group. As one can see from (20.3.58), the generators can be expressed also in the form ∂ , j = 1, 2,..., m . ∂a j

Xj =

(20.3.59')

The formalism used in the previous subsection is generalized by introducing m generating functions U k (q , p ) , while from (20.3.47') we obtain ∂U j ∂ ∂U j ∂ − = (U j , ∂qi ∂ pi ∂ pi ∂qi

Xj =

),

j = 1, 2,..., m .

(20.3.60)

corresponding to the operator (19.1.58). Let us consider the product of two generators applied to the function F ( x ) Xp Xr F (x ) = =

∂u r uqp i ∂ xq i =1 q = 1 m m

∑∑

∂F + ∂ xi

∂

s

∂

s

∑ uqp ∂xq ∑ uir ∂xi F (x )

q =1 s

i =1

∂ 2F

s

∑ ∑ uqp uir ∂xq ∂ xi

, p , r = 1, 2,..., s ;

i =1 q =1

using Schwarz’s theorem (from which, in case of functions of class C 2 , the mixed derivatives of second order do not depend on the order of differentiation), we obtain s

=

s

m

∂F

∑ ∑ ckpr uik ∂xi

i =1 k =1

=

s

∂u r

⎛

∑ ∑ ⎜⎝ uqp ∂xiq

( Xp Xr − Xr Xp ) F (x ) =

i = 1 q =1 s ∂ k c pr uik ∂ xi k =1 i =1 m

∑

∑

− uqr

F (x ) =

∂uip ⎞ ∂ F ∂ xq ⎟⎠ ∂ x i

m

∑ ckpr Xk F (x ) ;

k =1

hence, the generators of the transformations group verify the relation X p X r − Xr X p ≡ ( X p , Xr ) =

m

∑ ckpr Xk

, p , r = 1, 2,..., m .

k =1

(20.3.61)

The Poisson bracket thus obtained is called the commutator of the operators X p and Xr . The quantities c kpr are called the structure constants of the group; the property of antisymmetry ( X p , Xr ) = − ( Xr , X p ) is reflected in these constants in the form k c kpr = −crp , while from the Poisson-Jacobi identity

( Xi , ( X j , X k ) ) + ( X j , ( Xk , X i ) ) + ( X k , ( Xi , X j ) ) = 0 , i , j , k

= 1, 2,..., m , (20.3.62)


319

which is easily verified, we obtain the relation m

∑ ( cils cljk

l =1

+ c sjl ckil + ckls cijl

) = 0 , i, j ,k,s

= 1, 2,..., m .

(20.3.62')

The structure constants of every Lie group are determined, univocally, by the structure of the group. The commutator of the generators can be evaluated by applying it to an arbitrary function f (q , p ) ; thus we have

( X j , Xk ) f

= ( X j Xk − Xk X j ) f = (U j , (U k , f ) ) − (U k , (U j , f

));

(20.3.63)

from the above relations of antisymmetry and from the relation (20.3.63) we see that the Poisson brackets are skew-symmetric ( (U j ,U k ) = − (U k ,U j ) ) and verify the Poisson–Jacobi identity

(U i , (U j ,U k ) ) + (U j , (U k ,U i ) ) + (U k , (U i ,U j ) ) = 0 , i , j , k

= 1, 2,..., m . (20.3.64)

If we set one of the generating functions U k = H we are led to the Jacobi–Poisson theorem (see Sect. 19.1.2.3) for the case when the time does not appear explicitly. If U i and U j are two conserved quantities ( (U i , H ) = (U j , H

) = 0 ), then (U i ,U j )

is also

a conserved quantity. Finally, from (20.3.61), (20.3.63) and (20.3.64) we obtain the equation for the structure constants cijk of the group m

(U i ,U j ) = ∑ cijk U k

.

(20.3.65)

k =1

20.3.3 Space-Time Symmetries. Conservation Laws In what follows we will consider some particular groups (the translations group, the rotations group, the time translation group and the Galileo group); various conservation laws (conservation of the generalized linear and angular momenta, conservation of the mechanical energy and the motion of the mass centre) are thus put in evidence. We determine then the general form of the Lagrangian for the invariance with respect to symmetry transformations, introducing thus the Galileo–Newton group. 20.3.3.1 Conservation Laws Let be a discrete mechanical system, formed by n particles P α of masses mα , α = 1, 2,..., n , described by the Lagrangian L ( q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) , where qk (t ) , k = 1, 2,..., s , are the generalized co-ordinates in the configuration space Λs . Note that the only independent variable is the time t, such that the generalized


320

co-ordinates play the rôle of state functions. If s = 3n , then the relation between the generalized co-ordinates qk and the Cartesian co-ordinates x jα , j = 1, 2, 3 , of the particle of index α is given by the relation k = 3( α − 1) + j . In terms of the Cartesian co-ordinates, the condition (20.3.13) which has to be satisfied by a transformation in order to be a symmetry transformation has the form

d ⎞ d ⎛ ∂ α ∂ α ∂ ⎜ δt ∂t + δx j ∂ x α + δx j ∂ x α + dt δt ⎟ L = − dt ( δΩ ) , j j ⎝ ⎠

(20.3.66)

where we have used the summation convention of dummy indices (from 1 to 3 and from 1 to n, for Latin and Greek indices, respectively). Consequently, the left side in (20.3.66) has to be the total derivative, with respect to the time variable, of the function Ω ( x jα ; t ) . Also, the relation of conservation (20.3.15), corresponding to the invariance of the Lagrangian with respect to a symmetry transformation, becomes d dt

∂L α ⎞ ∂L ⎡⎛ ⎤ α ⎢ ⎜ L − ∂ x α x j ⎟ δt + ∂ x α δx j + δΩ ⎥ = 0 , j j ⎣⎝ ⎠ ⎦

(20.3.67)

which is equivalent to ∂L α ⎞ ∂L ⎛ α ⎜ L − ∂ x α x j ⎟ δt + ∂ x α δx j + δΩ = const . j j ⎝ ⎠

(20.3.67')

The equations (20.3.66) and (20.3.67') represent the analytic form of Noether’s theorem (see the Theorem 20.3.1) in classical mechanics. Thus, to every symmetry transformation of a mechanical system (that is an infinitesimal transformation which satisfies the condition (20.3.66)) it corresponds a conservative law expressed by (20.3.67'). 20.3.3.2 Translations Group. Conservation of the Generalized Momentum The simplest symmetry transformations are the translations of the origin of the frame of reference (that is of the Cartesian co-ordinates) by a time-independent vector ( δx1 , δx 2 , δx 3 ). These transformations are defined by the relations (the space translations group T) t ′ = t , δt = 0 , x j′α = x jα + δx j , x j′α = x jα , j = 1, 2, 3 , α = 1, 2,...,n .

If we choose δΩ = 0 , then (20.3.66) takes a simpler form 3

⎛

n

∂L ⎞

∑ ⎜⎝ ∑ ∂x αj ⎟⎠ δx j j =1

α =1

= 0;

(20.3.68)


321

because the infinitesimal displacements δx1 , δx 2 , δx 3 are independent, it follows that n

∂L

∑ ∂x αj

α =1

= 0 , j = 1, 2, 3 .

(20.3.69)

Now, from (20.3.67) we obtain d n ∂L d n ∂L ∑ α δx j = δx j α = 0; dt α =1 ∂ x j dt α∑ = 1 ∂ x j

consequently, the generalized momentum P of components Pj =

n

∑ p jα

α =1

=

∂L

n

∑ ∂x αj

, j = 1, 2, 3 ,

(20.3.70)

α =1

is conserved if the conditions (20.3.69) are satisfied. The relations (20.3.69) represent not only a sufficient condition, but also a necessary ones, because taking into account Lagrange’s equations of motion, the assumption of conservation of the generalized momentum leads to 0=

dPj d n ∂L = α = dt dt α∑ = 1 ∂ x j

n

d ⎛ ∂L ⎞

n

∂L

∑ dt ⎜⎝ ∂x αj ⎟⎠ = ∑ ∂x αj ,

α =1

α =1

j = 1, 2, 3 .

20.3.3.3 Rotations Group. Conservation of the Generalized Moment of Momentum Let us consider now a rotation of the system of orthogonal co-ordinates, defined by the relations xi′ = aij x j , i = 1, 2, 3 ,

(20.3.71)

where a ∈ SO ( 3, \ ) is an orthogonal matrix. In case of an infinitesimal rotation, this matrix has the form a = E + ak δθk ,

(20.3.72)

where ak are the matrices of the generators of the rotations group SO ( 3, \ ) , δθk , k = 1, 2, 3 , represent the infinitesimal variations of the rotation parameters and E is the identity matrix. Hence, we have aij = δij + ( ak δθk )ij = δij + δaij , δaij = − δa ji , i , j = 1, 2, 3 ,

(20.3.72')

δij being the Kronecker symbol. We assume that the transformation (20.3.71) is of the

form (20.3.2), such that, by substituting (20.3.72') into (20.3.71) we have δxi = x j δaij = − x j δa ji .


322

It follows that the variations of the particle co-ordinates are of the form δxiα = − xkα δaki , δxiα = − xkα δaki , i = 1, 2, 3 , α = 1, 2,...,n ,

(20.3.73)

while the independent variable t remains unchanged, that is t ′ = t , δt = 0 . Thus, from the condition (20.3.66), with δΩ = 0 , we obtain ⎛ α ∂L α ∂L ⎞ ⎜ x j ∂ x α + x j ∂ x α ⎟ δa jk = 0 . ⎝ k k ⎠

The matrix δa is skew-symmetric and the variations δθk are independent, such that we can rewrite this result as a system of equations x αj

∂L α ∂L α ∂L α ∂L = 0 , j , k = 1, 2, 3 . α − xk α + x j α − xk ∂xk ∂x j ∂ xk ∂ x αj

(20.3.74)

If these equations are satisfied, then the transformation (20.3.71) is a symmetry transformation and (20.3.67) yields the equation d ⎛ ∂L α d ⎡ 1 ⎛ ∂L α ∂L α ⎞ ⎤ ⎞ x δa = x − α x k ⎟ ⎥ δa jk = 0 , ∂ x j dt ⎜⎝ ∂ xkα j jk ⎟⎠ dt ⎢⎣ 2 ⎝⎜ ∂ xkα j ⎠⎦

which is equivalent to the system of equations x αj

∂L ∂L − x kα = x αj pkα − x kα p αj , j , k = 1, 2, 3 . ∂ xkα ∂ x αj

Therefore, the conserved quantity is the generalized moment of momentum K, which is a skew-symmetric tensor of rank two, of components

K jk = x jα pkα − xkα p jα , j , k = 1, 2, 3 .

(20.3.75)

Conversely, assuming the conservation of the generalized moment of momentum and taking into account Lagrange’s equations of motion, one can write d ⎛ α ∂L d ⎛ ∂L ⎞ ∂L ⎞ ∂L ∂L − x kα = x αj + x αj − xkα x ∂ x αj ⎟⎠ ∂ xkα ∂ x αj dt ⎜⎝ j ∂ xkα dt ⎜⎝ ∂ xkα ⎟⎠ d ⎛ ∂L ⎞ α ∂L α ∂L α ∂L α ∂L − x kα , j , k = 1, 2, 3 . ⎜ α ⎟ = x j α − xk α + xj α − xk dt ⎝ ∂ x j ⎠ ∂ xk ∂ x j ∂xk ∂ x αj

0=

dK jk dt

=

Therefore, the relations (20.3.74), which represent sufficient conditions for the conservation of the generalized moment of momentum K, are necessary conditions in this sense too.


323

20.3.3.4 Time Translations Group. Conservation of the Generalized Mechanical Energy The time translations are defined by the equations t ′ = t + δt , xk′α = x kα , δxkα = 0 , xk′α = xkα , δxkα = 0 , k = 1, 2, 3 , α = 1, 2,...,n ,

(20.3.76)

and form a one-parameter Abelian group (the time translations group T ). If we choose δΩ = 0 for δt = const , the condition (20.3.66) is reduced to ∂L = L = 0 , ∂t

(20.3.77)

that is the Lagrangian does not depend explicitly on time. Also, the equation of conservation takes the form d ⎛ ∂L α ⎞ x − L ⎟ = 0 . dt ⎜⎝ ∂ xkα k ⎠

It follows that the generalized mechanical energy

E =

∂L α x − L , ∂ xkα k

(20.3.78)

corresponding to the relation of definition (18.2.68), is a conserved scalar quantity (Jacobi’s first integral or a first integral of Jacobi’s type). Reciprocally, if the conservation relation (20.3.77) holds, then we can write 0=

dE d ⎛ ∂L α d ⎛ ∂L ⎞ = x − L ⎟ = ⎜ α dt dt ⎜⎝ ∂ xkα k ⎠ dt ⎝ ∂ xk

∂ L α dL ⎞ α ⎟ xk + ∂ x α xk − dt = − L . ⎠ k

Here, we have used the relation

dL ∂L α ∂L α x + x = L + dt ∂ x kα k ∂ xkα k and Lagrange’s equations of motion. Hence, the condition (20.3.77) is not only a sufficient condition, but also a necessary condition for the conservation of the mechanical energy.

20.3.3.5 The Galileo Group. Theorem of the Mass Centre Motion Let be two reference frames: one considered to be a fixed frame of reference and the other moving uniformly along a straight line relative to the first one, with the velocity v = (v1 , v2 , v3 ) . The transformation connecting the co-ordinates of a particle in the two reference frames is of the form


324

xk′ = xk + vk t , k = 1, 2, 3 , t ′ = t , δt = 0 ,

(20.3.79)

where the primed co-ordinates correspond to the fixed frame of reference. The operator associated to this transformation acts in the space of the vectors ( r ; t ) = ( x1 , x 2 , x 3 ; t ) and has the associated matrix ⎡1 ⎢0 t ( v ) = ⎢⎢ 0 ⎢ ⎢⎣ 0

0 0 v1 ⎤ 1 0 v2 ⎥ ⎥. 0 1 v3 ⎥ ⎥ 0 0 1 ⎥⎦

(20.3.80)

The set of transformations (20.3.79) forms an Abelian group with three parameters (the components v1 , v2 , v 3 of the velocity), called the Galileo group (denoted here by Γ ). The matrices (20.3.80) satisfy the relation t ( u + v ) = t ( u )t ( v ) ,

where u and v are two arbitrary velocities. Therefore, the set of matrices (20.3.80) forms a representation of Γ . The infinitesimal transformation corresponding to (20.3.79) has the form t′ = t , xk′ = xk + t δvk , xk′ = xk + δvk , k = 1, 2, 3 ,

(20.3.81)

while the variations of the co-ordinates of the system of particles are given by the relations δt = 0 , δxkα = t δvk , δxkα = δvk , k = 1, 2, 3 , α = 1, 2,...,n .

(20.3.81')

With these notations, the condition (20.3.66) becomes ∂L

∑ ⎛⎜⎝ t ∂xkα n

α =1

+

∂L ⎞ d δv = − ( δΩ ) ; ∂ xkα ⎟⎠ k dt

(20.3.82)

assuming that (20.3.69) is satisfied (i.e., the generalized momentum is conserved), we have n d ∂L ( δΩ ) = − ∑ α δvk = − Pk δvk ∂ dt α =1 xk

(20.3.82')

and the right member has to be the total derivative with respect to time of a function of co-ordinates xk .


325

The centre of mass of the discrete mechanical system has the co-ordinates ξk =

1 mα xkα , k = 1, 2, 3 , M

(20.3.83)

where M =

n

∑ mα

α =1

is the mass of the mechanical system; therefore, taking into account the definition (20.3.70), we obtain

M ξk = mα xkα = Pk , k = 1, 2, 3 .

(20.3.83')

Hence, up to an additive constant, we may choose δΩ = − M ξk δvk ,

(20.3.82'')

such that the equation of conservation (20.3.67) takes the form d ⎡ ⎛ n ∂L t α − M ξk dt ⎢⎣ ⎝⎜ α∑ =1 ∂ xk

⎞ ⎤ ⎟ δvk ⎥ = 0 ⎠ ⎦

and, by the relation (20.3.70), we may write d [ ( Pk t − M ξk ) δvk ] = 0 . dt

Because the variations δvk are independent, we get d ( P t − M ξk ) = 0 , k = 1, 2, 3 , dt k

or Pk t − M ξk = const ; the constant is determined by the conditions

( Pk t − M ξk ) t = 0 = − M ξk0 , k = 1, 2, 3 . Finally, we obtain ξk0 = ξk −

Pk t , k = 1, 2, 3 . M

(20.3.83'')

Taking into account the conservation of the generalized momentum, it follows that the centre of mass of the considered mechanical system has a uniform rectilinear motion with respect to both frames of reference. It is a characteristic of classical mechanics


326

that, in case of a Lagrangian invariant with respect to space translations, not only the generalized momentum is conserved but the equation (20.3.83'') holds too (the law of motion of the centre of mass).

20.3.3.6 Lagrangians with Certain Symmetry Properties We will focus now on implications of the reciprocal of Noether’s theorem in classical mechanics. We restrict our considerations to two-particle interactions, thus neglecting higher order interactions, and make use of the superposition principle. Therefore, we assume that the Lagrangian is a sum of Lagrangians, some of them associated with single particle motions, while the others correspond to two-particle interactions, that is

L =

n

n

n

∑ Lα ( xkα , xkα ; t ) + ∑ ∑ Lαβ ( xkα , xkβ , xkα , xkβ ; t ) ,

α =1

(20.3.84)

α =1 β =1

where x α = ( x1α , x 2α , x 3α ) are the Cartesian co-ordinates of the particle P α , and similarly for x β . On this general form of the Lagrangian we impose the conditions (20.3.69), (20.3.74) and (20.3.77), that is we want the Lagrangian be invariant with respect to space translations, space rotations and time translations. In other words, we consider that the space, in which we study the evolution of our discrete mechanical system, is homogeneous (in both space and time variables) and isotropic. In fact, these conditions represent restrictions with respect to the functional form of the Lagrangian in the variables xkα , xkα , t . Firstly, the condition (20.3.77) leads to

L =

n

n

n

∑ Lα ( xkα , xkα ) + ∑ ∑ Lαβ ( xkα , xkβ , xkα , xkβ ) .

α =1

(20.3.84')

α =1 β =1

Then, for every index k = 1, 2, 3 , the condition (20.3.69) can be regarded as a partial differential equation, the solutions of this equation being the first integrals of the system of differential equations (the Lagrange–Charpit sequence) dx k1 dx 2 dx n = k = ... = k = dλ , 1 1 1

where λ is an arbitrary parameter. Thus, we obtain n (n − 1)/ 2 first integrals of the form ϕkαβ = xkα − xkβ = const , α > β , α, β = 1, 2,...,n , of which only n − 1 are linearly independent. Consequently, if the Lagrangian is of the form (20.3.84'), then Lα must be independent of xkα , while Lαβ has to depend on xkα and xkβ only by the agency of the functions ϕkαβ . From the condition (20.3.74), it follows that the Lagrangian

( xkα

− xkβ

has

)( xkα

to

− x kβ ) ,

depend

( xkα

only

− xkβ

)( xkα

on

the

dot

− xkβ ) ,

( xkα

products

− xkβ

)( xkα

xkα xkα ,

− xkβ ) ,

xkα xkα ,

without

summation with respect to α and β , which are invariant with respect to the space rotations. Hence, this particular Lagrangian has to be of the form


L =

n

n

n

∑ Tα ( xkα xkα ) − ∑ ∑ Vαβ ( ( xkα

α =1

α =1 β =1

where we have introduced the notations

− x kβ

327

)( xkα

− x kβ ) , xkα xkα , xkβ xkβ ) ,

(20.3.84'')

Lα = Tα and Lαβ = −Vαβ .

In classical mechanics, the equations of motion are invariant with respect to the Galileo group (Galilean invariance of non-relativistic mechanics), such that, by assuming that Vαβ does not depend on the velocities xkα and xkβ , the condition (20.3.82') becomes d ∂Tα ( δΩ ) = − α δvk . ∂ xk dt

Here, Ω is a function of xkα , k = 1, 2, 3 , α = 1, 2,...,n , and the right side is a total derivative of a function of this kind, only if the functions Tα are proportional to xkα xkα , without summation with respect to α. Finally, the Lagrangian which describes the motion of a system of n particles, invariant with respect to the group of space translations, the group of space rotations, the group of time translations and also conform to the Galilean invariance, has to be of the form

L =

1 n mα xkα xkα − 2 α∑ =1

n

n

∑ ∑ Vαβ ( ( xkα

α =1 β =1

− x kβ

)( xkα

− x kβ

)) ,

(20.3.85)

where we have introduced the proportionality factors mα / 2 , α = 1, 2,...,n , to set up the dependence of the functions Tα on xkα xkα , without summation with respect to α; actually, mα is the mass of the particle P α , α = 1, 2,...,n . In case of such a Lagrangian, the generalized momentum of this particle has the components pkα =

∂L = mα xkα , k = 1, 2, 3 , ∂ xkα

(20.3.86)

while the components of the vector associated to the generalized moment of momentum of the particle are Ki =

1 ∈ K =∈ijk xkα pkα , k = 1, 2, 3 , 2 ijk jk

(20.3.87)

both without summation with respect to α.

20.3.3.7 Application of the Reciprocal of Noether’s Theorem The example in the previous subsection shows, in a very explicit manner, the close relation existing between the motion laws in classical mechanics, inferred directly from the Lagrangian, and the geometrical properties of the space-time, which impose the analytical form of the Lagrangian. Now, we will use the form (20.3.85) of the


328

Lagrangian to illustrate how to find the symmetry group corresponding to a conservation law, that is how to apply the reciprocal of Noether’s theorem. To simplify the formulation, we will consider the motion of a single particle described by the Lagrangian 1 2

L = m ( x12 + x22 + x 32 ) − V ( x1 , x 2 , x 3 ) ,

(20.3.88)

where V ( x ) is the potential energy of the field of forces acting upon the particle. The Hessian of this Lagrangian with respect to velocities and its inverse are given by the identity matrix E of rank three (abstraction of a multiplicative factor) ⎡1 0 0⎤ ⎡1 0 0⎤ ⎢ ⎥ ⎥ 1 ⎢ −1 H = m ⎢0 1 0⎥ , H = ⎢0 1 0⎥ . m ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0 0 1 ⎥⎦ ⎢⎣ 0 0 1 ⎥⎦

(20.3.88)

Under the assumption that the momentum of the particle of position vector r is conserved, we obtain three first integrals (integrals of motion) Dk ( r , r ) = mxk = const , k = 1, 2, 3 ,

(20.3.89)

and from (20.3.27') it follows δx k =

∂ Dk H jk−1δτk = δτk , ∂ x j

]

(20.3.89')

without summation with respect to k = 1, 2, 3 . Hence, the corresponding infinitesimal transformation with respect to which the Lagrangian remains invariant is of the form xk′ = xk + δτk , k = 1, 2, 3 ,

(20.3.90)

that is an infinitesimal transformation of the form (20.3.68). Consequently, the conservation of the momentum leads to the invariance of the Lagrangian with respect to the translations of the Euclidean space E 3 . The conservation of the angular momentum provides three first integrals Dk ( r , r ) = m ∈ijk xi x j , k = 1, 2, 3 .

The first integral D3 ( r , r ) leads to variations of co-ordinates of the form δx1 = − x 2 δτ 3 , δx 2 = x1 δτ 3 , δx 3 = 0

and to the infinitesimal transformation x1′ = x1 − x 2 δτ 3 , x 2′ = x 2 + x1 δτ 3 , x 3′ = x 3 ,

(20.3.91)


329

respectively. In matric notation, the transformation takes the form ⎡ x1′ ⎤ ⎡ 1 0 0 ⎤ ⎡ x ⎤ ⎡ 0 −1 0 ⎤ ⎢ ⎥ ⎢ ⎥⎢ 1 ⎥ ⎢ ⎥ ⎢ x 2′ ⎥ = ⎢ 0 1 0 ⎥ ⎢ x 2 ⎥ + ⎢ 1 0 0 ⎥ δτ 3 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎣ x 3′ ⎥⎦ ⎢⎣ 0 0 1 ⎥⎦ ⎣ x 3 ⎦ ⎢⎣ 0 0 0 ⎥⎦

(20.3.92)

3 δτ 3 , r ′ = Er + X

(20.3.92')

or

3 is the generator of the group of transformations where E is the identity matrix, and X

associated to the first integral D3 ; by the same method we obtain for the first integrals D1 ( r, r ) and D2 ( r , r ) the generators ⎡ 0 0 1⎤ ⎡0 0 0 ⎤ ⎢ ⎥ ⎢ ⎥ 1 = ⎢ 0 0 −1 ⎥ , X 2 = ⎢ 0 0 0⎥ . X ⎢ ⎥ ⎢ ⎥ ⎣⎢ −1 0 0 ⎦⎥ ⎣⎢ 0 1 0 ⎥⎦

(20.3.93)

The product of the three transformations obtained above leads to an infinitesimal transformation defined by the matrix δτ , u ( δτ1 , δτ2 , δτ 3 ) = E + X k k

(20.3.94)

i , which is a square, orthogonal non-singular matrix u ∈ SO(3, \ ) . The matrices X i = 1, 2, 3 , satisfy the commutation relations

i X j − X jX i =∈ X , i , j = 1, 2, 3 , X ijk k

(20.3.95)

and can be identified with the matrices of the generators of the SO(3, \ ) group in exponential parameterization. Consequently, the transformation defined by matrices of the form (20.3.93) form a group isomorphic to SO(3, \ ) , and the relations expressing the conservation of the angular momentum lead to the invariance of the Lagrangian with respect to the proper rotations of the Euclidean space E 3 . At the same time, the property of invariance imposes a restriction on the form of the potential energy V ( x ) in (20.3.88). This function has to depend on the variables x1 , x 2 and x 3 in the form V ( x12 + x 22 + x 32 ) , the norm of the vector r (i.e., ( r ⋅ r )1/ 2 = ( x12 + x 22 + x 32

)1/ 2 )

being invariant with respect to the proper rotations of E 3 . Indeed, from the condition (20.3.27) it follows that V ( r ) is a solution of the partial differential equations


330

∂ Dk ∂V ∂ Dk ∂V ∂ Dk ∂V + + = 0 , k = 1, 2, 3 , ∂ x1 ∂ x1 ∂ x 2 ∂ x 2 ∂ x 3 ∂ x 3

or ∈ijk x j

∂V = 0 , i = 1, 2, 3 , ∂ xk

which admits the general solution V = V (ϕ ) , where ϕ = xk xk . The method described above can be generalized to systems of n particles because for ∂L / ∂t = 0 and ∂V / ∂ xk = 0 , in the case L = T − V , the function T is a homogeneous function of second degree, respectively a positive definite quadratic form, which can be reduced to its canonical form (i.e., a sum of squares) by means of a point transformation. Hence, the Hessian of the transformed Lagrangian with respect to the generalized velocities is a diagonal matrix, and the relations of conservation of the generalized angular momentum lead to the generators of the SO(3, \ ) group. We have shown in Sect. 20.3.3.4 that whenever the Lagrangian does not explicitly depend on time ( ∂L / ∂t = 0 ) along a system trajectory (i.e., when Lagrange’s equations of motion are satisfied), the generalized mechanical energy D ( r , r ) = E =

∂L x − L ∂x j j

is conserved ( ∂E / ∂t = 0 ). In our case, the only independent variable is the time t, and we consider an infinitesimal transformation of the form t ′ = t + δt , where δt is an arbitrary infinitesimal function of class C 2 . If we choose δΩ = 0 , then the equation of conservation (20.3.67) becomes

−E

d( δt ) d ⎛ ∂L ⎞ δx + = 0, dt dt ⎜⎝ ∂x j j ⎟⎠

which is identically satisfied for δt = const , that is the considered transformation is a time translation. Note that, for time translations δx k (t ) = x k′ (t ′) − x k (t ) = 0 , k = 1, 2, 3 ,

(20.3.96)

because xk′ (t ′) and xk (t ) have to represent the same point in the space of coordinates. Consequently, we also have δxk = 0 ; it follows that the symmetry transformation corresponding to the conservation of energy is the time translation for a one-particle system t ′ = t + δt , xk′ = xk , δx k = 0 , xk′ = xk , δxk = 0 , k = 1, 2, 3 .

(20.3.96')


331

20.3.3.8 The Galileo–Newton Group Let E 4 be the Euclidean space spanned by the vectors x = ( x1 , x 2 , x 3 ; t ) ≡ ( r ; t ) . The Galileo–Newton group G is the group of endomorphisms of E 4 , of the form r ′ = Rr + vt + a , t ′ = t + τ

(20.3.97)

and it consists of a time translation, defined by τ, a translation in E 3 ⊂ E 4 , defined by a ∈ E 3 , a transformation of the form (20.3.79) from the Galileo group Γ and a rotation R ∈ SO(3, \ ) of the space E 3 . The Galileo-Newton group is a subgroup of the affine group in four dimensions or, in homogeneous co-ordinates, a subgroup of the general linear group GL ( 5, \ ) . The time interval between two events in the space E 4 , Δt = t2 − t1 , as well as the spatial distance between two simultaneous events Δr = r2 − r1 , t2 = t1 , are invariant with respect to the transformations of G. Reciprocally, the Galileo–Newton group is the most general group of linear transformations in E 4 , which preserves the intervals Δt and Δr . The elements of the Galileo-Newton group G depend on ten parameters, and are denoted by g ( τ , a, v , R ) . The induced composition law is defined by g ( τ , a ′, v ′, R ′ ) g ( τ , a, v , R ) = g ( τ ′ + τ , a ′ + R ′a + τ v ′, v ′ + R ′v , R ′R ) .

(20.3.98)

The identity element of the group is g ( 0, 0, 0, 1 ) , where 1 is the identity element in SO(3, \ ) , while the inverse of a given element is g −1 = g ( τ , a, v , R )−1 = = g ( − τ , R −1 ( a − τ v ), − R −1 v, R −1 ) . The Galileo–Newton group is isomorphic to the

group of matrices v1 ⎡ ⎢ R v2 ⎢ v 3 g → t( g ) = ⎢ ⎢ ⎢0 0 0 1 ⎢0 0 0 0 ⎣⎢

a1 ⎤ a2 ⎥ ⎥ a3 ⎥ ⎥ τ ⎥ 1 ⎥⎦⎥

(20.3.99)

of order five, which depend on ten parameters and satisfy the relation t ( g1 g 2 ) = t ( g1 ) t ( g2 ) ,

(20.3.99')

identical with the internal composition law (20.3.98). We will outline now some results from the theory of the Galileo–Newton group. Thus, it can be shown that the group G is a non-compact Lie group. The most important subgroups of the Galileo–Newton group are:


332

T = { g ( τ , 0, 0, 1 ) }

(a) (b) (c) (d)

T = { g ( 0, a, 0, 1 ) } Γ = { g ( 0, 0, v , 1 ) } SO(3, \ ) = { g ( 0, 0, 0, R ) }

– the group of time translations, – the group of space translations in E 3 , – the Galileo group, – the group of proper rotations in E 3 .

The direct product of the groups T × T forms an invariant Abelian subgroup of G, and the corresponding factor group is the semi-direct product Γ ∧ SO(3, \ ) , that leads to the decomposition

G = (T × T ) ∧ (Γ ∧ SO(3, \ )) .

(20.3.100)

Note that there are also other decompositions of G, like the following one

G = (T × Γ ) ∧ (T ∧ SO(3, \ )) ,

(20.3.100')

where T × Γ is the maximal invariant Abelian subgroup of G. The basis of the Lie algebra associated to G comprises ten elements corresponding to the subgroups T, T, Γ , and SO(3, \ ) . Consequently, the first integrals obtained from the invariance of the Lagrangian with respect to these subgroups, determine the generators of the Lie algebra, corresponding to the relation (20.3.60). Thus, we obtain: (α ) (β ) (γ ) (δ )

H Pk , k = 1, 2, 3 , ξk0 , k = 1, 2, 3 , Ki = (1/ 2) ∈ijk K jk , i = 1, 2, 3 ,

– the generator of T, – the generators of T, – the generators of Γ , – the generators of SO(3, \ ) .

Hence, the generator of an infinitesimal transformation of G has the form X=

1 K δε + Pi δai + ξi0 δvi + Hδτ = K i δθi + Pi δai + ξi0 δvi + Hδτ . 2 ij ij

(20.3.101)

The matric form of these generators follows from the representation (20.3.99). We can also obtain these generators in the form of differential operators. Thus, we consider the space of square integrable functions f ( r ; t ) , defined on E 4 . Then, we obtain the regular representation of G as the set of differential operators u acting in the space of the functions f, defined by the equation u ( τ , a, v , R ) f ( r ; t ) = f ( R −1 ( r − vt − a + vτ ), t − τ ) .

(20.3.102)

Finally, we obtain the basis of the Lie algebra associated to G, as the set of generators H=

∂ , P = ∇ , ξ0 = t ∇ , K = r × ∇ , ∂t

(20.3.103)


333

that satisfy the commutation relations

) =∈ijk ξk0 , ( Ki , Pj ) =∈ijk Pk , ( ξi0 , H ) = Pi , ( Ki , H ) = ( ξi0 , ξ j0 ) = ( ξi0 , Pj ) = ( Pi , Pj ) = ( Pi , H ) = ( H, H ) = 0 . ( Ki , K j ) =∈ijk

Kk , ( Ki , ξ0j

(20.3.103')

The subgroup structure of G can be determined by means of the Lie subalgebras which may be identified on the basis of these commutation relations. From (20.3.103') we can also obtain the two invariants of the Galileo–Newton group

P 2 = P2 , N 2 = ( ξ 0 × P ) . 2

(20.3.104)

The invariance of the Lagrangian of a mechanical system with respect to the Galileo–Newton group is the necessary and sufficient condition for the conservation of the generalized linear and angular momenta, of the generalized mechanical energy and of the uniform rectilinear motion of the centre of mass. The corresponding Lagrangian has the form (20.3.85). It is easy to show that, under the transformations of the GalileoNewton group, the mechanical energy and the momentum of a free particle are transformed according to the relations 1 E ′ = E + v ⋅ ( Rp ) + m v2 , p ′ = Rp + m v . 2

(20.3.105)

It follows that V = E′ −

1 1 2 p ′2 = E − p , 2m 2m

(20.3.106)

i.e. V is invariant with respect to the transformation of G. To prove (20.3.106) we used the invariance of the dot product with respect to SO(3, \ ) , that is ( Rp ) ⋅ ( Rp ) = p ⋅ p . From a mechanical point of view, V represents the difference between the mechanical energy and the kinetic energy of the particle, that is the energy of the particle at rest ( p = 0 ); hence, we may say that V represents a sort of “intrinsic energy” (of the nature of a potential energy) of the particle. It is worthwhile to mention that, while in the relativistic mechanics the existence of the rest energy of a particle is an obvious result, a similar energy appears here in the context of classical mechanics, from considerations on the symmetry properties of the Lagrangian in a four-dimensional Euclidean space-time.

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Chapter 21 Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems Closely connected to the canonical transformations, one can introduce the notions of integral invariant and of ergodicity. The action-angle variables, useful in the study of periodic motions, are then considered. The methods of exterior differential calculus prove their importance to mathematical modelling of the phenomena of nature, in particular to mathematical modelling of mechanics; in this order of ideas, we will present some elements of invariantive mechanics. Besides the formalisms in the spaces Λs and Γ 2s , other formalisms can be used in various cases; we mention thus the Birkhoffian formalism, applicable to a larger circle of problems than of the Lagrangian and Hamiltonian formalisms. The theory of control will allow then a profound insight into optimal trajectories.

21.1. Integral Invariants. Ergodic Theorems After some results concerning the integral invariants of order 2s , one passes to a study of the integral invariants of first order. As well, one makes some general considerations concerning ergodic theorems (Arnold, V.I., 1976).

21.1.1 Integral Invariants of Order 2s In what follows one introduces the notion of integral invariant and one gives some corresponding general results; one passes then to the case of the absolute integral invariant of order 2s . 21.1.1.1 Notion of Integral Invariant. General Considerations We return to the system of differential equations (15.1.22), written in the form x j = X j ( x1 , x 2 ,..., x 2 s ; t ) , j = 1,2,..., 2s

(21.1.1)

for n = 2s + 1 ; if the functions X j and ∂X j / ∂xk , j , k = 1, 2,..., 2s , are definite and continuous in a neighbourhood U of the point ( x10 , x 20 ,..., x 20s ; t0 ), then there exists a neighbourhood U ⊂ U in which the solutions of this system of differential equations is obtained by means of the first integrals (15.1.23). The initial conditions of Cauchy type


335


336

allow to determine the integration constants so that through any point P 0 (x10 , x 20 ,..., x 20s ) ∈ D0 does pass an integral curve (according to the theorem of existence and uniqueness) x k = x k (t ; x10 , x 20 ,..., x 20s ), k = 1, 2,..., 2s ,

(21.1.2)

and only one; vectorially, we can write (we consider 2s -dimensional vectors) r = r (t ; r0 ) .

(21.1.2')

⎡ ∂ ( x1 , x 2 ,..., x 2 s ) ⎤ J = det ⎢ ⎥ ≠ 0, 0 0 0 ⎣ ∂ ( x1 , x 2 ,..., x 2 s ) ⎦

(21.1.2'')

As well,

so that we also may calculate

x k0 = x k0 (t ; x1 , x 2 ,..., x 2 s ), k = 1,2,...,2s ,

(21.1.2''')

r0 = r0 (t , r ) ,

(21.1.2IV)

or

Fig. 21.1 Transformation of the domain

D0 into the domain D

A representative point P 0 corresponds to the state of the mechanical system for t = t0 , in the phase space Γ 2s , so that D0 represents a set of possible states of the system. If we fix r0 , then the trajectory of the representative point which passes through P 0 is given by (21.1.2'). If we fix t , while r0 travels through the domain D0 , then (21.1.2'') corresponds to a continuous and twice differentiable transformation of the domain D0 into a domain D ⊂ Γ 2 s (Fig. 21.1); thus, the domain D represents the set of the states of the mechanical system at the moment t , every state corresponding to a state at a point of the domain D0 . In a physical (mechanical) modelling one can imagine that the representative points are the particles of a fictitious

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems

337

fluid which occupies the domain D 0 at the moment t0 and the domain D at the moment t . The transformation (21.1.2) is one-to-one and bicontinuous (hence topological), so that a point of D0 passes at a point of D , while any point of D is the image of only one point of D0 ; hence, the transformation (21.1.1) preserves the individuality of the points, as well as the domain D as a material variety. Indeed, the domain D is formed only of the points which have been in D0 at the initial moment. In general, the transformation (21.1.1) conserves any material variety: curves (linear varieties), surfaces (two-dimensional varieties) etc. Let us consider an integrable function F = F ( r ; t ) of class C 1 definite on the set of possible states D of the system at the moment t . We say that the quantity I2s =

∫D F (r ; t ) δV2 s ,

(21.1.3)

where the element of volume δV2 s = δx1 δx 2 ...δx 2 s (we use the operator δ because we calculate for a fixed t ) is an integral invariant of the system (21.1.1) if (δV20s = δx10 δx 20 ...δx 20s )

∫D F (r ; t ) δV2 s

=

∫D

F ( r0 ; t ) δV20s 0

(21.1.3')

for any moment t ; in this case, I 2s is an invariant of the transformation (21.1.2). In general, the integral I 2s depends on t (directly, through the agency of r and of D ); hence, it is necessary that dI 2 s / dt = 0 . Let be a curve of possible states of the system at the moment t and the form of first degree ω1 =

2s

∑ Fi δxi ,

i =1

definite on Γ ; one obtains an integral invariant

∫Γ ω1 = ∫Γ

∫Γ ω1

if

ω1 . 0

(21.1.4)

One can make an analogous statement for a surface of simultaneous states. Let be a m -dimensional manifold Vm and let be Im =

N

k ∫V k∑=1 Fk ( r ; t )δVm , m

(21.1.5)


338

where δVmk = δx i1 δx i2 ...δx im , the indices i1 , i2 ,..., im taking all the values from 1 to 2s , while the index k (which specifies the volume element) takes all the values

from 1 to N = C 2ms (the number of volume elements); I m is an integral invariant if N

k ∫V k∑=1 Fk ( r ; t )δVm m

=

N

0k ∫ k∑=1 Fk ( r0 ; t0 )δVm Vm0

(21.1.5')

at any moment t . In general, let be a m -chain Vm , 1 ≤ m ≤ 2s , of possible states at the moment t ; N

∑ Fk (r ; t ) δxi

if one gives the form ωm =

1

k =1

∧ δx i2 ∧ ... ∧ δx im , defined on Vm ,

where the indices i1 , i2 ,..., im take all the values from 1 to 2s , the sum having C 2ms terms and where we have introduce the exterior product (see App., Subsec. 1.2.1), then we say that the quantity ∫ ωm is an integral invariant if (Vm0 is the m -chain of states Vm

at the initial moment, from which result, by trajectories, the states of Vm )

∫V

m

ωm =

∫V

0 m

ωm .

(21.1.4')

The order of the invariant is specified by the degree of the form. The integral invariant is called absolute (denoted by I m ) if Vm is an open set or relative (denoted by Im ) if Vm is a closed set (with frontier). Using Stokes’s theorems, we notice that

Im = I m +1 ;

(21.1.6)

one can state Theorem 21.1.1 A relative invariant of order m is equivalent to an absolute invariant of order m + 1 . Thus, in case of a relative invariant of first order we have (see. App. Subsec.2.3.2) 2s

∫∂V k∑=1 Fk δxk 2

=

2s

⎛ ∂F

k ⎜ ∫V j ∑ ,k = 1 ⎝ δx j 2

j 0 , is an invariant domain, which contains only one

and

clockwise.

The

circle

trajectory. The domain R1 < r < R2 is an invariant domain too. The motion of the fluid is a motion in which the fluid is rotating as a rigid solid. Hence, the domain Ω may be a circle, a circular disc or a circular annulus. 21.1.3.2 Ergodic Theorems Poincaré’s theorem states the existence of the motions in which the movable point re-enters in ω an infinity of times. Further, following problem is put: How long is the movable point in ω ? As well, the problem is put in case of discrete intervals of time: In what proportion (of time) is the movable point in ω ? The theorems connected to such problems and to analogous ones are called ergodic theorems. In connection with such problems appears the necessity of integration on a set of points; one must use Lebesgue’s measure of a set of points instead of the simpler set of volume or extension, sufficient till now, and the integrals will be Lebesgue integrals instead of Riemann ones (sufficient, usually, in classical mechanics). Let be the transformations Tt defined by the solutions of the autonomous system (21.1.36) of zero divergence; let be also an invariant domain Ω of finite measure mΩ .


356

We consider a function of position f (P ) , definite and summable on Ω . We denote by

Pt the point which is reached by the point P by means of the transformation Tt ; in other words, the movable point which starts from P0 at the moment t0 reaches Pt at the moment t . We will be interested on the mean value (with respect to time) of the function f (P ) on a portion of the trajectory (let be the trajectory which starts from the point A ), travelled through by the movable point from the moment t = a to the moment t = b ; let be μab (A) =

1 b −a

b

∫a f (At )dt .

(21.1.38)

The existence of the mean value μab ( A) for nearly all the points A ∈ Ω results from the summability of f (P ) , according to Fubini’s theorem; we exclude from the above considerations the set of null measure of the points A , for which the above mean value does not exist. We can state Theorem 21.1.9 (ergodic). The mean value μab ( A) tends to the limit ϕ (A) together with b → ∞ for almost all the points A ∈ Ω The proof of the theorem is made in two steps; one effects firstly a passing to limit by entire values and then one passed continuously to the limit. If ϕ (A) exists for a particular value of A , let be ϕ ( A0 ) , then it exists for all the points A , on the trajectory through A , having the same value in all these points. In certain hypotheses, we can go farther, stating that ϕ ( P ) is constant not only along the trajectory, but is constant on Ω too. This property of invariant domain is fundamental in statistical mechanics. It does not state for Hamilton’s equations in classical dynamics; to take place in this case too, Ω must be indecomposable (one must not have Ω = Ω1 ∪ Ω2 , where Ω1 and Ω2 are disjoint invariant domains of positive measure).

21.2 Periodic Motions. Action-Angle Variables After the presentation of periodic and quasi-periodic motions, one introduces the action-angle variables, useful for solving the respective problems. The important rôle played by the adiabatic invariance is then put in evidence (Pars, L., 1965; Santilli, R.M., 1984).

21.2.1 Periodic Motions. Quasi-Periodic Motions The motions of planets and the atomic phenomena have, in general, a character of periodicity; the study of the periodic motions has thus a great importance, both to celestial and quantum mechanics. Other natural motions are only quasi-periodic (conditionally-periodic). The respective problems play an important rôle in classical quantum mechanics.


357

21.2.1.1 Periodic and Multiply Periodic Motions Let firstly be the case of a dynamical system which has only one degree of freedom, to which corresponds the variable of position q . Let us suppose that one can make a canonical transformation (q , p ) → (w , α ) , so that the canonical system be reduced to α = const , w being a circular function of time; one obtains thus a periodic motion. If there exists a w so that q ( w + ω ) = q (w ) ,

(21.2.1)

then one obtains a phenomenon of libration (the generalized co-ordinate q oscillates between two limit positions), e.g., the oscillatory motions of the pendulum. If the mechanical system returns to the same position after a certain variation of q (let be 2π ), which is effected repeatedly, in the same interval of time, so that q (w + ω ) = q (w ) + 2 π ,

(21.2.1')

then the motion is a rotation (e.g., the motion of rotation of the pendulum). In the following, we consider separable and periodic, generalized conservative mechanical systems. Because H = 0 , the complete integral of the Hamilton-Jacobi equation (19.2.9) will be given by (19.2.15'), where as = h ; by separation of variables, the function S will be of the form (19.2.41). On the other hand, it results p j = p j (q j ; a1 , a2 ,..., as ), j = 1, 2,..., s , as = h .

(21.2.2)

Fig. 21.5 Motion of libration (a); motion of rotation (b)

We say that the motion is multiply periodic if the conjugate canonical co-ordinates q j (t ) and p j (t ) admit the same period Tj (the projection of the representative point on the plane (q j , p j ) describes a closed curve C j , corresponding to a motion of libration (Fig. 21.5a)) or if any generalized momentum p j is a periodic function of q j , with the period Q j (the projection of the representative point on the plane (q j , p j ) describes a periodic open curve C j , corresponding to a motion of rotation (Fig. 21.5b)). For instance, in case of an anisotropic linear oscillator, the components of the elastic force are − k j x j (!) ; there results the Hamiltonian


358 H =

1 2m

s

∑ p2j j =1

+

1 s k j x 2j . 2∑ j =1

(21.2.3)

The relations (21.2.2) are given by p 2j + mk j x 2j = a j , j = 1, 2, 3, a1 + a2 + a 3 = 2mh ,

(21.2.3')

the obtained closed curves C j being ellipses (motion of libration). In case of a mathematical pendulum, Hamilton’s function is given by H =

1 pθ2 − mgl cos θ , 2ml 2

(21.2.4)

the generalized momentum being (H = h ) pθ =

2ml 2 ( h + mgl cos θ ) ;

(21.2.4')

if h > mgl , then the generalized momenta pθ vary periodically and θ increases unlimited (motion of rotation). 21.2.1.2 Linear Periodic Motions In the case in which we have only one degree of freedom, the canonical equations are q =

∂H ∂H , p = − ∂p ∂q

(21.2.5)

with Hamilton’s function H =

1 f (q ) p 2 − U (q ) , 2

(21.2.5')

where U (q ) is the potential of forces; there results the reduced Hamilton–Jacobi equation 2

⎛ ∂S ⎞ f (q ) ⎜ ⎟ = 2(U + h ) , ⎝ ∂q ⎠

(21.2.6)

h being the constant of energy. Integrating this equation, we get

S =

∫

2(U + h ) dq , f (q )

(21.2.6')

so that t − t0 =

∂S = ∂h

∫

dq . 2(U + h ) f (q )

(21.2.6'')


359

The study of the motion depends on the study of the equation ϕ (q ) ≡ 2(U + h ) f (q ) = 0 .

(21.2.7)

One can show that q cannot pass beyond a value q 0 , a real root of the equation ϕ (q ) = 0 (we have at any moment q < q 0 or q > q 0 ). Often, q oscillates between two

limit values α < q < β ; in general, taking into account the preceding result, if α and β > α are two real roots of the mentioned equation, then q can vary in the interval ( α, β ) or can remain outside this interval, as – at the initial moment – q belongs or not to the respective interval. If q1 and q2 are multiple roots of order of multiplicity m and n , respectively, we can write q2 = ϕ (q ) = (q − α )m ( β − q )n ψ (q ) ,

(21.2.7')

where ψ (q ) does not have roots in the neighbourhood of α and β . A change of independent variable t , by the relation dw 1 = ψ (q ) , dt ν

(21.2.8)

where ν is a non-determinate constant, allows a more convenient analysis of the differential equation (21.2.7'). One takes the sign + before the radical, so that w does vary in the same sense as t ; as well ϕ > 0 , because it cannot change of sign in the considered interval, being a continuous and finite function of q . One obtains dq = ν (q − α )m / 2 ( β − q )n / 2 , dw

(21.2.8')

from (21.2.7') and (21.2.8). We get the integral q (w ) = α cos2

ν ν w + β sin2 w , 2 2

(21.2.9)

in the particular case m = n = 1 ; hence, q is a periodic function of w , of period 2 π / ν . The constant ν is determined by the condition that q be a periodic function of time, of period T =

2π/ν

∫0

ν dw = ψ (q )

q2

∫q

1

dq , (q − α )( β − q )ψ (q )

(21.2.9')

where q is given by (21.2.9). It results thus a motion of libration, the limits q1 and q2 being reached by q .


360

If m and n differ from unity, then one can make an analogous study; in this case, dq / dw does not change of sign, q nearing asymptotically one of the two frontiers ( q1 and q2 ), without reaching it in a finite time. 21.2.1.3 Quasi-Periodic Motions Let us consider now a generalized conservative dynamical system with s > 1 degrees of freedom; assuming that the system is with separate variables, we use Stäckel’s theory (see Sect. 19.2.2.3). Let α j and β j be two isolated consecutive zeros of multiplicity m j and n j , respectively, of the function ϕ j (q j ) , specified by (19.2.58), so that the initial values do satisfy the relations α j < q j0 < β j , j = 1, 2,..., s . In this case ϕ j (q j ) = (q j − α j )m j ( β j − q j )n j ψj (q j ), j = 1, 2,..., s ,

(21.2.10)

with ψj > 0 and without any other zero in the considered interval. Let us make the change of variable dq j = (q j − α j )m j / 2 ( β j − q j )n j / 2 du j , j = 1, 2,..., s ;

(21.2.11)

as well, let us denote (see Sect. 19.2.2.3) f jk (q j ) =

λjk (q j ) ψj (q j )

, j , k = 1, 2,..., s ,

(21.2.12)

According to the lemma in the preceding subsection, the variables q j remain in the interval ( α j , β j ) , the motion being thus a libration. We consider thus the more simple case in which m j = n j = 1 , case in which all variables have a variation of libration. We obtain q j = α j cos2

uj uj + β j sin2 , j = 1, 2,..., s , 2 2

(21.2.13)

in this case; introducing in (21.2.12), we can calculate χ jk =

∫ fjk (q j )du j .

The quantity

2 ω jk = χ jk (u j + 2 π ) − χ jk (u j ) is constant, because q j is a periodic function of u j ; we have

(21.2.12')


ω jk =

π

∫0 fjk (q j )du j

=

βj

∫α

j

λjk (q j )dq j , j , k = 1, 2,..., s . (q j − α j )( β j − q j )ψj (q j )

361

(21.2.14)

We introduce the new variables w j , j = 1, 2,..., s , by means of the relations s

∑ ω jk w j j =1

= bk , j , k = 1, 2,..., s − 1,

s

∑ ω js w j j =1

= t + bs ,

(21.2.15)

as in Sect. 19.2.2.3. Assuming that det[ ω jk ] ≠ 0 , it results q j = q j (w1 , w2 ,..., ws ) , with w j = ν j t + γ j , ν j , γ j = const, j = 1, 2,..., s . If the frequencies ν j are commensurable (are of the form ν j = n j ν , n j ∈ ` ), then the motion is periodical; otherwise, the motion is similar to a Lissajous one. In such a motion, the trajectory passes as much as possible near to any point in Λs ; it satisfies the quasi-ergodic hypothesis: the trajectory fills everywhere dense the whole domain of co-ordinates. The motions which fulfil this condition care called quasi-periodic; (conditional periodic) motions; as examples of problems which lead to such motions, we mention: the problem of two particles, the problem of two centres, the spherical pendulum etc.

21.2.2 Action-Angle Variables In what follows, we introduce the action variables and the angle variables, useful to solve the above mentioned problems; we present then some applications. 21.2.2.1 Action Variables. Angle Variables We introduce the variables (without summation) Jj =

v∫ C

p j dq j , j

(21.2.16)

of the form J j = J j (a1 , a2 ,..., as ), j = 1, 2,..., s (taking into account (21.2.2)), called action variables; obviously, these integrals are areas (see Fig. 21.5a). There result ak = ak (J ), J ≡ {J 1 , J 2 ,..., J s }, k = 1, 2,..., s , so that the complete integral of the reduced Hamilton-Jacobi equation becomes S = S (q , J ) ,

(21.2.17)

Hamilton’s function (due to the first integral of the energy, we have H = h ) will be of the form H = H (J ) = h ,

(21.2.17')

where we took into account the modality to introduce the constants ak . We notice that the function (21.2.17) plays the rôle of a function S1 generating canonical


362

transformations (see Sect. 20.2.1.6). The formulae (20.2.33) suggest the introduction of the new variables wj =

∂S , j = 1, 2,..., s , ∂J j

(21.2.18)

called angle variables. Thus, the canonical transformation is completed. The new Hamilton function is (21.2.17'), so that we can write the new canonical equations in the form w k =

∂H ∂H = νk (J ), Jk = − = 0, k = 1, 2,..., s . ∂J k ∂wk

(21.2.19)

Hence, the angle variables are linear functions of time wk = νk t + γk , k = 1, 2,..., s ,

(21.2.19')

with νk , γk = const . One can thus show that the action and the angle variables are canonically conjugate ( (J j , wk ) = δ jk ). Imposing a complete cycle of variation C j to the co-ordinate q j , the other co-ordinates remaining unchanged, and denoting by ΔC j wk the variation of the co-ordinate wk on this cycle, we can write (without summation) ΔC j wk =

v∫ C

j

∂wk dq j = ∂q j =

∂ ∂J k

v∫ C

j

v∫ C

j

∂2 S ∂ dq j = ∂q j ∂J k ∂J k

v∫ C

j

∂2 S dq j ∂q j

∂J j p j dq j = = δ jk ; ∂J k

hence, only the angle variable w j has a variation on the cycle of variation C j ( ΔC j w j = 1 ), the other variables having null variations ( ΔC j wk = 0, k ≠ j ). If the period of w j is Tj , we have, obviously, ΔC j w j = ν jTj ; hence, ν j are variation frequencies with respect to q j . It results that, in case of a libration, for a complete cycle of qk (it returns to the initial value), wk increases with a unity; the function qk is periodical with respect to wk and we can use Fourier’s formalism of representations. If the motion with respect to qk is a rotation, then wk does no more return to the initial value (for a unitary variation of wk ), being non-periodical; the function qk − wk Qk ( Qk is the increment of qk ), which allows a Fourier representation, is periodical in this case. As we have seen in the preceding subsection, if the frequencies are commensurable, then the motion is purely periodical, the system being degenerated.


363

21.2.2.2 Applications In case of the anisotropic linear oscillator, considered in Sect. 21.2.1.1, the relation (21.2.3') allows to write (without summation) Jj =

v∫ C

p j dx j =

v∫ C

aj

j

The substitution x j = Jj =

1− j

mk j 2 x dx , j = 1, 2,..., s . aj j j

(21.2.20)

a j / mk j sin ϕ leads to aj

2π

∫0

mk j

cos2 ϕdϕ =

πa j mk j

, j = 1, 2,..., s .

Taking into account (21.2.3), (21.2.3'). it results H (J 1 , J 2 ,..., J s ) =

1 s kj Jj , 2π ∑ j =1 m

so that

νj =

1 kj , j = 1, 2,..., s . 2π m

(21.2.20')

Let be now the case of a free particle of mass m , subjected to the action of a conservative force of potential (19.2.79); assuming that U ϕ (ϕ ) = U θ ( θ ) = 0 and U r (r ) = k / r , k > 0 , corresponding to a force of Newtonian attraction, we obtain the

generalized momenta (19.2.80'). We may calculate

Jϕ =

v∫ pϕ dϕ =

2m v ∫ a1 dϕ = 2 π 2ma1 .

(21.2.21)

To calculate the action variable Jθ =

v∫ pθ dθ =

2m v ∫ a2 −

a1 dθ , sin2 θ

(21.2.22)

we consider the equation sin θ0 = a1 / a2 ; if θ1 and θ2 > θ1 are the first two solutions of this equation, then it is necessary that θ1 ≤ θ ≤ θ2 to have J θ real. It results J θ = 2 2m ∫

θ2 θ1

a2 −

a1 dθ = 2 2 ma2 sin2 θ

= 2 2ma2

θ2

∫θ

1

θ2

∫θ

1

cos θ0 − cos θ 2

sin2 θ − sin2 θ0

2

dθ . sin θ

dθ sin θ


364

The substitution cos θ = cos θ0 cos u leads to cos2 θ0 sin2 u ∫0 1 − cos2 θ0 s cos2 u du π du = 2 π 2ma2 − 2 2ma2 sin2 θ0 ∫ 0 1 − cos2 θ s cos2 u 0 π π u d du ⎛ ⎞. = 2 π 2ma2 − 2ma2 sin2 θ0 ⎜ ∫ +∫ ⎟ 0 0 u u 1 cos cos 1 cos cos θ θ + − ⎝ ⎠ 0 0 π

J θ = 2 2ma2

Observing that dx ∫ a + b cos x =

a 2 − b 2 tan

2 a 2 − b2

arctan

a +b

x 2 , a 2 > b2 ,

(21.2.23)

we can write π

dx

∫0 a + b cos x

2

=

a −b 2

2

, a 2 > b2 .

(21.2.23')

In this case J θ = 2 π 2ma2 (1 − sin θ0 ) = 2 π ( a2 − a1 ) .

(21.2.22')

As well, to calculate the action variable Jr =

v∫ pr dr =

2m v∫ h +

k a2 − dr , r r2

(21.2.24)

we introduce the apsidal distances rmin and rmax (at the pericentre and the apocentre, respectively), so that J r = 2 −2mh ∫

rmax rmin

( rmax

− r ) ( r − rmin )

dr . r

(21.2.25)

Observing that r = a (1 − e cos u ), rmax − r = ae (1 + cos u ), r − rmin = ae (1 − cos u ) , where a is the semi-major axis of the ellipse, e is the eccentricity and u is the eccentric anomaly (see Sect. 9.2.1.3), we can write Jr =

−2mh ∫

2π 0

( ae sin u )2

a (1 − e cos u )

du = a −2mh ∫

2π 0

(1 + e cos u ) du

2π

du = 2 πa −2 mh 1 − e cos u π π du du −a (1 − e 2 ) −2 mh ∫ +∫ 0 1 − e cos u 0 1 + e cos u −a (1 − e 2 ) −2 mh ∫

(

0

(

)

= 2 πa −2mh 1 − 1 − e 2 ,

)


365

where we took into account (21.2.24'). By means of the relations a k rmax + rmin = 2a = − , rmax rmin = a 2 (1 − e 2 ) = − 2 , h h

(21.2.26)

we obtain

2ma2 −2mh , mk

mk , 1 − e2 = −2mh

a −2mh = so that Jr = 2 π

(

mk − −2mh

)

2ma2 .

(21.2.25')

There results, finally, H (J ϕ , J θ , J r ) = −

2 π 2 mk 2

( J ϕ + J θ + J r )2

=h.

(21.2.27)

The three frequencies are equal, in this case, and we can write ν =

∂H ∂H ∂H 4 π 2 mk 2 . = = = ∂J ϕ ∂J θ ∂J r ( J ϕ + J θ + J r )3

(21.2.28)

If h < 0 , then −2mh is real and the motion (on an ellipse) is periodical. Eliminating the sum J ϕ + J θ + J r between (21.2.27) and (21.2.28), taking into account (21.2.26) and observing that ν = 1/T , where T is the period, we get Kepler’s third law (see Sect. 9.2.1.4 too, formula (9.2.17'')) T2 m = 4 π2 . 3 k a

(21.2.29)

21.2.3 Adiabatic Invariance If the parameters on which depends a mechanical system have a sufficient slow motion, then the action variables preserve a practically constant value (a property of adiabatic invariance). In connection to this property, the quantification criteria of Sommerfeld, as well as the Burgers and the Gibbs–Hertz theorems are presented. 21.2.3.1 Sommerfeld’s Quantification Criteria In case of a holonomic and scleronomic mechanical system with only one degree of freedom, the first integral of energy ( H (q , p ) = h = const ) determines, in the phase space Γ 2 , a closed curve C , without double points, which bounds a domain the area of which is given by the integral


366

v∫ pdq

= const .

(21.2.30)

If the mechanical system has s degrees of freedom, then one uses the methods of calculation foreseen for the quasi-periodic motions (see Sect. 21.2.1.3), when the variables can be separated. In this case, beside the first integral of energy, can intervene other m , m < s , ones ( pk = ck = const, k = 1, 2,..., m ), corresponding to the apparition of cyclic co-ordinates; we remain with s − m degrees of freedom, while the phase space Γ 2( s − m ) will be of canonical co-ordinates qm + 1 , qm + 2 ,..., qs , pm + 1 , pm + 2 ,..., ps . Assuming that one can write the first integral of energy H (qm + 1 , qm + 2 ,..., qs , pm + 1 , pm + 2 ,..., ps , c1 , c2 ,..., cm ) = h = const ,

in a closed manifold V2( s − m ) , the integrals

∫V

2 ( s −m )

dqm + 1 dqm + 2 ...dqs dpm + 1 dpm + 2 ...dps ,

(21.2.31)

are integral invariants, corresponding to the phase volume (see Sect. 21.1.1.3 too). Let be a separable canonical system for which, beside the first integral of energy, there are known other s − 1 first integrals. These last first integrals allow us to express the generalized momenta p1 , p2 ,..., ps −1 as functions of ps (we solve a system of s − 1 equations); replacing in H , it results a new Hamiltonian H ′ . We obtain thus the canonical system qs =

∂H ′ ∂H ′ , , ps = − ∂ps ∂qs

(21.2.32)

which corresponds to only one degree of freedom, so that one can impose a condition of the form (21.2.30). Writing such equations for all the other s − 1 generalized momenta, we obtain – totally – s conditions of the form (21.2.30), corresponding to the s degrees of freedom of the mechanical system. If the mechanical system is an atom with s degrees of freedom, then the conditions of the form (21.2.30) which are verified by the action variables will be A. Sommerfeld’s quantification conditions. 21.2.3.2 Adiabatic Invariants The integrals of the form (21.2.30), which express quantic conditions are – in fact – adiabatic invariants. A mechanical system is not, in general, isolated. For instance, an atom (or a set of atoms) is continuously subjected to external influences: masses, electric fields, thermic fields, electromagnetic forces, radiation fields etc.; these influences are exerted, usually, as a continuous, very slow perturbation and can be represented by parameters which vary slowly in time. Thus, in Hamilton’s function H = H (q , p ; a1 , a2 ,..., am ) the adiabatic parameters ak (t ), k = 1, 2,..., m , vary slowly with the time t , as the adiabatic processes in case of phenomena of thermodynamic nature; if the qualitative aspect of the mechanical system is not altered, then the respective process is adiabatic.


367

Let be thus a mechanical system for which Hamilton’s function H = H (q , p ; λ ) depends on the parameter λ = λ (t ) ; we assume that the function λ is known, varying slowly in a time interval T . If we can use action-angle variables (for a constant λ ), then, in the interval T , the variation of J is in direct proportion to Δλ /T where Δλ is the variation of the parameter λ ; it can be thus noticed that, if T does not pass beyond a certain limit, but is sufficiently great, then the variation of J can be made as small as necessary. Introducing the action variable (21.2.16), we determine the constants ak (J ; λ ) , k = 1, 2,..., s , and then the function S (q , J ; λ ) , which generates the canonical transformation (q , p ) → (w , J ) , by the relations pj =

∂S ∂S , wj = , j = 1, 2,..., s ; ∂q j ∂J j

(21.2.33)

the transformation remains canonical if λ varies in time, Hamilton’s function being given by ∂S H = H + S = H (J ; λ ) + Sλ λ , Sλ = ∂λ

. q =q ( w ,J ;λ )

(21.2.34)

In this case, the equations of motion will be w j =

∂S ∂S ∂H ∂H = + λ λ = ν j (J ; λ ) + λ λ , ∂J j ∂J j ∂J j ∂J j ∂S ∂H J j = − = − λ λ , ∂w j ∂w j

(21.2.35)

where ν j are the frequencies of the system with λ = const . We assume that λ and λ are constant on T (as it was evaluated by Kneser for the mechanical systems with a single degree of freedom). The variation of the action J j is thus given by ΔJ j = J j

T 0

T ∂Sλ dt . = − λ ∫ 0 ∂w j

(21.2.36)

One can show that the integral which intervenes in (21.2.36) remains bounded how great could be T ; if sup | λ | is sufficiently small, then ΔJ j remains small. We say that a variable which has this property is called adiabatic invariant; the action variables have this property. Thus, P. Ehrenfest notices that the quantities which define “the stable trajectories” in an atom (the mechanical energy and the action variables of the form (21.2.30)) must be

368


invariant not only by canonical transformations but also by variations imparted to this mechanical system (supposed to be of Liouville type) by adiabatic processes. 21.2.3.3 The Gibbs–Hertz Theorem. Burger’s Theorem P. Ehrenfest, in 1916, and then T . Levi-Civita, in 1928, have considered the problem of adiabatic invariants for canonical systems which do not depend explicitly on time, but in which the time appears through the agency of a parameter a (t ) , which varies very slowly with t ( H = H (q , p ; a (t )) ). In connection to these systems, one can state Theorem 21.2.1 (Gibbs–Hertz). If the mechanical system is quasi-ergodic, existing only a single uniform first integral, then the volume V2 s −1 bounded in the phase space Γ 2 s by the manifold H = H (q , p ; a (t )) = h = const is an adiabatic invariant.

We notice that this one is, in fact, the only adiabatic invariant in the mentioned conditions. In this case, each trajectory in V2 s −1 passes as much as possible near to each point V2 s −1 . As well, we can state Theorem 21.2.2 (Burgers). In case of a quasi-periodic system of Stäckel type, with s degrees of freedom, the s cyclic integrals of the form (21.2.30) of Sommerfeld are adiabatic invariants. Between these two limit cases is situated the intermediary case in which, besides the first integral of energy, there exist still m , m < s , uniform and independent first integrals; such systems have been called by Levi-Civita primitive systems of order m . These theorems have interesting applications both in celestial mechanics and in atomic mechanics; as well, they can be used in problems concerning mechanical systems of variable mass.

21.3 Methods of Exterior Differential Calculus. Elements of Invariantive Mechanics After applying some methods of exterior differential calculus to mechanical systems, one passes – in this order of ideas – to the presentation of some elements of invariantive mechanics; some of the results thus obtained are applied to the study of the motion of a mechanical system. The invariantive mechanics has been elaborated by O. Onicescu. beginning with 1954, by a number of papers; these have been continued by other ones, as well as in some monographs. We will present in this paragraph the most important results in this direction, using the expositions made by the author of this mathematical model of mechanics (Onicescu, O., 1974).

21.3.1 Methods of Exterior Differential Calculus By introducing the fundamental differential form, one can enounce Cartan’s principle; this allows a study of the inertial motion of a particle.


369

21.3.1.1 The Fundamental Differential Form. Cartan’s Principle In Sect. 19.1.1.9 we have introduced the bilinear covariant (19.2.41') of a Pfaff form ω (the differential form (19.1.40)); if this bilinear covariant is equated to zero, then the Pfaffian is an exact differential, corresponding to the associate differential system (19.1.44). The differential system associated to the Pfaff form of Hamilton (19.1.45) is the system of canonical equations of motion of the representative point in the space Γ 2 s , to which is associated the energetical relation dH / dt = H . We mention the integral invariant Poincaré-Cartan (21.2.31), studies in Sect. 21.1.2.4 and obtained by E. Cartan by extending Poincaré’s integral invariant to contours of non-simultaneous states. We consider thus that Pfaff’s form of Hamilton is the fundamental differential form which allows the setting up of mathematical models of mechanical systems; we write this differential form in the form ( q is the generalized position vector and p is the generalized momentum vector) ω = p ⋅ dq − (T − W )dt ,

(21.3.1)

making an extension to the case in which the mechanical system is non-natural (the given forces are not conservative or quasi-conservative, as it has been considered by Cartan). We introduce the notion of exterior derivative in the sense of Elie Cartan and calculate (see App., Subsec. 1.2.2) Dωδ = dωδ − δωd = d[ p ⋅ δq − (T − W )δt ] − δ[ p ⋅ dq − (T − W )dt ]

(

= dp ⋅ δq − δp ⋅ dq − (dT − dW )δt + ( δT − δW )dt

)

(

)

dp dq = ⎡ + gradqT − Q ⋅ δq − − gradpT ⋅ δp − (dT − dW )δt ⎤ dt , ⎢⎣ dt ⎥⎦ dt

taking into account that T = T ( q, p ) and δW = Q ⋅ δq , where Q is the generalized force vector. If the generalized momentum is given by p = gradqT ,

(21.3.2)

where q is the generalized velocity vector, then Lagrange’s equations read dp ⋅ δq = (gradqT + Q ) ⋅ δq ; dt

(21.3.3)

replacing in the expression of the calculated differential, we obtain Dωδ = [2gradqT ⋅ δq − q ⋅ δp + gradpT ⋅ δp − (dT − dW ) δt ]dt = [ δT + gradqT ⋅ δq − δ( q ⋅ p ) + p ⋅ δq − (dT − dW )δt ]dt ,

where we took into account δT = gradqT ⋅ δq + gradpT ⋅ δp . According to the theorem of kinetic energy ( dT = dW ), it results


370

Dωδ = [ δT + gradqT ⋅ δq − δ( q ⋅ p ) + p ⋅ δq ]dt .

Taking into account (21.3.2), of canonical equation q = − gradpT ,

(21.3.4)

Euler’s relation p ⋅ gradpT = 2T ( T is a quadratic form in the generalized momenta) and the relation δT = gradqT ⋅ δq + gradq T ⋅ δq , we obtain Dωδ = dωδ − δωδ = 0 .

(21.3.5)

As well, starting from (21.3.5), we find again the equations of motion of the representative point. We can state Theorem 21.3.1 (E. Cartan). The canonical equations of motion of a mechanical system and the corresponding theorem of kinetic energy are a consequence of the relation (21.3.5). Thus, the relation (21.3.5) can be seen as a principle: Cartan’s principle, the fundamental differential form being given by (21.3.1). We can state that these equations are a consequence of the principle according to which the motion is definite by the property of invariance of the integral

∫C [ p ⋅ dq − (T

− W ) δt ] ,

(21.3.6)

for any δ . By δ has been denoted the elementary displacement on the curve C , while d is the displacement on trajectories (from C 0 to C ), corresponding to the equation of motion (see Fig. 21.4). 21.3.1.2 Inertial Motion of the Particle As mathematical model for a mechanical system, a particle P is characterized, after O. Onicescu, by two quadrivectors of a space with four dimensions (the space M4 = E 3 × T ): (i) The quadrivector position-time ( r , t ), r being the position vector of components x1 , x 2 , x 3 in E 3 ; the frames of reference for space and time are considered to be inertial (Newtonian). (ii) The quadrivector momentum-energy ( p ; E ) , the generalized momentum p

having the components p1 , p2 , p3 , while E = E (t ) is a function not yet specified of invariant α =

1 2 1 p = ( p12 + p22 + p32 ) . 2 2

(21.3.7)

To describe the motion of the particle P , one introduces the geometric differential of the quadrivector ( p ; E ) , using the exterior derivative of the fundamental differential form


ωδ = p ⋅ δr − E δt ;

371 (21.3.8)

we obtain thus Dωδ = dωδ − δωd = dp ⋅ δr − δp ⋅ dr − dE δt + δEdt

or Dωδ = dp ⋅ δr + [ E ′( α )pdt − dr ] ⋅ δp − dE δt ,

(21.3.8')

where we have put in evidence the derivative of the function E ( α ) with respect to α . Hence, the components of the geometric differential D ( p ; E ) are dp, E ′( α )pdt − dr and dE . We can enounce Principle 21.3.1 (principle of inertial motion of the particle). The inertial motion of a particle is definite by equating to zero the geometric differential of the quadrivector momentum-energy D( p ; E ) = 0 ,

(21.3.9)

dp dp dp dE = 0, E ′( α )p = = v, = E ′( α )p ⋅ = 0. dt dt dt dt

(21.3.9')

hence, by the equations

In an equivalent manner, we can enounce Principle 21.3.1' The inertial motion of a particle is definite by equating to zero the exterior derivative of the fundamental form ω for any δ (equation (21.3.5)). The last equation (21.3.9') is verified together with the first one, hence p = p 0 , E ( α ) = E ( α0 ), α0 =

1 2 p , 2 0

(21.3.10)

the momentum being constant during the inertial motion. Denoting by m = 1/ E ′( α ) , E ′( α ) ≠ 0 , the mass of the particle P , the second equation (21.3.9') shows that the

momentum is given by p = mv .

(21.3.10')

The fundamental form ωδ is thus

(

ωδ = m v ⋅ δr −

)

E δt , m

(21.3.11)

containing all the elements of the motion. The space M4 being uniform and isotropic, besides position and velocity, the motion of the particles differs only by mass, so that


372

the expression v ⋅ δr − (E / m ) δt is the same for any particle. We can thus postulate that the ratio E / m is a universal constant; this constant being of the nature of a square of velocity, from a dimensional point of view, we denote E / m = εω2 , ε = ±1 , where ω is a velocity, so that

E = εm ω2 .

(21.3.12)

According to the principle (21.3.9'') and taking into account (21.2.8') and (21.3.10), we can write v ⋅ δp = E ′( α )p ⋅ δp = E ′( α )δα = δE ( α ) ; by means of the relations (21.3.10'), (21.3.12) too, we may write p ⋅ δp = εω2 m ⋅ δm or δ( p2 − εω2 m 2 ) = 0 . Denoting by m0 the mass of rest (the mass which corresponds to the velocity v 0 = 0 , hence to the generalized momentum p0 = 0 ), we obtain

p2 − εω2 m 2 = − εω2 m02 ;

(21.3.13)

hence, it results m =

m 02 +

ε 2 p . ω2

(21.3.14)

Taking into account (21.3.10'), the relation (21.2.13) leads to m =

m0

.

v2 1−ε 2 ω

(21.3.15)

Choosing ε = 1 , we get m =

m0 v2 1− 2 ω

,

(21.3.15')

ω being a superior limit velocity (to have a positive quantity under the radical). Corresponding to the experimental data, we choose ω = c , where c is the velocity of propagation of light in vacuum; thus, we find again the formula m =

m0 v2 1− 2 c

,

(21.3.15'')

obtained by Einstein in the frame of the relativistic model of mechanics for the dependence of mass on velocity. The relation (21.2.14) becomes m =

m 20 +

1 2 p , c2

(21.3.14')


373

and the relation (21.3.12) leads to E = E ( α ) = mc 2 = c 2 m 20 +

1 2 2α p = c 2 m 20 + 2 . 2 c c

(21.3.12')

We can also calculate E (α ) =

1 2α m 20 + 2 c

=

1 1 m 0 + 2 p2 c

≠ 0,

2

verifying thus the condition previously imposed. The choice ε = 1 and ω = c puts in evidence the structural characteristic of matter in motion, hence a relation between matter, space and time; the inertia of matter is thus closely connected to the structure of space and time. We notice that m ≠ 0 if m0 ≠ 0 , the velocity v being constant during the motion, which is rectilinear and uniform. The energy of the particle is, as well, constant, being given by (we use the relation (21.3.15')) E = mc 2 = m 0c 2 +

1 m v2 + 2 0

η v2 1− 2 c

, 0 0, x α′

∂Λ = Λ, ∂x α′

(21.4.2)

the last relation corresponding to Euler’s theorem; in what concerns the smoothness, we assume the existence and the continuity of any derivative which intervenes. We have the relation of invariance Λ (x , x ′) = Λ(x , x ′) by a change of co-ordinates (x , x ′) → (x , x ′) . The Lagrangian action on the curve C from u = u1 to u = u2 is given by

AΛ (C ) =

u2

∫u

Λ(x , x ′)du ,

1

(21.4.3)

so that AΛ (C ) is a functional of the curve C , independent of the parameterization (due to the homogeneity). The elementary Lagrangian action is dAΛ (x , x ′)du = Λ( x , dx ) ; thus, the space of the events becomes a Finsler space. If Λ( x , dx ) is the square root of a homogeneous quadratic form in differentials, then the space (q , t ) is a Riemannian space. Choosing u = t , we can define Lagrange’s function L (q ; t ; q ) by

L = (q1 , q2 ,..., qs ; t ; q1 , q2 ,..., qs ) = Λ(q1 , q2 ,..., qs ; t ; q1 , q2 ,..., qs ,1) ,

(21.4.4)

the elementary Lagrangian action being dAL = L (q ; t ; q )dt . Let us pass, by a synchronous variation δ , from the curve C to a curve C , so that


392 δAΛ =

AΛ (C ) − AΛ (C ) =

u2

∫u

1

⎛ ∂Λ δx + ∂Λ δx ′ ⎞ du . α α ⎟ ⎜ ∂x α′ ⎝ ∂x α ⎠

Assuming that δx α |u = u1 = δx α |u = u2 = 0 and observing that the variations δx α , α = 1, 2,..., s + 1 , are independent, it results that the variational equation δAΛ = δ ∫

u2 u1

Λ( x , x ′ ) du = 0 ,

(21.4.5)

is equivalent to Euler–Lagrange equations d ⎛ ∂Λ ⎞ ∂Λ = 0, α = 1, 2,..., s + 1 . ⎜ ⎟− dt ⎝ ∂x α′ ⎠ ∂x α

(21.4.6)

Obviously, the principle (21.4.5) is equivalent to Hamilton’s principle (20.1.38'), while Euler–Lagrange equations (21.4.6) are equivalent to Lagrange’s equations (18.2.38); it is true that the system of equations (21.4.6) has an equation more than the system of equations (18.2.38'), but between the first mentioned equations takes place the linear relation d ⎛ ∂Λ ⎞ ∂Λ ⎤ d ⎛ ∂Λ ⎞ ∂Λ ∂Λ dΛ d Λ − = − x α′′ − x α′ = − = 0, x α′ ⎡ x′ ⎢⎣ du ⎝⎜ ∂x α′ ⎠⎟ ∂x α ⎦⎥ du ⎝⎜ α ∂x α′ ⎠⎟ ∂x α′ ∂x α du du

which justifies the affirmation thus made. Taking into account that the equations (18.2.38'), for which the values q 0 and q0 at the moment t0 are known, lead to a unique solution, we can say, corresponding to the affirmation made above, that through a point P of the space (q , t ) passes only one trajectory of direction specified by dx1 , dx 2 ,..., dxs + 1 . We call Λ -dynamic the theory based on the variational equation (21.4.5) and on the extremal equations (21.4.6); as well, the α -dynamics will be the theory based on Hamilton’s principle (20.1.38') and on the extremal equations (18.2.38). These dynamics are equivalent, the L -dynamics being a form of the Λ -dynamics, in which the time t is at the same time a co-ordinate in the space (q , t ) and a parameter on a curve in this space; as a matter of fact, both dynamics are Lagrangian dynamics. Instead to impose a Lagrangian Λ( x , x ′) , we can impose an equation of energetic nature Ω (x , y ) = 0 ,

(21.4.7)

which links the co-ordinates x α of a point in the space (q , t ) to the components yα of an associate (s + 1) -dimensional vector; this equation can be seen – from a geometric point of view – as associating to each point of (q , t ) a s -dimensional hypersurface in a (s + 1) -dimensional space, tangent to (q , t ) , yα being co-ordinates in this tangent space. It is convenient – sometimes – to solve the equation (21.4.7) with respect to ys + 1 , so that


393

ys + 1 + ω (x1 , x 2 ,..., xs +1 , y1 , y2 ,..., ys ) = 0 .

(21.4.7')

y j = p j , j = 1, 2,..., s , ys +1 = − H

(21.4.8)

If we define by

the generalized momenta and Hamilton’s function, respectively, we find a relation of the form H = ω (q1 , q2 ,..., qs ; t ; p1 , p2 ,..., ps ) ,

(21.4.7'')

H = H (q ; t ; p ) ,

(21.4.7''')

hence of the form

between the 2s + 2 quantities q j , t , p j , H . Obviously, one obtains thus the canonical form of Hamilton’s principle, which reads δA = δ ∫ yα dx α = 0, Ω (x , y ) = 0 ,

(21.4.9)

in the co-ordinates x , y , the integration being effected along a curve C with fixed ends; as well, the extremal curves will be given by the canonical equations ∂Ω dyα ∂Ω dx α . = = − , ∂yα dw ∂x α dw

(21.4.10)

where dw is an infinitesimal multiplier of Lagrange. A curve x α = x α (w ) in (q , t ) and the associate vector field yα = yα (w ) will form a trajectory.

21.4.1.2 Description of the Motion of a Mechanical System in the Momentum-Energy Space ( p , H ) Being situated in the (s + 1) -dimensional space ( p , H ) , which is called the momentum-energy space, we use the co-ordinates y α , definite by the relations (21.4.8); the energetic equation (21.4.7) leads us to the energy (21.4.7''). For a curve C of equations yα = yα (u ), α = 1, 2,..., s + 1 , in ( p , H ) , we define a new type of action by the integral

A =

u2

∫u

x α dy α

1

(21.4.11)

where x α (u ) are arbitrary functions, excepting (21.4.7); we can write

A = ∫ ( q j dp j − tdH ) ,

(21.4.11')


394

too. By variation of the curve C we obtain δA = x α δyα

u2 u1

+

u2

∫u

( δx α dyα − δyα dx α ) .

1

In case of varied curves with fixed ends (δyα |u = u1 = δyα |u = u2 = 0) , we get the variational equation

δA = δ ∫

u2 u1

x α dy α = 0 ,

(21.4.12)

which, together with the condition (21.4.7), leads to the same canonical equations (21.4.10). It is obvious that the dynamics in the space ( p , H ) , based on the equation (21.4.7) and on the variational equation (21.4.12), is equivalent to the dynamics in the space (q , t ) , based on the same equation (21.4.7) and on Hamilton’s principle (21.4.9) for varied curves with fixed ends. The two actions which intervene are linked by the relation

A + A = ∫ yα dx α + ∫ x α dyα = x α yα − x 0α y α0 ,

(21.4.13)

where x α0 , yα0 are referring to the initial moment t = t0 . In the space (q , t ) , x α are the co-ordinates of the representative point, while y α are the components of an associate field of vectors; these rôles are inverted in the space ( p , H ) . Thus, between the two formalisms there exists a formal duality. For instance, we could use the technics in the preceding subsection to define a homogeneous Lagrangian in ( p , H ) , as a function of the arguments yα and yα′ = dy α / du , α = 1, 2,..., s + 1 , passing then to a classical Lagrangian, function of pα , H and dpα / dH . In general, the space ( p , H ) has a secondary importance. It is useful if, e.g., the particles of a discrete mechanical system, at the initial moment in free motion, begin to interact, returning then to a free motion. When the particles are moving freely (before and after interaction), the representative point P maintains itself a fixed position in ( p , H ) , the interaction having as consequence the motion of this point from a fixed position to another fixed one.

21.4.2 Formalism in Spaces with 2s + 1 or with 2s + 2 Dimensions Starting from the results in the preceding subsection, we consider the (2s + 1) -dimensional space (q , t , p ) and the (2s + 2) -dimensional space (q , t , p , H ) ; by this occasion we put in evidence also the utility of such formalisms.

21.4.2.1 Description of the Motion of a Mechanical System in the Space of States (q , t , p ) The most used representative space is perhaps, the phase space (q , p ) . The totality of the trajectories, in this space, appears as a congruence of curves, so that, in case of given conservative forces, through each point passes only one curve; but the problem becomes


395

more intricate in case of given non-conservative forces, because through each point passes a simple infinity of trajectories. In the space (q , t , p ) , which we call the space of states, the time t is considered the same as the generalized co-ordinates and the generalized momenta; Hamilton’s function H (q , t , p ) is a function of position in this space. If the given forces are non-conservative, then the image of the trajectories is more simple that in the space (q , p ) , because through each point passes one curve of this congruence. We also notice that, from a mathematical point of view, this (2s + 1) -dimensional space is the only representative space which has always an odd dimension. The canonical equations of motion are (19.1.14); for an equal treatment of all the co-ordinates, these equations – written in the form (19.1.19) – put in evidence the natural congruence of the trajectories in the space (q , t , p ) , through each point passing only one curve. On the other hand, these equations imply the energetic condition (19.1.25). We define the circulation on a curve C by κ(C ) =

v∫ C ( p j δq j

− H δt ) ;

(21.4.14)

giving an infinitesimal displacement (not necessarily along the natural congruence) and integrating by parts, we can write dκ(C ) = =

⎡

⎛

v∫ C ⎣⎢ δq j ⎝⎜ dp j

+

v∫ C ( dp j δq j

− dq j δp j − dH δt + dt δH

(

)

)

∂H ∂H ∂H ⎞ ⎛ ⎞ ⎤ dt ⎟ + δp j ⎜ − dq j + dt ⎟ + δt − dH + dt ⎥ . ∂q j ∂ ∂ p t j ⎠ ⎝ ⎠ ⎦

If the displacement takes place along the natural congruence, then the equations (19.1.14), (19.1.25) lead to dκ (C ) = 0 .

(21.4.15)

On the other hand, if the relation (21.4.15) takes place for an arbitrary curve C and for a displacement along an arbitrary congruence, then this congruence must satisfy the equations (19.1.19), (19.1.25), being thus a natural congruence. As a matter of fact, the condition (21.4.15), which must be verified by the circulation, is equivalent to the canonical equations (19.1.14). Let be the transformation of co-ordinates (q , p , t ) → (x ) , so that x j = q j , xs + j = p j , j = 1, 2,..., s ,

x 2 s +1 = t ;

(21.4.16)

generically, we denote such a co-ordinate in the form x A , where the capital letters take values form 1 to 2s + 1 in the frame of a convention of summation. In this case, we can write

p j δq j − H (q ; t ; p )δt = X A δx A , where

(21.4.17)


396

X j = p j , Xs + j = 0, j = 1, 2,..., s ,

X 2 s + 1 = − H (q ; t ; p ) ;

(21.4.16')

it results κ(C ) =

v∫ C XA δx A ,

(21.4.17')

wherefrom dκ(C ) =

v∫ C ( dXA δx A

− dx A δX A ) .

Denoting ∂X A / ∂x B = XA,B , we can also write dκ(C ) =

v∫ C XA,B ( dx B δx A

− dx A δx B ) =

v∫ C ( XA,B

− X B ,A ) dx B δx A ;

if dx A is along the natural congruence, then the relation (21.4.15) takes place for an arbitrary curve C . Hence, the trajectories must satisfy the equations

( XA,B

− XB ,A ) dx B = 0, A = 1, 2,..., 2s + 1 .

(21.4.18)

Because X A,B − X B ,A are the components of an antisymmetric tensor, while dx B are the components of a vector, we can affirm that the equations (21.4.18) are vector equations; these equations are satisfied for the co-ordinates (21.4.16), (21.4.16'), so that they are satisfied in any system of co-ordinates. One can build up thus a new mathematical formalism of mechanics, starting from the Pfaff form X A δx A ; this form is determined, for the same trajectories, excepting an exact differential.

21.4.2.2 Description of the Motion of a Mechanical System in the Space of States and Energy (q , t , p , H ) The (2s + 2) -dimensional representative space (q , t , p , H ) , which we call the space of states and energy, allows the most general formalism for the dynamics of a mechanical system; the time and the energy H , in this space, are treated as the generalized co-ordinates and the generalized momenta, obtaining thus formally a complete symmetry. The 2s + 2 co-ordinates can be divided in two groups: the group (q , t ) and the group ( p , H ) , corresponding to the space of events and to the momentum-energy space, respectively, the rôle of which can be – in general – inverted. It is convenient to use an equation of energy which imply all the 2s + 2 co-ordinates of the space (q , t , p , H ) to can preserve the symmetry; this equation defines a (2s + 1) -dimensional hypersurface in the space of states and energy, while the representative point P must be on this surface. It is often better to use a function of energy instead of an equation of energy, to can imply the whole space (q , t , p , H ) , not only the mentioned hypersurface. The space of states and energy corresponds very well to Hamilton’s optical method; all the theories developed in this representative space can


397

be transferred, in an isoenergetical dynamics, in the phase space, by a convenient reduction of dimensions. Let be the co-ordinates x α and yα , α = 1, 2,..., s + 1 , called conjugate, which have been previously introduced. To a energy function Ω (x , y ) which we consider, there corresponds only one energy surface Ω (x , y ) = 0 ; to a given energy surface there corresponds an infinity of energy functions. By means of the variational formalism (21.4.9) or of the variational formalism (21.4.12), we may obtain canonical equations of the form (21.4.10). The solutions of the system (21.4.10) fill up the space (q , t , p , H ) by a natural congruence of trajectories, one for each point. Thus, the totality of the dynamical trajectories, including those on the energy surface, presents a geometric image much more simple than that of the space (q , t ) , where there exists only one trajectory which passes through a point in a given direction. One takes again – in the representative space (q , t , p , H ) – the whole theory of the space (q , t ) , but where each quantity and each mathematical or mechanical fact has a richer significance.

21.4.3 Notions on the Inverse Problem of Mechanics and the Birkhoffian Formalism In what follows, we present firstly the inverse problem of the Newtonian mechanics; is a larger sense, in connection to this problem, we consider some notions concerning the Birkhoffian formalism too.

21.4.3.1 Inverse Problem of Newtonian Mechanics In the formulation of the direct problem of Newtonian mechanics, starting from the motion of a given mechanical system, one obtains a function of Lagrange, which leads to the equations of motion in one of the representative spaces considered in the previous sections; these equations are the extremal equations of an action which can be definite corresponding to the space in which we are situated and to the used formalism. Starting from these considerations, we can pass to the formulation of the inverse problem of Newtonian mechanics. Thus, giving the equations of motion, the problem of existence (including the conditions in which such a thing can take place) of one or several functions of Lagrange is put, so that the associate Euler–Lagrange equations do coincide with the given equations of motion or be equivalent to them. Obviously, in this case, the problem of the effective determination of these Lagrangians is put. Once, it has been considered that only the natural systems (especially the conservative system) admit a variational principle, which leads to the associate Euler– Lagrange equations, equivalent to the Newtonian equations. As well, it has been shown that, being given a Lagrangian L , any other Lagrangian L ′ = L + dϕ / dt , where ϕ = ϕ (q ; t ) (obtained by a gauge transformation), leads to the same dynamical system; in other words, the Euler–Lagrange systems of equations associated to the two Lagrangians coincide (see Sect. 18.2.3.2 too). Reciprocally, if two Lagrange’s functions L and L ′ lead to the same Euler–Lagrange equations, hence if they describe the same dynamical system, then L ′ is of the form mentioned above. But there exist also dynamical systems associated to two Lagrangians which have not the mentioned property; e.g., to the system of equations


398

q1 + aq1 + bq1 = 0, q2 − aq2 + bq2 = 0, a , b = const ,

(21.4.19)

one can associate the Lagrangians a 2

L = q1q2 + ( q1q2 − q2q1 ) − bq1q2 , 1 2

1 2

L ′ = eat ( q12 − bq 12 ) + e− at ( q 22 − bq 22 ) ,

(21.4.19')

which cannot be obtained one from the other by a gauge transformation. We mention also that there are known non-conservative systems which admit a variational principle and have Lagrangians; among them there is also the system considered above. But there are also dynamical (non-conservative) systems for which one cannot write Euler– Lagrange equations, corresponding to a function of Lagrange L = L (q , q ; t ) . We are thus led to a classification of the non-conservative systems after the property of admitting or not Lagrangians of the above mentioned form. We can consider a class of dynamical systems (to which belong all the conservative systems) formed by those systems which have Lagrangians and for which the Euler–Lagrange equations are just the equations of motion of the system. In case of another class of dynamical systems one obtains the previous situation after multiplying by an integrant factor. Hence, the inverse problem is reduced – finally – to the determination of an integrant factor. Let be thus the equations of motion mi qi − Fj (q , q ; t ) = 0, (!), i = 1, 2,..., s ,

(21.4.20)

in Newtonian formulation; it is thus put the problem to determine a non-singular matrix [ gij ] , for which gij = gij (q , q ; t ) , so that s

∑ gij ( m j qj j =1

− Fj

)=

d ⎛ ∂L ⎞ ∂L − , i = 1, 2,..., s , dt ⎜⎝ ∂qi ⎟⎠ ∂qi

(21.4.20') ,

for a certain function L = L (q , q ; t ) , as well as the problem of the conditions in which one can find such a matrix. Starting from the conditions given by H. von Helmholtz in 1887 (for the particular case gij = Fj = 0 and without showing their sufficiency), we write the conditions in which one can solve the problem put above in the form given by R.M. Santilli in 1984, that is Aij = Aji ,

∂Aij ∂Aik = , ∂qk ∂q j

∂Bi ∂B j ∂ ⎞ ⎛ ∂Bi ∂B j 1⎛ ∂ − = ⎜ + qk − ∂q j ∂qi ∂qk ⎠⎟ ⎝⎜ ∂q j ∂qi 2 ⎝ ∂t ∂Bi ∂B j ∂ ⎞ ⎛ ∂ A , + = 2⎜ + qk ∂q j ∂qi ∂qk ⎟⎠ ij ⎝ ∂t

⎞ ⎟, ⎠

(21.4.21)


399

with summation with respect to k and where we have denoted Aij = m j gij , without summation, and Bi = − gij Fj , with summation with respect to j . In 1891, G. Darboux showed that any dynamical system with one degree of freedom admits a variational principle in certain conditions of continuity and regularity, which are – in general – satisfied; other properties have been put in evidence by D.G. Currie and E.J. Saletan, in 1966, by J.A. Kobussen, in 1979, and by W.I. Sarlet, in 1982. In 1941, I. Douglas dealt with the case of two degrees of freedom, the problem being taken again, in 1979–1983 by L.Y. Bahar, H.G.Kwatny and W.I. Sarlet; as principal mathematical tool, one uses variational methods of self-adjunction. Caviglia showed, in 1896, that – by introducing additional variables – some of Helmholtz’s conditions become equations of Euler–Lagrange type. Not all dynamical systems with two degrees of freedom admit an integrant factor gij , det[ gij ] ≠ 0 , hence a variational principle; as simple example, we mention the system x1 − x 2 = 0, x2 − x 2 = 0 ,

(21.4.22)

considered by E.T. Whittaker in 1904, as well as the system x1 + x 2 = 0, x2 + x 2 = 0 ,

(21.4.22')

considered by S. Hokjman and L.F. Urrutia in 1981. In such a situation, one passed from a Lagrangian formulation to a Hamiltonian one, by Legendre’s formulation. There have been searched the conditions in which one can obtain an integrant factor so that the equation of motion, multiplied by this one, do lead to equations of Hamilton type; these conditions (necessary and sufficient) are just Helmholtz’s conditions.

21.4.3.2 Notions on the Birkhoffian Formalism In the case in which one cannot determine (one does not know or there does not exist) a function of Lagrange or a function of Hamilton such that the dynamical system become a Lagrangian system or a Hamiltonian system, respectively, one searches a generalization of the latter ones; thus, the equations of motion in generalized Hamiltonian form (equivalent to the equations of motion in Newtonian form) must be obtained by a variational principle too. This represents the Birkhoffian generalization of the Hamiltonian mechanics (given by G.D. Birkhoff in 1927). A system of differential equations of first order constitutes Birkhoff’s equations of a dynamical system if there exist a function B = B (a ; t ) , a ≡ {a1 , a2 ,..., as } , called Birkhoff ’s function (Birkhoffian) of the mechanical system, and an exact Symplectic 2 -form Ω = Ωμν da μ ∧ daν (we use the exterior product and the convention of summation with respect to Greek indices), so that Ω = dθ , with θ = Rμ da μ ( Rμ (a ; t ) are the components of Birkhoff’s vector); the system can be written in the form ⎛ ∂Rν ∂Rμ ⎜ ∂a − ∂a ν ⎝ μ

∂Rμ ⎞ ⎞ ⎛ ∂B ⎟ aν − ⎜ ∂a + ∂t ⎟ = 0 . ⎠ ⎝ μ ⎠

(21.4.23)


400 The tensor

Ωμν (a , t ) =

∂Rν ∂Rμ − , μ, ν = 1, 2,..., s , ∂a μ ∂aν

(21.4.24)

is called Birkhoff’s tensor; Birkhoff’s system reads ∂Rμ ⎞ ⎛ ∂B Ωμν aν − ⎜ − =0 a ∂ ∂t ⎟⎠ μ ⎝

(21.4.23')

too, being – in fact – a generalized Hamiltonian system (in particular, Hamilton’s equations can be written in this form). The system (21.4.23) is called autonomous if the functions Rν and B do not depend explicitly on time ( R ν = 0, ν = 1, 2,..., s , B = 0 ); such a system reads Ωμν (a )aν −

∂B (a ) = 0, μ = 1, 2,..., s . ∂a μ

(21.4.25)

The system (21.4.23) is called semi-autonomous if only the conditions R ν = 0 , ν = 1, 2,...,s , take place; in this case Ωμν (a )aν −

∂B (a ; t ) = 0, μ = 1,2,..., s . ∂a μ

(21.4.25')

In general, the system is non-autonomous. A Birkhoffian system is called regular if det[ Ωμν ] ≠ 0 ; it is singular if det[ Ωμν ] = 0 . Let be the equations of motion (21.4.20) in Newtonian formulation; these equations can be written as a system of first order in the form dqi − qi dt = 0, mi dqi − Fi dt = 0, (!), i = 1, 2,..., s .

(21.4.26)

Let us consider the application qi → pi = mi qi , without summation, by which the system becomes

dq i −

1 p dt = 0, (!), dpi − Fi dt = 0, , i = 1,2,..., s ; mi i

(21.4.26')

we can write aμ − Aμ (a ; t ) = 0, μ = 1, 2,..., s ,

(21.4.27)

too, with the notations ⎡⎡ p ⎤⎤ ⎡ [q ] ⎤ [a ] = ⎢ ⎥ , [ A ] = ⎢ ⎣ m ⎦ ⎥ , ⎢ [F ] ⎥ ⎢⎣ [ p ] ⎥⎦ ⎣ ⎦

(21.4.27')


401

where

{

}

p1 p2 p ps ≡ , ,..., , m m1 m2 ms p = { p1 , p 2 ,..., ps }, F = {F1 , F2 ,..., Fs }.

q ≡ {q1 , q2 ,..., qs },

Eliminating the derivatives aμ between the equations (21.4.23) and the equations (21.4.27), we get ∂Rμ ⎛ ∂Rν ⎜ ∂a − ∂a ν ⎝ μ

∂Rμ ∂B ⎞ ⎟ Aν = ∂a + ∂t , μ = 1, 2,..., s , μ ⎠

(21.4.28)

hence a system of 2s partial differential equations with s + 1 unknown functions. By integrating this system of equations, one obtains the potential Rμ and Birkhoff’s function B , the functions Aμ (considered to be differentiable) being arbitrary; as in case of a non-autonomous solution, as well as is case of a semi-autonomous (eventually autonomous) one, by applying the Cauchy–Kovalevsky theorem of existence and uniqueness, the obtainment of those solutions is ensured. We can thus state Theorem 21.4.1 (universal theorem of Birkhoffian representation). Any Newtonian system admits locally a Birkhoffian representation. Let be a dynamical system which is neither a Lagrangian, hence nor a Hamiltonian one; one can put the problem (the inverse problem of Newtonian mechanics in Birkhoffian formulation) to find a Birkhoff’s function B , a Symplectic structure associated to the system and Legendre’s transformations, so that the equations of motion be obtained in the form of Birkhoff’s equations. The practical method to determine the Symplectic structure and the Birkhoffian consists in searching an integrant factor and in imposing some conditions of self-adjunction; one can determine a Pfaffian from which, according to a variational principle, one obtains Birkhoff’s equations. This Pfaffian generates a Lagrangian of second order, that is of the form

L (q , q, q; t ) = V (q , q ; t )q + W (q , q ; t ) ,

(21.4.29)

which leads to the associate equations of Euler–Lagrange type

d2 ⎛ ∂L ⎞ d ⎛ ∂L ⎞ ∂L = 0, j = 1, 2,..., s ; ⎜ ⎟+ ⎜ ⎟− dt 2 ⎝ ∂qj ⎠ dt ⎝ ∂q j ⎠ ∂q j

(21.4.29')

these equations are linear combinations between Newton’s equations and their derivatives, assuming as solutions the solutions of the given Newtonian system. We can thus state that to any Newtonian dynamical systems one can associate a functional of action type, the condition of extremum of which defines the trajectories of the system.


402

21.5 Control Systems The study of dynamical systems can be effected in closed connection to the study of control systems; the methodology linked to these latter systems and – especially – the aspects of optimization of the trajectories can give precious information for the mechanical systems. All these reasons make necessary and useful the presentation of some elements concerning the control systems and the optimal trajectories.

21.5.1 Control Systems After some preliminary considerations concerning the control systems, one presents Bolza’s problem, as well as some particular cases; we mention that various forms of the conditions of minimum are put in evidence.

21.5.1.1 Preliminary Considerations A control system is a mathematical object represented by a system of equations of state in the form x j′ = fi ( x1 , x 2 ,..., x n , u1 , u2 ,..., um ; t ), i = 1, 2,..., n ,

(21.5.1)

where x i (t ), i = 1, 2,..., n , are variables of state (eventually, generalized co-ordinates in the space of configurations or in the phase space), while u j (t ), j = 1, 2,..., m are control parameters; t ∈ [t0 , t1 ] for all these quantities. The fundamental problem consists in the determination of the control parameters so that a certain functional (called performance index) I = G ( x (t1 ), u (t1 ); t1 ) +

t1

∫t

f0 ( x (t ), u (t ); t ) dt ,

(21.5.2)

0

where u (t ) ≡ {u1 (t ), u2 (t ),..., um (t )} is the control function, while x (t ) ≡ {x1 (t ), x 2 (t ),..., x m (t )} is a function of state, does realize a maximum or a minimum. For instance, if a control function so that the distance between two points be travelled through in a minimal time is searched, then we must have G ( x (t1 ), u (t1 ); t1 ) = 0 , f0 ( x (t ), u (t ); t ) = 1 . We must notice that the state variable cannot be completely arbitrary (e. g., the velocity cannot pass beyond certain limits); to simplify the problem, we assume that the domain of definition of the vector x is \n . In that concerns the control vector, there can appear – as well – restrictions (e. g., if u1 , u2 , u3 are direction cosines we have u 12 + u 22 + u 23 = 1 ), superior or inferior limits, discontinuities etc. If U is the domain of definition, independent on t , of the vector u , then we have u ∈ U ⊆ \m . We say that the functional u is an admissible control function if: (i) the function U is definite and piecewise continuous on the interval [t0 , t1 ] ; (ii) u ∈ U ;


403

(iii) for any point of discontinuity we have

u ( τ ) = u ( τ − 0) =

lim u (t ) ;

t →τ , t 0

(21.5.15)

be fulfilled for any unit vector α of components α1 , α2 ,..., αm . This condition does not exclude the possibility of existence of a relative minimum. To have an absolute minimum (with respect to the whole set U ) it is necessary that Weierstrass’s condition

E ≥ 0, E = L ( x , u , t , μ, λ , x ) − L ( x , u , t , μ, λ , x ) −

n

∑ ( x

i =1

− xi )

∂L , ∂xi

(21.5.16)

where we have introduced Weierstrass’s function for any set ( x , u , t , x ) ≠ ( x , u , t , x ) , which satisfies the equations (21.5.12'), ∀t ∈ [t0 , t1 ] , be verified. From (21.5.16) one obtains Clebsch’s (or the Legendre–Clebsch) condition, according to which the inequality (21.5.15) must take place for α1 , α2 ,..., αm , which verify the equations m

∂h j

∑ ∂uk αk

k =1

= 0, j = 1, 2,..., p .

(21.5.17)

The performance index of the optimal control function is realized by passing the system from a point of the manifold S 0 to a point of the manifold S1 ; one determines the moments t0 and t1 , so that the equations (21.5.3'') which define a ( n + 1 − r0 )-dimensional or a ( n + 1 − r1 )-dimensional manifold, respectively, be verified. Taking into account the conditions at the ends, one obtains the transversality conditions

Other Considerations on Analytical Methods in Dynamics of Discrete Mechanical Systems n ⎡ ⎛ ⎢ dG + ⎜ f0 − ∑ λi fi ⎣ ⎝ k =1

⎞ ⎟ dt + ⎠

⎤

n

∑ λi dxi ⎦⎥

i =1

= 0, s = 0,1 ,

t = ts

407

(21.5.18)

which give 2n + 2 − (r0 + r1 ) equations; associating the conditions (21.5.3''), one obtains 2n + 2 equations at the ends. The first equations, so that the theorem of implicit functions can determine the unknowns uk and μj as function of λi and x i . Substituting in the rest of the equations (21.5.12), (21.5.12'), it results a system of 2n differential equations of first order in the unknowns λi and x i ; the general solution of this system depends on 2n + 2 constants (including t0 and t1 ), which are specified by the 2n + 2 conditions at the ends. The problem is thus entirely solved. If τ is a point of discontinuity of the control function u (t ) , then it is shown that the Weierstrass-Erdmann conditions ∂L ∂xi ⎡ ⎢ −L + ⎣

t =τ −0

=

∂L ∂xi

, t =τ +0 n

∂L ⎤ ⎡ = ⎢ −L + ∑ ∂xi xi ⎥⎦ ⎣ i =1 t =τ −0 n

∂L ⎤ ∑ ∂xi xi ⎥⎦ i =1 t =τ +0

(21.5.19)

must be verified.

21.5.2 Optimal Trajectories In what follows we consider the case of problems of Lagrange type without constraints, being thus led to Pontryagin’s principle of maximum; we present then some applications.

21.5.2.1 Problems of Lagrange Type Without Constraints In case of a problem of Lagrange type without constraints (for which h j = 0 , j = 1, 2,..., p ), Lagrange’s function (21.5.11) has a much more simple form n

L = f0 + ∑ λi ( xi − fi ) ; i =1

(21.5.20)

the equations (21.5.12), (21.5.12') are reduced to ∂L = 0, k = 1, 2,..., m , ∂uk

(21.5.21)

d ⎛ ∂L ⎞ ∂L ∂L − = 0, = 0, i = 1, 2,..., n . dt ⎜⎝ ∂xi ⎟⎠ ∂x i ∂λi

(21.5.21')

The equations (21.5.21') become xi =

∂H ∂H ,λ = − , i = 1, 2,..., n , ∂λi i ∂x i

(21.5.22)


408

where we have introduced Hamilton’s function

H = − f0 +

n

∑ λi fi

;

k =1

(21.5.23)

as well, the equations (21.2.21) read

∂H = 0, k = 1, 2,..., m . ∂uk

(21.5.22')

Weierstrass’s condition (21.5.16) takes the form

H ( x , u ;t ;λ ) ≤ H ( x , u ;t ;λ ) ,

(21.5.24)

for any u which satisfies the first equations (21.5.3') and u (t ) ≠ u (t ), ∀t ∈ [ t0 , t1 ] .

21.5.2.2 Principle of Maximum The conditions (21.5.22') and (21.5.24) can be written together in the form

H ( x (t ), u (t ); t ; λ (t ) ) = sup [ x (t ), u ; t ; λ (t ) ] ; u ∈U

(21.5.25)

the necessary condition of optimality expresses Pontryagin’s principle of maximum. A control autonomous system is described by equations of the form xi = fi (x , u ), i = 1, 2,..., n ,

(21.5.26)

where the time does not appear explicitly; the functions fi and ∂fi / ∂x j are considered as defined on the set \n × U . Thus, for a given control function u = u (t ) and for given initial conditions x (t0 ) = x 0 , the system (21.5.26) admits a unique solution. If the trajectory x = x (t ) is definite for t ∈ [t0 , t1 ] and x (t1 ) = x 1 , then we say that the control function u (t ) transfers the system from the position x 0 to the position x 1 . Considering the performance index ( f0 has the same properties as f1 ) I =

t1

∫t

f0 (x , u )dt ,

(21.5.27)

0

we can enounce the problem of optimality is the following form: Being given two points x 0 ∈ S 0 and x 1 ∈ S1 and the admissible control functions u = u (t ) (if there exist such functions), which transfer the material system from the position x 0 to the position x 1 , one asks to determine that admissible control function for which the index of performance (21.5.27) takes the minimal value. The moment t1 is determined by the condition x (t1 ) = x 1 and by the second condition (21.5.3'').


409

We introduce the co-ordinate x 0 definite by the equation x 0 = f0 ( x , u ) , so that the differential system (21.5.26) does take place for i = 0 too; we denote x ≡ {x 0 , x } . Let be also the system n

λi = − ∑ λj j =0

∂f j , i = 1,2,..., n , ∂x i

(21.5.28)

which defines the vector function λ (t ) ≡ {λ0 (t ), λ1 (t ), λ2 (t ),..., λn (t )} for the trajectory x (t ) and an admissible control function u (t ) . By means of Hamilton’s function

H ( x, u , λ ) =

n

∑ λi fi ( x, u ) ,

i =0

(21.5.29)

the equations (21.5.26), (21.5.28) are written in the canonical form (21.5.22) (for i = 0 too). We may thus state Theorem 21.5.1 If u (t ) is an optimal control functions, then there exists a continuous and non-zero vector function λ (t ) so that: (i) x(t ) and λ (t ) are solutions of the canonical system (21.5.22); (ii) ∀t ∈ [t0 , t1 ] the function H ( x(t ), u , λ (t ) ) , with respect to the variable u ∈ U , reaches its maximum for u (t ) (principle of maximum)

H ( x(t ), u (t ), λ (t ) ) = sup H ( x(t ), u (t ), λ (t ) ) ; u ∈U

(21.5.30)

(iii) ∀t ∈[t0 , t1 ] we have λ0 (t ) = const ≤ 0, H ( x(t ), u (t ), λ (t ) ) = 0 .

(21.5.30')

The equations thus established allow the complete determination of the unknown functions x i (t ), λi (t ), i = 1, 2,..., n , u j (t ), j = 1, 2,..., m , and of the moment t1 . In case of non-autonomous systems one can make a convenient substitution, which formally reduces the problem to that corresponding to autonomous systems.

21.5.2.3 Applications As applications to the results presented above we can study the motion of a rocket, modeled as a particle of variable mass (see Sect. 10.3.1.1). We consider thus Meshcherskiĭ’s equation m v = F + m ( u − v ), m < 0 ,

(21.5.31)

where v is the velocity of the particle of mass m with respect to an inertial frame of reference, u is the absolute velocity of the detached mass and F is the resultant of the given forces; obviously, w = u − v is the relative velocity of the detached mass with respect to a non-inertial frame of reference, attached to the particle in motion, while R = m w is the reactive force. We assume that the motion of the rocket can be guided


410

by modifying the force of propulsion (which includes the reactive force), after an imposed program; because m < 0 , the force of propulsion is guided in the sense of the motion, accelerating it. We denote by u1 , u2 , u3 the direction cosines corresponding to this sense (with respect to an inertial frame), by u4 = −m > 0 the velocity of variation of mass with changed sign and by u5 =| w |=| u − v | the magnitude of the relative velocity; these variables are control variables. The variables of state are the co-ordinates x1 , x 2 , x 3 of the particle (of the rocket), the components x 4 = v1 , x 5 = v2 , x 6 = v3 of its velocity and the mass x 7 = m . With respect to an inertial frame of reference, the equation (21.5.31) becomes xi = x i + 3 , i = 1, 2, 3, xi + 3 =

1 [ F ( x, t ) + ui u4 u5 ], x7 i x7 = − u 4 .

(21.5.32)

We suppose that F depends only on the variables of state and time. The problem which is put leads to the definition of the set U by the conditions u 12 + u 22 + u 23 = 1, 0 ≤ u 4 ≤ u4 , 0 ≤ u5 ≤ u5 .

(21.5.33)

Among the problems which can be put, we mention: the optimization of the force of propulsion (in particular, in case of the motion in a homogeneous gravitational field), the finding of an optimal control function to realize a maximal distance of flight, the determination of an optimal control function to realize a minimal consumption of combustible in case of a rocket of constant power etc. Another interesting problem which leads to optimization problems is that of airplane flight. One can mention other problems concerning the modern industry and technology, in connection also with the idea of automatization. All these aspects have led to a chapter in the theory of differential equations, which is now – by its importance and its development – independent.

Chapter 22 Dynamics of Non-holonomic Mechanical Systems The theory of non-holonomic mechanical systems appeared when it was seen that the classical Lagrangian formalism (corresponding to the holonomic mechanical systems) cannot be applied in case of some very simple problems (e.g., the rolling without sliding of a rigid solid on a fixed plane). After that Lindelöf’s error in this direction has been detected by Chaplygin, one has obtained many formulations and there have been made many studies by Appell, Bobylev, Chaplygin, Tsenov, Hamel, Hertz, Maggi, Voronets, Zhukovskiĭ and others; the actual research in this direction is very rich. The development of dynamics of holonomic mechanical systems puts in evidence the closed connection between the study of these systems and the study of the holonomic ones; moreover, the geometric treatment of these problems led to the creation of the non-holonomic geometry by Vrănceanu, Scouten, Wagner and others. On the other hand, there appeared also many specific aspects (e.g., non-linear non-holonomic constraints considered by Gibbs, Appell, Chetaev, Hamel, Johnson, Novoselov etc.); as well, other types of non-classical constraints have been put in evidence. We mention also that, in the last time, an analog between these systems and the electromechanical ones has been developed. After passing in review some elements of kinematical nature, one presents Lagrange’s equations, as well as other equations of motion (especially the Gibbs-Appell equations and Chaplygin’s equations) and various applications; one considers then also other problems concerning the dynamics of non-holonomic mechanical systems (Neĭmark, Ju.I. and Fufaev, N.A., 1972; Pars, L., 1965).

22.1 Kinematics of Non-holonomic Mechanical Systems After some general considerations, one presents some conditions of holonomy and one introduces the notion of quasi-co-ordinate. A special attention is given to the geometrization of the problem of non-holonomic mechanical systems.

22.1.1 General Considerations One presents, in the following, some preliminary aspects concerning the differential forms, the number of degrees of freedom of the mechanical systems, the representative spaces which are used etc. One considers then various cases, especially the rolling of a rigid solid over another rigid solid.


411


412 22.1.1.1 Preliminary Notions

In the study of non-holonomic mechanical system we use, as till now, a discrete modelling, their state being specified – at a given moment – by a finite number of parameters. We will consider further systems formed by particles and rigid solids. It is necessary to use the notions of real displacement dr , possible displacement Δr and virtual displacement δr introduced in Sect. 3.2.1.1 for the space E 3 , in Sects. 18.1.1.2 and 18. 2.1.2 for the spaces E 3n , and Λs , respectively. Let be a mechanical systems S of particles Pi , of position vectors ri , subjected to constraints expressed, in general, by real displacements dri , i = 1, 2,..., n , in the form (3.2.13), if the constraints are bilateral, or in the form (3.2.14), if these ones are unilateral; excepting special cases, we will consider only bilateral constraints. The constraint relations are expressed in the form (3.2.15), using virtual displacements δri . If the differential forms of first degree by which these relations are expressed are (locally) non-integrable, hence if there do not exist integrant factors for the corresponding equations, then the respective constraints are non-holonomic (kinematic) constraints. Passing to the space Λs , the constraint relations expressed by means of the generalized displacements dq j , j = 1, 2,..., s , are of the form (see Sect. 18.2.1.2 too) akj (q ; t )dq j + ak 0 (q ; t )dt = 0, k = 1,2,..., m ,

(22.1.1)

akj (q ; t )q j + ak 0 (q ; t ) = 0, k = 1,2,..., m ,

(22.1.1')

or of the form

where q ≡ {q1 , q2 ,..., qs } are generalized co-ordinates. If ak 0 = 0 , then the constraints are catastatic (and homogeneous) and if we have also akj = 0, k = 1, 2,..., m , j = 1,2,..., s , then these constraints are even scleronomic. As well, the constraints expressed by means of these relations are non-holonomic (anholonomic) if there does not exist an integrant factor for the corresponding differential equations. In Sect. 3.2.2.6, Frobenius’s theorem, which specifies the necessary and sufficient conditions of holonomy for a constraint system, is given. Using the virtual generalized displacements δq j , the constraint relations read akj (q ; t )δq j = 0, k = 1,2,..., m .

(22.1.2)

The existence of non-holonomic constraints has not been known by Lagrange, nor by other researchers; only in 1894, H. Hertz made distinction between holonomic and non-holonomic constraints. A simple example (the motion of a rigid skate on the ice plane) has been given in Sect. 3.2.2.6. Let be now a circular disc, of centre O and radius R , which is rolling on the plane Π (not necessarily horizontal), where we situate the system of axes O ′x1′x 2′ ; its position is specified by the co-ordinates x1′ and x 2′ of the contact point I , by the angle ψ between the tangent to the disc at I and the O ′x1′ -axis, by the inclination angle θ of the plane of

Dynamics of Non-holonomic Mechanical Systems

413

the disc with respect to the plane Π and by the angle ϕ , which gives the position of a point P of the contour with respect to the point I (Fig. 22.1, where the sense of the angles is put into evidence). The infinitesimal displacement of the disc is characterized by the real displacements dx1 , dx 2 , dψ, dθ and dϕ . In case of a rolling without sliding of the disc on the plane Π , case in which the relative velocity of the contact point I vanishes, one can easily see that (the projections of the arc Rdϕ on the two axes of co-ordinates) dx1′ + R cos ϕdϕ = 0, dx 2′ + R sin ϕdϕ = 0

(22.1.3)

x1′ = − Rϕ cos ψ , x2′ = − Rϕ sin ψ .

(22.1.3')

or

We mention that the five generalized co-ordinates can take any value among the set of possible values, hence that the disc can have any position with respect to the plane. Indeed, starting from an initial position x1′0 , x 2′0 , ψ 0 , θ 0 , ϕ 0 , one can reach any other

position x1′1 , x 2′1 , ψ1 , θ1 , ϕ1 . Firstly, the disc rolls from the contact point I 0 ( x1′0 , x 2′0

)

till

the contact point I 1 ( x 1′1 , x 2′1 ) , along a curve of length R ( ϕ1 − ϕ 0 + 2k π ) , k ∈ ` , then it rotates about the normal at I 1 to the plane Π , till ψ = ψ1 , and – finally – it inclines till θ = θ1 . Hence, the conditions (22.1.3) or (22.1.3') do not impose any restrictions to the values which can be taken by the considered generalized co-ordinates; these constraints are non-holonomic and scleronomic.

Fig. 22.1 Rolling of a circular disc on a plane Π

In the case of a system of differential forms of first degree, it is not sufficient that only one form be non-integrable so that the corresponding mechanical system be non-holonomic. Let be, e.g., the differential forms ω1 = ( x12 + x 22 ) dx1 + x1x 3 dx 3 = 0, ω2 = ( x12 + x 22 ) dx 2 + x 2 x 3 dx 3 = 0 ;


414

each of these forms, taken apart, is non-integrable, but combinations of them lead to

( x12

+ x 22 ) d ( x 21 + x 22 + x 23

) = 0, ( x12

+ x 22 ) dln

x1 =0 ; x2

The number r of kinematic degrees of freedom of a mechanical system is equal to the number of linear independent virtual displacements of this system. In case of a holonomic mechanical system, this number equates the number of the generalized co-ordinates ( r = s ), while in case of a non-holonomic one we have r = s − m . Let be the circular disc which rolls without sliding on a horizontal plane, considered above (Fig. 22.1); the constraint relations are written in the form δx1′ + R cos ψδϕ = 0, δx 2′ + R sin ψδϕ = 0 .

(22.1.3'')

These equations admit three independent systems of solutions, i.e.: (i)

δx1′ = − R cos ψδϕ , δx 2′ = − R sin ψδϕ , δϕ , δθ = δψ = 0 ,

(ii)

δx1′ = δx 2′ = 0, δψ = δϕ = 0, δθ ,

(iii)

δx1′ = δx 2′ = 0, δϕ = δθ = 0, δψ ,

corresponding to the virtual rotations of the disc about an axis normal to the disc at the point I , about the tangent to the disc at the contact point, as well as about an axis normal to the plane Π at the point I , respectively. 22.1.1.2 Representative Spaces We have introduced the space of configurations Λs in Sect. 18.2.1 the representative point P being specified by a minimal parameterization q j , j = 1, 2,..., s . In a qualitative dynamical theory, the topological structure of this space plays an important rôle; in this order of ideas, we consider some particular cases.

Fig. 22.2 Frictionless sliding of a ball S along the axis of a rigid pendulum

Let be a physical pendulum for which the axis of suspension passes through O and the position of which with respect to a vertical axis is specified by the angle θ ; we consider a rectilinear guide rail along the pendulum (the Ox -axis) on which slides


415

frictionless a small body (a ball) S , its position being thus given by the generalized co-ordinates x and θ (Fig. 22.2). The correspondence between the point P in the plane of configurations ( x , θ ) and the configuration of the mechanical system is not one-to-one, because to two points ( x , θ ) and ( x , θ + 2k π ), k ∈ ] , corresponds only one position of the considered mechanical system; to have a one-to-one correspondence, we must limit ourselves to the strip 0 ≤ θ < 2 π (Fig. 22.3a). But we

Fig. 22.3 Physical pendulum; plane (a strip) (a) and cylindrical (b) space of configurations

notice that to two close configurations of the mechanical system S , i.e. ( x , ε ) and ( x , 2 π − ε, ε > 0 , arbitrary small), correspond two points of the strip, which are distant one of the other; hence, to a continuous motion of the system S does not correspond a continuous displacement of the representative point. To restore the continuity, we roll the strip on a circular cylinder of radius equal to unity and “glue” the opposite sides. The cylinder can be thus considered to be the space of configurations corresponding to the mechanical system S , the correspondence between the positions of the mechanical system and the points of the cylinder being one-to-one and continuous (Fig. 22.3b).

Fig. 22.4 Double pendulum; plane (a square) (a) and a three-dimensional (torus) (b) space of configurations

In case of a double pendulum, the position of which is specified by the angles θ1 and θ2 (see Sect. 17.1.1.2, Fig.17.1a), the one-to-one correspondence is ensured if one takes as space of configurations a square of side 2 π (0 ≤ θ1 , θ2 < 2 π ) (Fig. 22.4a). If one rolls the square on a cylinder, gluing the opposite sides θ1 = 0 and θ1 = 2 π , and then one bends the cylinder, gluing the opposite faces θ2 = 0 and θ2 = 2 π , one obtains a

416


torus (Fig. 22.4b); this is the space of configurations in case of the double pendulum, the continuity being – as well – ensured.

Fig. 22.5 Rigid skate – three-dimensional space of configurations

In Sect. 3.2.2.6 we have considered the motion of a rigid skate AB on the ice plane (plane Ox1x 2 , Fig. 3.16). The position of the segment AB is specified by the co-ordinates x1 and x 2 of the middle C of this segment and by the angle θ made by it with the Ox1 -axis; one has the constraint relation (3.2.31) between these generalized co-ordinates. The space of configurations is three-dimensional and is formed by the stratum 0 ≤ θ < 2 π , two opposite points of the planes θ = 0 and θ = 2 π being considered to be united (Fig. 22.5); one cannot obtain a “gluing together” in the three-dimensional space.

Fig. 22.6 Spaces of configurations: rigid solid with a fixed point (a); circular disc rolling without sliding on a horizontal plane (b)

One can be show that to a rigid solid with a fixed point corresponds as space of configurations the interior (with the frontier) of a sphere of radius π ; the diametrical opposite points on the sphere must be considered united (Fig. 22.6a). As well, in case of a circular disc constraint to roll slidingless on a horizontal plane (Fig. 22.1), the space of configurations is formed from the interior (with the frontier) of a cube of side equal to 2 π , two opposite points (of opposite faces) of which being united (Fig. 22.6b), together with the plane of the contact points ( x1′ , x 2′ ); one obtain thus a pentadimensional space of configurations (the space of configurations is the same if the motion takes place with sliding).


417

The state of a mechanical system S is specified by the generalized co-ordinates q j and by generalized velocities q j , j = 1, 2,..., s . If there exist m relations of non-holonomic constraints of the form (22.1.1) or (22.1.1'), then it is sufficient to use s − m generalized velocities, determining the other generalized velocities starting from these relations; instead of generalized velocities one can use kinematic characteristics of the form ωi = αij q j + αi 0 , i = 1, 2,..., s − m ,

(22.1.4)

i.e. linear combinations of generalized velocities with αij = αij (q1 , q2 ,..., qs ; t ) , j = 1, 2,..., s , where det [ αij

]≠0

and α j 0 = α j 0 (q1 , q2 ,..., qs ; t ) . The mechanical system S

(22.1.4') is thus specified by

2s − m co-ordinates, i.e. s generalized co-ordinates and s − m characteristics, which form the phase space Γ 2 s − m .

kinematic

Fig. 22.7 Space phase of a physical pendulum

For instance, the state of a physical pendulum is specified by an angle θ and by a generalized angular velocity θ ; the corresponding phase space is a circular cylinder, θ being along the director circle, while θ is along the generatrix, the correspondence with the state of the physical pendulum being one-to-one and continuous (Fig. 22.7). In the problem of the rigid skate on the ice plane (known as Chaplygin’s problem) can be introduced the kinematic characteristic v = x1 cos θ + x2 sin θ instead of x1 and x2 , hence the velocity of the middle C of the segment AB ; the co-ordinates x1 , x 2 , θ , v , θ lead thus to a pentadimensional phase space. This space cannot be

418


visualized, but one can visualize sections in this space, e.g. x1 = const and θ = const . 22.1.1.3 Kinematics of the Rolling of a Rigid Solid on Another One Let be two rigid solids S and S ′ , bounded by two convex surfaces S and S ′ , respectively, which – at any moment t have the same tangent plane at an ordinary common point P ≡ P ′, P ∈ S , P ′ ∈ S ′ (see Sect. 5.3.3.1, Fig. 5.29); to fix the ideas, we admit that the rigid solid S ′ is fixed, while the rigid solid S is movable, remaining permanently in contact with the rigid solid S ′ . The velocity vQ (t ) of a movable point Q which coincides with P ≡ P ′ at any moment t is the velocity of transportation (velocity of sliding) with respect to the fixed surface S ′( vQ = vP′ − vP ) , contained in the common tangent plane. If vQ = 0 , then the surface S rolls without sliding over the surface S ′ . The motion of the rigid solid S with respect to the rigid solid S ′ is specified if the angular velocity vector ω(t ) , which passes through the point P is given too; hence, this motion is characterized by a translation of velocity vQ (t ) and by a rotation of angular velocity ω(t ) . The vector ω(t ) can be decomposed in two components: an angular velocity ωn (t ) along the normal to the tangent plane, which characterizes a pivoting about the respective axis, and an angular velocity ωt (t ) , contained in the tangent plane, which characterizes a rolling about the corresponding axis. In general, the motion of the rigid solid S over the rigid solid S ′ takes place so that the surface S is rolling and pivoting with sliding on the surface S ′ . If vQ (t ) = 0 , then the motion of the rigid solid

S with respect to the rigid solid S ′ is an instantaneous rotation (pivoting and rolling) about an instantaneous axis of rotation which passes through the point of contact. The fixed axoid Af intersects the surface S ′ along the curve C ′ (the locus of the point Q with respect to the surface S ′ ), while the movable axoid Am intersects the surface S along the curve C (the locus of the point Q with respect to the surface S ); in this case vP = vP ′ , so that – during the motion – the curve C rolls without sliding over the curve C ′ . In particular, if ωn = 0 ( ω = ωt ) , then the surface S is rolling slidingless

over the surface S ′ (pure rolling; e.g., the rolling of a cylinder, when the instantaneous axis of rotation is the contact generatrix); analogously, if ωt = 0 ( ω = ωn ) , then the surface S is pivoting without sliding over the surface S ′ (pure pivoting, e.g., the rotation of a sphere on a horizontal plane about its vertical diameter). If vQ (t ) ≠ 0 , then the particular rotations considered above are associated with a sliding. To can establish the constraint relations which take place in the considered motion, one must determine the connection between the angle of rotation dθ = ωdt and the arcs ds and ds ′ , described by the contact point Q on the curve C and on the curve C ′ , respectively. Introducing the angle of rotation dθr , corresponding to the pure rolling, and the angle of rotation dθ p , corresponding to the pure pivoting, we may write dθ = dθr + dθ p ;

(22.1.5)


419

we notice that the arc elements ds and ds ′ depend only on dθr , because the point Q does not move during the pivoting. Let P and P be two successive contact points on the surface S , along the curve p = ds , while n and n = n + dn are unit vectors normal to S at these C , so that PP

points, respectively (Fig. 22.8a); analogically, corresponding to the surface S ′ , we consider the points P ′ , P ′ and the unit vectors n ′ and n ′ = n ′ + dn ′ . Because n × n = n × dn and n ′ × n ′ = n ′ × dn ′ ≅ n × dn ′ , we can write dθr = n × dn + n × dn ′ = n × ( dn + dn ′ ) .

(22.1.6)

Fig. 22.8 Geometric elements at two successive points (a) or at a point (b) of a surface

Let us denote by dϕ and dϕ ′ the angles made by the normals n and n and by the normals n ′ and n ′ , respectively; we can write (we notice that n ⋅ dn = 0 ) d ϕ = n × dn , d n = d ϕ × n , dϕ ′ = n ′ × dn ′, dn ′ = dϕ ′ × n ′.

(22.1.7)

We decompose dϕ along the tangent to the curve C and along the normal to it ( dϕ = dϕ τ + dϕ β ; Fig. 22.8a); the vector dϕβ is contained in the tangent plane (as well as the vector dϕ ). The angle dϕ β is the angle of curvature, while the angle dϕ τ is the angle of torsion. If the curve C is a line of curvature of the surface S , then the torsion vanishes and the angle of curvature is given by dϕβ = ds / R , where R is

the corresponding principal radius of curvature. Let τ1 and τ2 be the unit vectors tangent to the lines of curvature at P , ds1 and ds2 the elements of arc along these lines and R1 and R2 the corresponding radii of curvature (Fig. 22.8b); we can thus write dϕ = −

ds 2 ds τ + 1 τ2 . R2 1 R1

(22.1.8)


420

Observing that dθr = dϕ + dϕ ′ , it results dϕr = −

ds 2 ds ds ′ ds ′ τ + 1 τ2 − 2 τ1′ + 1 τ2′ , R2 1 R1 R2′ R1′

(22.1.9)

where we use analogous notations for the surface S ′ . Let be dτ and dτ ′ the projections on the common tangent plane of the variations of the unit vectors tangent to the curves C and C ′ at the point Q , respectively; we notice that dθp = dτ − dτ ′ . The velocity of pivoting is thus given by ωp =

dθp = ( K g − K g ′ ) vQ , dt

(22.1.10)

where vQ = ds / dt , while K g = dτ / ds and K g ′ = dτ ′ / ds are the geodesic curvatures of the curves C and C ′ , respectively, at the same point Q .

Fig. 22.9 Rolling of a sphere on a horizontal plane

In particular, let us consider the rolling of a sphere of radius R on a horizontal plane with a constant angular velocity ω ; the principal radii of curvature of the sphere being equal to R , while those of the plane being infinite, it results dθr =

1 ( τ ds − τ1 ds2 ) , R 2 1

(22.1.11)

wherefrom the angular velocity of the pure rolling is given by ωr = dθr / dt . The velocity of displacement of the contact point is normal to the angular velocity ωr and has the magnitude vQ = Rωr . The curve C ′ in the fixed plane is a straight line, while the curve C on the sphere is a circle of constant geodesic curvature given by Kg =

1 1 ωp . ωp = vQ R ωr

(22.1.12)


421

To calculate this curvature, we notice that 1/ K g = ρ / cos θ , ρ = R sin θ , where ρ is the radius of the circle C , while the angle θ specifies the inclination of the plane (Fig. 22.9). Eliminating the angle θ , we get K g2 =

1 1 − . ρ2 R2

(22.1.12')

In connection with the motion of a rigid solid which slides frictionless on a fixed plane, we mention the considerations in Sect. 17.1.2.3. In particular, one considers the case of a heavy homogeneous rigid solid of rotation (in Sect. 17.1.2.4), the case of a heavy gyroscope (in Sect. 17.1.2.5), the case of a rigid cylindrical solid (in Sect. 17.1.2.6) and the case of a sphere (in Sect. 17.1.2.7), the fixed plane being horizontal. The motion without sliding of a sphere on a fixed plane is studied in Sect. 17.1.2.7 too; in Sect. 17.1.2.8 has been considered the case of a hollow homogeneous sphere, the plane being horizontal. As well, the motion without sliding of a heavy circular disc has been presented in Sect. 17.1.2.9.

22.1.2 Conditions of Holonomy. Quasi-co-ordinates. Non-holonomic Spaces In what follow, we will present firstly some conditions of holonomy, we introduce then the notion of quasi-co-ordinate, the corresponding relations of transposition being also established. As well, we make considerations concerning the introduction of the non-holonomic spaces too.

22.1.2.1 Conditions of Holonomy Let be a mechanical system S , the representative point (q ; t ) of which is subjected, in general, to the constraint relations (22.1.1), written in the form ( t = q 0 , for the uniformity of the notation) s

∑ akj ( q ; t ) dq j

j =0

= 0, k = 1, 2,..., m .

(22.1.13)

From the geometric point of view, these conditions show that a point in the space (q ; t ) cannot be displaced arbitrarily, but only along a curve which is tangent at any point of it to a (s − m ) -dimensional hyperplane, which contains all the vectors dq 0 , dq1 , dq2 , ..., dqs , which satisfy the conditions (22.1.13); if the system (22.1.13) is not integrable, then the representative point can occupy any point in the space (q ; t ) , although the imposed restrictions. First of all, let us consider the particular case s = 3 , with a scleronomic constraint relation a1 ( q1 , q2 , q 3 ) dq1 + a2 ( q1 , q2 , q 3 ) dq2 + a 3 ( q1 , q2 , q 3 ) dq 3 = 0 ;

(22.1.14)


422

it means that the point (q1 , q2 , q 3 ) cannot move on any curve in Λ3 , but only along some curves which – at any point of them – are normal to the vector (a1 , a2 , a 3 ) at the respective point. As we have seen in Sect. 3.2.2.6 (formula (3.2.30'')), the necessary and sufficient conditions of holonomy is (a1 , a2 , a 3 ) ⋅ curl(a1 , a2 , a 3 ) = 0 .

(22.1.14')

As it has been shown by Carathéodory, if we have (a1 , a2 , a 3 ) ⋅ curl(a1 , a2 , a 3 ) ≠ 0 in a given domain, then any two points of this domain can be linked by an admissible curve (even if the constraint relation (22.1.14) takes place). In the general case of the constraint relations (22.1.13'), A. M. Lipschitz has introduced a linear transformation in any point of the space (q ; t ) , obtaining a necessary and sufficient condition of holonomy of the form δMdq − dM δq = 0 ,

(22.1.15)

for any displacements in the (s − m ) -dimensional hyperplane; this condition is equivalent to the conditions expressed in the Theorem 3.2.2 of Frobenius (see Sect. 3.2.2.6). We are thus led to the necessary and sufficient conditions s

∑ ( δakj dq j

j =0

− dakj δq j

) = 0.

(22.1.15')

Returning, in particular, to the constraint relation (22.1.14), we obtain the necessary and sufficient conditions δa1 dq1 + δa2 dq2 + δa 3 dq 3 − da1 δq1 − da2 δq2 − da 3 δq 3 = 0

or, in a developed form, ⎛ ∂a2 − ∂a 3 ⎞ δq dq − δq dq ⎛ ∂a 3 − ∂a1 ⎞ δq dq − δq dq ⎜ ∂q ⎟( 3 2 ⎟( 1 3 2 3 )⎜ 3 1 ) ⎝ 3 ∂q2 ⎠ ⎝ ∂q1 ∂q 3 ⎠ ∂a ∂a + ⎛⎜ 1 − 2 ⎞⎟ ( δq2 dq1 − δq1 dq2 ) = 0, ⎝ ∂q2 ∂q1 ⎠

for any δq1 , δq2 , δq 3 and dq1 , dq2 , dq 3 which equate to zero the Pfaffian (22.1.14); this condition can be written also in the form curl(a1 , a2 , a 3 ) ⋅ [ δ(q1 , q2 , q 3 ) × d(q1 , q2 , q 3 ) ] = 0 .

But the displacements δ( q1 , q2 , q 3 ) and d(q1 , q2 , q 3 ) satisfy the conditions (a1 , a2 , a 3 ) ⋅ δ(q1 , q2 , q 3 ) = 0, (a1 , a2 , a 3 ) ⋅ d(q1 , q2 , q 3 ) = 0


423

being normal to the vector (a1 , a2 , a 3 ) ; hence, the vector δ( q1 , q2 , q 3 ) × d(q1 , q2 , q 3 ) is collinear with the vector (a1 , a2 , a 3 ) , obtaining again the condition (22.1.14'). In case of a disc which rolls on a horizontal plane, e.g., one writes the constraint relations (22.1.3''); we mention that the bilinear covariants R sin ψdψδϕ − R sin ψdϕδψ , − R cos ψdψδϕ + R cos ψdϕδψ

can vanish only if sin ψ = 0, cos ψ = 0 , the corresponding constraints being thus non-holonomic. We say that a function f (q 0 , q1 , q2 ,..., qs ) is a first integral of the system (22.1.13) if for any dq 0 , dq1 , dq2 ,..., dqs which satisfy this system we have df =

∂f ∂f ∂f ∂f dq 0 + d q1 + dq2 + ... + dqs = 0 , ∂q 0 ∂q1 ∂q2 ∂qs

hence if df = 0 . Obviously, each integral of the system of equations (22.1.13) allows to reduce by a unity the number of co-ordinates which determine the position of the representative point; as well, if the number of the independent co-ordinates may be reduced due to the existence of a functional relation f = 0 between the co-ordinates q1 , q2 ,..., qs and the time q 0 , then the function f is an integral of the system of equations (22.1.13). Hence, the problem to determine the number of co-ordinates and the number of constraints of a mechanical system S is reduced to the problem of finding all the functional independent integrals of the system (22.2.13). Finally, the system of constraint relations (22.1.13) is equivalent to m ≤ m holonomic constraints and to m − m non-holonomic (non-integrable) constraints; hence, the representative point (the mechanical system S ) is specified by s − m generalized co-ordinates and has s − m + m degrees of freedom.

22.1.2.2 Quasi-co-ordinates. Relations of Transposition The velocity of the representative point corresponding to a given configuration of the mechanical system S at a certain moment can be specified by means of the generalized velocities or by the kinematical characteristics (22.1.4), introduced in Sect. 22.1.1.2, where we take i = 1, 2,..., s . These characteristics are total derivatives with respect to time only if ∂αij ∂α ∂αi 0 − ik = 0, α ij = , i , j , k = 1, 2,..., s , ∂qk ∂q j ∂q j

(22.1.16)

case in which the generalized co-ordinates can be obtained by a simple integration; otherwise, the determination of the generalized co-ordinates needs the integration of a system of linear differential equations, which is – in general – a difficult problem. We introduce thus the quantities πi =

t

∫0 ωi dt , i

= 1, 2,..., s ,

(22.1.17)


424

which have a certain analogy with the co-ordinates, but are not functions of the position of the mechanical system S ; therefore, these quantities are called quasi-co-ordinates (if ωi = qi , then it results πi = qi ) and form a corresponding representative space Λs′ . We can write dπi = ωi dt = αij dq j + αi 0 dt , i = 1,2,..., s ,

(22.1.18)

too. Taking into account (22.1.4'), we also have dqk = βki dπi + βk 0 dt , k = 1, 2,..., s ,

(22.1.18')

αij βki = δ jk , βk 0 = − αi 0 βki , j , k = 1,2,..., s .

(22.1.18'')

where

By introducing the kinematic characteristics (and the quasi-co-ordinates), we can identify m of them with the left member of the constraint relations (22.1.1') ωk = akj q j + ak 0 , k = 1, 2,..., m ,

(22.1.19)

the last relations being thus written in the form ωk = 0, k = 1, 2,..., m .

(22.1.20)

Thus, by using quasi-co-ordinates, one can establish a generalized form of Lagrange’s equations. The representative point P of co-ordinates q1 , q2 ,..., qs describes, during the motion of the mechanical system S , a curve C , in the space of configurations Λs , while the point Π of quasi-co-ordinates π1 , π2 ,..., πs describes a curve Γ in an analogous space Λs′ of the quasi-co-ordinates; both curves describe – completely – the motion, so that there is not any difference between the generalized co-ordinates and the quasi-co-ordinates. However, to any point P corresponds only one position of the mechanical system S , but to a point Π can correspond any position of this system. We notice thus that a correspondence between a point Π and an arbitrary position of the mechanical system S can be established only if, after travelling through a closed curve Γ , the mechanical system would return to the same position. Let be Δπi =

v∫ C ωi dt , i

= 1, 2,..., s ,

(22.1.21)

the variations of the quasi-co-ordinates corresponding to the travelling through a closed curve C by the representative point P ; we notice thus that to a contour C in the space


425

of configurations Λs corresponds – in general – an open contour Γ in the space of quasi-co-ordinates Λs′ , because the kinematic characteristics are not integrable. Using the virtual generalized displacements, we can write δπi = αij δq j , i = 1, 2,..., s ,

(22.1.22)

δqk = βki δπi , k = 1, 2,..., s .

(22.1.22')

Applying the operator d to the relations (22.1.22), it results dδπi = αij dδq j +

∂αij dqk δq j + α ij dt δq j ; ∂qk

as well, by applying the operator δ to the relations (22.1.18), we get δdπi = αij δdq j +

∂αij ∂α δqk dq j + i 0 δqk dt . ∂qk ∂qk

Subtracting the two relations one of the other, we can write

dδπi − δdπi = αij ( dδq j − δdq j

⎛ ∂α

) + ⎜ ∂qij ⎝

k

−

∂αik ⎞ ∂α ⎛ dqk δq j + ⎜ α ij − i 0 ⎟ ∂q j ⎠ ∂q j ⎝

⎞ ⎟ δq j dt . ⎠

Taking into account (22.1.18'), (22.1.22') and denoting

i γm

∂α ⎞ ⎛ ∂αij i γlm =⎜ − ik ⎟ βkl β jm , i , l , m = 1, 2,..., s , ∂q j ⎠ ⎝ ∂qk ∂α ⎞ ∂α ⎞ ⎛ ∂αij ⎛ =⎜ − ik ⎟ βk 0 β jm + ⎜ α ij − i 0 ⎟ β jm , i , m = 1, 2,..., s , ∂ ∂ ∂q j ⎠ q q j ⎠ ⎝ k ⎝

(22.1.23)

the relations of transposition read dδπi − δdπi = αij ( dδq j − δdq j

i dπl δπm ) + γlm

i + γm δπm dt , i = 1, 2,..., s .

(22.1.23) The permutability relations of the operators d and δ applied to the generalized co-ordinates ( dδq j = δdq j ) lead to the relations of transposition (the transitivity relations of K. Heun) i i dδπi − δdπi = γlm dπl δπm + γm δπm dt , i = 1, 2,..., s ;

(22.1.24)

in the particular case in which αi 0 = 0 (we have βi 0 = 0 too) and α ij = 0 , i , j = 1, 2,..., s , hence in the case in which the kinematic characteristics do not depend i = 0 , while the relations (22.1.24) read explicitly on time, we obtain γm


426

i dδπi − δdπi = γlm dπl δπm , i = 1, 2,..., s .

(22.1.24')

The obvious relations i i γlm = − γml , γlli (!) = 0, i , l , m = 1, 2,..., s ,

(22.1.23'')

take place. Replacing in (22.1.24') the symbol of variation δ by the differential d , it results i γlm dπl dπm = 0, i = 1, 2,..., s ,

(22.1.25)

putting thus in evidence the gyroscopic nature of the non-holonomic terms in the equations of a non-holonomic mechanical system. We mention that the relations i i and γm . (22.1.24) and (22.1.24') allow to determine, easily, the coefficients γlm As an example of kinematic characteristics frequently used, we mention the components of the angular velocity ω of a rigid solid along the principal axes of inertia, expressed by means of Euler’s angles (generalized co-ordinates) and by means of their derivatives with respect to time (generalized velocities), given by the relations (5.2.35). The variations (22.1.21) of the quasi-co-ordinates are

v∫ C ω1 dt = v∫ C ( cos ϕdθ + sin θ sin ϕdψ ), = v∫ ω2 dt = v∫ ( − sin ϕdθ + sin θ cos ϕdψ ) , C C Δπ3 = v∫ ω3 dt = v∫ ( dϕ + cos θdψ ); C C

Δπ1 = Δπ2

(22.1.26)

we get thus Δπ1 = ( δd − dδ ) π1 = ( δϕdψ − δψdϕ ) sin θ cos ϕ + ( δθdψ − dθδψ ) cos θ sin ϕ + ( δθdϕ − dθ δϕ ) sin ϕ, Δπ2 = ( δd − dδ ) π2 = ( δψdϕ − dψδϕ ) sin θ sin ϕ

(22.1.26')

+ ( δθdψ − dθδψ ) cos θ cos ϕ + ( δθdϕ − dθ δϕ ) cos ϕ , Δπ3 = ( δd − dδ ) π3 = ( δψdθ − dψδθ ) sin θ .

Corresponding to the relation (5.2.35), we will have ( q1 = ψ , q2 = θ , q 3 = ϕ ; αi 0 = 0, i = 1, 2, 3 ) α11 = sin θ sin ϕ, α12 = cos ϕ, α13 = 0, α21 = sin θ cos ϕ, α22 = − sin ϕ, α23 = 0, α31 = cos θ , α32 = 0, α33 = 1;

as well, the relations (14.1.15) allow to write ( βk 0 = 0, k = 1, 2, 3 )


427

β11 = sin ϕ cosec θ , β12 = cos ϕ cosec θ , β13 = 0, β21 = cos ϕ, β22 = − sin ϕ, β23 = 0, β31 = − sin ϕ cot θ , β32 = − cos ϕ cot θ , β33 = 1. i The first formula (22.1.23) leads to γlm =∈iml , where ∈iml is Ricci’s symbol (2.1.29). In case of a regular precession, defined by

ψ = mt + ψ0 , θ = θ0 , ϕ = mt + ϕ0 , ψ0 , θ0 , ϕ0 = const ,

(22.1.27)

the trajectory of the representative point in the space of configuration is a straight line in the plane θ = θ0 . In the space of quasi-co-ordinates we have m sin θ0 [ cos ( nt + ϕ0 ) − cos ϕ0 ] , n m π2 = − sin θ0 [ sin ( nt + ϕ0 ) − sin ϕ0 ] , n π3 = ( m cos θ0 + n ) t ,

π1 = −

(22.1.27')

the trajectory of the point Π being a spiral on a circular cylinder; this last representation is useful for the study of the motion of the angular velocity vector ω .

22.1.2.3 Non-holonomic Spaces. Vrănceanu’s Theorems Being situated in the space of configurations Λs , let be a holonomic and catastatic (hence, scleronomic) mechanical system, for which the kinetic energy is of the form T =

1 g (q )qi q j ; 2 ij

(22.1.28)

we organize this space as a Riemannian space, defined by the element of arc ds 2 = gij (q )dqi dq j .

(22.1.28')

Corresponding to the Theorem 20.1.12 of Jacobi (see Sect. 22.1.4.2), being given a discrete mechanical system S , subjected to holonomic and scleronomic ideal constraints, acted upon by no one force, the trajectory of the point P in the space Λs , endowed with the metrics (22.1.28'), is a geodesic of this space. Let be v j , j = 1, 2,..., s , the components of a vector with the origin at the point P , in the space Λs . The formulae ⎧⎪ j ⎫⎪ dv j = − ⎨ ⎬ vk dql , j = 1, 2,..., s . ⎩⎪ k l ⎪⎭

(22.1.29)

where one uses Christoffel’s symbol of the second kind, define the parallel displacement in Levi-Civita’s sense of the vector (of components) v j with the origin at


428

the point ( q1 , q2 ,..., qs ), the vector displaced being v j + dv j with the origin at the point ( q1 + dq1 , q2 + dq2 ,..., qs + dqs ). In this case, the geodesics of the space Λs are the curves along which the tangent vector is displaced by parallelism; these curves are called autoparallel curves too. We can state Theorem 22.1.1 The trajectory of the representative point P in the space Λs , endowed with the metrics (22.2.28'), corresponding to the motion of a discrete mechanical system S , subjected to holonomic and scleronomic, ideal constraints and acted by no one force is an autoparallel curve of this space. One obtains thus a generalization of the principle of inertia, the straight lines in the space E 3 being just its autoparallel curves. Let us consider now, in the Riemannian space Λs , a Pfaff system

dsk = λjk dq j = 0, k = 1, 2,..., s ;

(22.1.30)

one defines thus a system of orthogonal congruencies if the relations λik λjk = gij

(22.1.30')

are satisfied. In this case d s 2 = ( ds 1 ) + ( d s 2 2

)2

+ ... + ( ds s )2 ,

(22.1.31)

the metrics of the space Λs being thus expressed in the form of a sum of squares. Solving the system (22.1.30) with respect to dq j , we obtain dq j = μkj ds k , j = 1, 2,..., s ,

(22.1.32)

where μkj is the normalized algebraic complement of the element λjk , so that λjk μki = δij . The system of congruencies is thus definite by the differential system d q1 dq d qs = 22 = ... = s , k = 1, 2,..., s , 1 μk μk μk

(22.1.33)

through each point of Λs passing one curve of each of the s families; μk1 , μk2 ,..., μks are the components of the tangent vector at a point of the corresponding curve which passes through that point. Multiplying both members of the relation (22.1.30') by μki μlj and summing, we get gij μki μlj = δkl ;

hence, the vectors { μ1k , μ2k ,..., μsk } are unitary and orthogonal two by two.

(22.1.34)


429

Being given a vector Vi , i = 1, 2,..., s , in Λs , by Levi-Civita’s parallel displacement one obtains the vector Vi + dVi , with ⎪⎧ j ⎫⎪ i dVi = − ⎨ ⎬Vj dqk = γ jkVj dqk , k l ⎪⎭ ⎩⎪

(22.1.35)

where l ∂μlj n ⎧ l ⎫ i m n ⎧ l ⎫ i m n ∂λm γ ijk = − ⎨ λl μj μk − λli μk = − ⎨ λl μj μk + μm μ n ⎬ ⎬ ∂qm ∂qn j k ⎩m n ⎭ ⎩m n ⎭ (22.1.35')

are Ricci’s coefficients of rotation. We assume that, in Λs , one gives a Pfaff’s system ds 1 = ds 2 = ... = ds m = 0 ,

(22.1.36)

which is not integrable; this system defines a non-holonomic space Λss − m . A displacement dq j is called interior to this space if ds α = μjα dq j = 0 for α = 1, 2,...,m ; more general, a vector is called interior to the non-holonomic space Λss − m if its components Vα , α = 1, 2,..., m , are all equal to zero. In this space too we introduce a parallel displacement characterized by the formulae (22.1.35), (22.1.35'), stating (Vrănceanu, Gh., 1936) Theorem 22.1.2 (Gh. Vrănceanu). To a non-holonomic space Λss − m , defined by the metrics (22.1.28') and by the equations (22.1.36), one can univocally associate a parallel displacement of the interior vectors, along the interior paths, maintaining the lengths. Let be the vectors ui = λji

dq j , i = 1, 2,..., s , dt

(22.1.37)

with uα = 0, α = 1, 2,..., m ; we can write dq j = dt

s

∑

k =m +1

μkj uk , j = 1, 2,..., s ,

(22.1.37')

where uk are the components of the vector tangent to the curve on the congruencies { λji } . The curve is autoparallel if the relations dui = γ ijk u j ds k hold for ui = ds i / dt ; hence, the autoparallel curves in the space Λss − m verify the equations dui = dt

s

∑

j ,k = m + 1

γ ijk u j uk , i = m + 1, m + 2,..., s .

(22.1.38)

430


We obtain thus a differential system (22.1.37'), (22.1.38) of the first order of 2s − m equations with 2s − m unknown functions q j , j = 1, 2,..., s , and ui , i = m + 1,

m + 2,..., s . A curve is definite if a point and a vector interior to the space Λss − m are given. If the equations (22.1.36) represent the constraints of a non-holonomic mechanical system S , then the equations (22.1.37'), (22.1.38) are equivalent to Lagrange’s equations in the quasi-co-ordinates ds i , which define the trajectories in the absence of the given forces. We can thus state Theorem 22.1.3 (Gh. Vrănceanu). The trajectory of the representative point P in the space Λss − m defined by the metrics (22.1.28') and the conditions (22.1.36), corresponding to the motion of a discrete mechanical system S subjected to non-holonomic and catastatic ideal constraints, non-acted upon by given forces, is an autoparallel curve of this space.

22.2 Lagrange’s Equations. Other Equations of Motion We considered, in Sect. 17.1.2, the motion of a rigid solid which slides without friction on a fixed plane, in particular the motion of a heavy homogeneous rigid solid of rotation, of a heavy gyroscope or of a cylindrical homogeneous rigid solid; as well, the motion without sliding of a sphere on a fixed plane, in particular that of a heavy homogeneous one on a fixed horizontal plane, has been considered. We mention also the results concerning the motion without sliding of a heavy circular disc on a fixed horizontal plane. These problems have been studied by means of the Newtonian equations of classical mechanics. In this order of ideas we consider the motion of an arbitrary rigid solid on a fixed surface too. After introducing Lagrange’s equations, one studies the above mentioned problems in the frame of Lagrangian mechanics. One presents then some other equations of motion.

22.2.1 Motion of a Rigid Solid on a Fixed Surface In what follows, we study first of all the motion of a rigid solid on a fixed horizontal plane, using Newtonian equations of classical mechanics; as well, we consider the motion of a sphere on a perfect rough fixed surface. In what concerns the kinematics of motion, we use the results obtained in Sect. 22.1.1.3.

22.2.1.1 Motion of a Rigid Solid on a Fixed Horizontal Plane Let be a rigid solid which moves on a horizontal plane Π , with support at three points. Two of the supports slide freely (without friction) on a plane, while the third one is a knife edge (a small wheel with a sharpened rim), rigidly connected to the solid in motion; one assumes that the contact point O of the knife can move freely in the plane Π along it, but not along a normal to its direction. This problem has been studied in 1911 by Chaplygin and solved by quadratures; therefore it will be called Chaplygin’s problem. Later, it was considered in detail, in a particular case, by Carathéodory, in 1933, the respective mechanical system being called a sleigh. Other results are due to Wagner.


431

We consider the fixed frame of reference O ′x1′x 2′ and the movable frame of reference Ox1x 2 , rigidly connected to the sleigh in the plane Π ; the Ox1 -axis is along the direction of the knife, the point O being the theoretical point of contact. The position of the movable frame is specified by the co-ordinates xO′ 1 , xO′ 2 of the point O and by the angle ϕ made by the Ox1 -axis with the O ′x1′ -axis (Fig. 22.10); thus, the problem is ′ = xOk ′ (t ), k = 1, 2 , and ϕ = ϕ (t ) . reduced to the determination of the functions xOk But it is more convenient to choose as unknowns the components vO′ 1 and vO′ 2 of the velocity along the axes of the movable frame of reference, as well as the angular velocity of rotation ω = ϕ of the rigid solid about a vertical axis. We denote by IC the moment of inertia of the rigid solid with respect to a vertical axis which passes through the mass centre C (of position vector ρ with respect to O ) of the rigid solid of mass M . The constraint force R , applied at O , is directed along the Ox 2 -axis. For the sake of simplicity, we assume that ρ2 = 0 , the centre of mass being projected at the point C ′( ρ1 ,O ) , on the O ′x1′ -axis.

Fig. 22.10 Motion of a rigid solid with supports at three points

The theorem of momentum (14.1.60) reads M [ v O′ + ω × vO′ + ω × ρ + ω × ( ω × ρ ) ] = R .

(22.2.1)

We have ( ω1 = ω2 = 0, ω3 = ω ) vO′ 1 − ωvO′ 2 − ω 2 ρ1 = 0, M ( vO′ 2 + vO′ 2 + ωρ 1 ) = R,

(22.2.1')

in projection on the axes of the movable frame. As well, the theorem of moment of momentum with respect to a vertical axis passing through C leads to IC ω = − Rρ1 .

(22.2.2)


432

We must have vO′ 2 = 0 , because the point O is moving along the axis Ox1 ; it results the constraint force R = M ( ωvO′ 1 + ωρ 1 ).

(22.2.3)

Taking into account (22.2.2) and denoting k2 = 1 +

IC , M ρ12

(22.2.4)

we can write the equations of motion in the form vO′ 1 = ω 2 ρ1 , vO′ 1 + k 2

ω ρ = 0; ω 1

eliminating the velocity vO′ 1 , we get k2

d ω + ω2 = 0 . dt ω

Multiplying by ω / ω and integrating, it results the differential equation of first order k2

( ωω )

2

+ ω 2 = k 2C 2 ,

(22.2.5)

where C is an integration constant. By a change of variable π π ω = kC cos ψ, ψ ∈ ⎡ − , ⎤ , ⎣ 2 2⎦

(22.2.5')

it results ψ 2 = C 2 cos2 ψ ; we choose the sign of the constant C in (22.2.5) so as to have ψ = C cos ψ . Hence, we can write Ct =

ψ

∫0

dα 1 1 + sin ψ = ln cosα 2 1 − sin ψ

and then sin ψ =

eCt − e −Ct 2 1 ; = tanh Ct , cos ψ = Ct = Ct −Ct −Ct cosh Ct e +e e +e

it results that t ∈ [ −∞, ∞ ] , because ψ ∈ [ − π / 2, π / 2] , the change of variable (22.2.5') covering the whole interval of time. Finally, we obtain ω = k ψ , wherefrom – by integration and by a convenient determination of the integration constant – we get ϕ = k ψ ; hence, the sleigh is rotating by the angle kπ for a variation of the time t from −∞ to ∞ . Finally,


ϕ = k arcsin ( tanh Ct ) .

433 (22.2.6)

Starting from vO′ 1 = − k 2

ω ρ = k 2 ρ1 tan ψψ = Ck 2 ρ1 sin ψ ω 1

and from xO′ 1 = vO′ 1 cos ϕ , xO′ 2 = vO′ 1 sin ϕ

and observing that dt = dψ /C cos ψ, ψ = ϕ / k , we can write the equations of motion of the point O with respect to the fixed frame in the form ψ

xO′ 1 = xO′01 + k 2 ρ1 ∫ tan ψ cos k ψ dψ , 0 ψ

xO′ 2 = xO′02 + k 2 ρ1 ∫ tan ψ sin k ψ dψ .

(22.2.7)

0

We can determine thus the trajectory of the point O in the plane Π for various values of the parameter k ; but we notice that for ρ1 = 0 (the projection of the mass centre C on the plane Π coincides with the contact point O of the knife with this plane) we have k = ∞ , so that the equation of motion can no more be written in the form (22.2.5). The equations (22.2.1'), (22.2.2) lead to ω = const , vO′ 1 = const ; hence, the point O has a uniform motion in a direction which is rotating with a constant angular velocity, describing thus a circle of radius r = ω / vO′ 1 .

Fig. 22.11 Trajectories of the contact point of the knife edge in the motion of a rigid solid on a horizontal plane

If ρ1 ≠ 0 , then the trajectory (22.2.7) has only a single point for ψ = 0 ; to determine the nature of the singularity, we consider expansions of the form xO′ 1 − xO′01 =

1 1 ρ k 2 ψ2 + ..., xO′ 2 − xO′02 = ρ1k 3 ψ 3 + ... , 2 1 3

(22.2.7')


434

which put in evidence the cuspidal point for ψ = 0 . The straight line xO′ 2 = xO′02 is a symmetry axis of this curve; observing that dxO′ 2 / dxO′ 1 = tan kψ and that ψ

(

lim ρ1k 2 ∫ tan ψ sin k ψ −

ψ →π /2

0

)

π dψ = α < ∞ , 2

we can state that xO′ 2 = xO′ 1 tan k

π +α 2

is an asymptote of the considered trajectory. The trajectories of the point O for k = 1 , k = 3 / 2 , k = 2 and k = 3 are represented in Fig 22.11a–d, respectively; the point O stops at the cuspidal point, at which the velocity vanishes.

22.2.1.2 Motion of a Sphere on a Rough Fixed Surface Let us consider a homogeneous sphere S of mass M , acted upon by forces the resultant of which passes through the centre O ; we assume that the sphere rolls on a perfect rough surface Σ . We use a movable frame of reference R of co-ordinate axes Ox1x 2 x 3 , so that Ox 3 passes through the contact point P between the sphere and the surface Σ . Let Ω and ω be the instantaneous angular velocities of the movable frame and of the sphere, respectively, v (v1 , v2 , 0) be the velocity of the point O with respect to a fixed frame and F and R (T1 ,T2 , N ) be the resultants of the given forces, applied at O , and of the constraint forces, applied at P , respectively (Fig. 22.12). The components of these quantities are taken along the axes of the frame of reference R .

Fig. 22.12 Motion of a sphere on a rough fixed surface

The equations of motion read M ( v1 − Ω3 v2 ) = F1 + T1 , M ( v2 − Ω3 v1 ) = F2 + T2 , M ( Ω1v2 − Ω2 v1 ) = F3 + N ,

(22.2.8)


435

I ( ω 1 − Ω3 ω2 + Ω2 ω3 ) = T2 r , I ( ω 2 − Ω1 ω3 + Ω3 ω1 ) = −T1r ,

(22.2.8')

ω 3 − Ω2 ω1 + Ω1 ω2 = 0,

where I is the moment of inertia of the sphere with respect to a diameter of it; the conditions of rolling without sliding are given by v1 − r ω2 = 0, v2 + r ω1 = 0 .

(22.2.9)

Eliminating the constraint forces T1 and T2 and the angular velocities ω1 and ω2 , it results r2 I F1 + Ω1r ω3 , I + Mr 2 I + Mr 2 r2 I v2 + Ω3 v1 = F2 + Ω2 r ω3 . I + Mr 2 I + Mr 2 v1 − Ω3 v2 =

(22.2.10)

These equations put in evidence the motion of the centre O of the sphere, the trajectory of which is on the surface Σ ′ , formed by the extremities of the normals, of length r , to the surface Σ ; the point O is acted upon by the given forces, reduced in the ratio Mr 2 /(I + Mr 2 ) , and by the forces I I M⎡ Ω1r ω3 + Ω3 v2 ⎤ , M ⎡ Ω2 r ω3 − Ω3 v1 ⎤ , ⎣⎢ I + Mr 2 ⎦⎥ ⎣⎢ I + Mr 2 ⎦⎥

respectively. We choose the axes Ox1 and Ox 2 along the lines of curvature of the surface Σ ′ , of radii of curvature of the normal sections to the surface ρ1 and ρ2 , respectively; in this case Ω1 =

v2 v , Ω2 = 1 . ρ2 ρ1

(22.2.11)

Meusnier’s theorem (4.1.19) leads to Ω3 =

v1 v tan θ1 + 2 tan θ2 . ρ1 ρ2

(22.2.11')

Taking into account the third equation (22.2.8'), it results ω 3 =

v1v2 ⎛ 1 1 ⎞ − . ⎜ r ⎝ ρ2 ρ1 ⎟⎠

(22.2.11'')

Let xO′ 1 , xO′ 2 , xO′ 3 be the co-ordinates of the point O ; the velocities v1 and v2 can be obtained, in this case, starting from the equation of the surface, as functions of


436

xO′ 1 , xO′ 2 , xO′ 3 , by decomposing the velocity along the axes of the frame of reference R . Eliminating the quantities v1 , v2 , Ω1 , Ω2 and Ω3 by means of the relations (22.2.10), (22.2.11), and (22.2.11'), we obtain three equations for the unknowns Ω1 , Ω2 , Ω3 and ω3 and their derivatives with respect to time; together with the equation of the surface Σ ′ , we have thus at our disposal a system of four equations, sufficient to specify the motion of the sphere. Obviously, in various particular cases one obtains important simplifications of calculation. For instance, in case of the rolling of a sphere on a plane we have Ω1 = Ω2 = 0 and, assuming that the frame of reference R is chosen so that Ω3 = 0 , it results v1 =

r2 r2 F , v = F2 . 1 2 I + Mr 2 I + Mr 2

(22.2.12)

Thus, the motion of the mass centre of the sphere does not depend on the component ω3 of the angular velocity ω , which is constant. As a matter of fact, this problem has been considered in Sect. 17.1.2.7 too. Other interesting particular cases are these in which the surface Σ is a sphere, a cylinder or a cone or – more general – a surface of rotation.

22.2.2 Lagrange’s Equations In what follows, we take again the equations with multipliers of Lagrange; as well, we establish Lagrange’s equations in quasi-co-ordinates.

22.2.2.1 Lagrange’s Equations with Multipliers In the frame of Lagrangian mechanics, one can make a systematic study of the motion of a non-holonomic mechanical system by means of Lagrange’s equations with multipliers (see Sect. 18.2.2.2) m d ⎛ ∂T ⎞ ∂T − = Q + ∑ λk akj , j = 1,2,..., s , ⎜ ⎟ dt ⎝ ∂q j ⎠ ∂q j k =1

(22.2.13)

where the last sums correspond to the components Rj , j = 1, 2,..., s , of the constraint generalized force, a study in this direction being made by Ed. Routh. To determine the s + m unknowns (the functions q j = q j (t ), j = 1, 2,..., s , and the parameters λk , k = 1,2,..., m ), we use the s Lagrange’s equations with multipliers and the m < s non-holonomic constraint relations (18.2.9'), which can be – conveniently – written in the form akj q j = −ak 0 , k = 1,2,..., m .

(22.2.14)

Applying the methodology in Sect. 18.2.3.4 to the equations (22.2.13), one obtains the relation


d ( T − T0 ) = Q j q j + Rj q j − T , dt 2

437 (22.2.15)

analogue to the relation (18.2.60'). We notice that Rj q j =

m

∑ λk akj qj

k =1

m

= − ∑ λk ak 0 , k =1

(22.2.16)

where we took into account (22.2.14). Hence, in case of catastatic constraints ( ak 0 = 0 , k = 1, 2,..., s ) we have Rj q j = 0 , so that the relation (22.2.15) is reduced to the relation (18.2.60'). We can thus state that, in case of non-holonomic and catastatic constraints, one obtains a first integral of Painlevé of the form (18.2.61), in the same conditions in which this is obtained in case of holonomic (in general, non-catastatic) constraints. As well, in case of a non-holonomic and catastatic mechanical system, for which the generalized forces are quasi-conservative, assuming a simple quasi-potential in the form (18.2.20), we can introduce the kinetic potential (18.2.34), obtaining the relation (18.2.62); if the kinetic potential L does not depend explicitly on time ( L = 0 ), we find again Jacobi’s first integral (18.2.63). Obviously, in the condition of some catastatic constraints (and only in such a condition) one can obtain all types of first integrals mentioned in Sect. 18.2.3.4 (e.g., a first integral of Jacobi type). But we must notice that this method of calculation is not completely satisfactory. Indeed, because the generalized co-ordinates are not independent, one must take into account the m constraint relations (22.2.14). But these relations contain generalized velocities, so that we cannot express m dependent generalized co-ordinates as functions of n − m independent generalized co-ordinates ; one cannot thus eliminate these co-ordinates between the respective relations and the equations (22.2.13). To realize this and to remove Lagrange’s multipliers from the equations of motion, one can introduce quasi-co-ordinates instead of generalized co-ordinates.

22.2.2.2 Lagrange’s Equations in Quasi-co-ordinates We have introduced in Sect. 22.1.2.2 the quasi-co-ordinates by the relations (22.1.17) or (22.1.18) or by the relations (22.1.18') or (22.1.18''), for which one can write the relations of transposition (22.1.24). As one can see, the quasi-co-ordinates allow a generalization of Lagrange’s equations, by unifying their form in case of non-holonomic systems. To put in evidence the importance of this generalization, from the practical as well as from the formal point of view, it is sufficient to mention that the choice of convenient unknown parameters (generalized co-ordinates and quasi-co-ordinates which determine the motion) plays a very important rôle in the study of some particular mechanical problems. We mention thus the problem of motion of the rigid solid with a fixed point, which has been formulated and studied by Euler in quasi-co-ordinates, although he did not use this motion. One can mention also other researches where the velocities of the points of a mechanical system in motion have been put in evidence (e.g., the non-holonomic problem of rolling of a non-homogeneous sphere on a plane, considered by Chaplygin), using the quasi-co-ordinates. But only at the beginning of last century the generalized co-ordinates and the kinematic characteristics have lead to a unified concept, that of

438


quasi-co-ordinates. In 1904, G. Hamel has deduced Lagrange’s equations in quasi-co-ordinates (which he called Lagrange–Euler equations), applicable to holonomic as well as to non-holonomic mechanical systems. Let be a mechanical system S definite by the generalized co-ordinates q j and by the quasi-co-ordinates π j , j = 1, 2,..., s , linked by the relations (22.1.18) or (22.1.18'), where we make αi 0 = 0 , hence βk 0 = 0, i , k = 1, 2,..., s (catastatic case) too; we can write π i = αij q j , q j = β jk π k , αij β jk = δik , i , j , k = 1,2,..., s ,

(22.2.17)

where αij = αij (q1 , q2 ,..., qs ), β jk = β jk (q1 , q2 ,..., qs ), i , j , k = 1, 2,..., s . Observing that the relation dπi = ( ∂πi / ∂q j )dq j = αij dq j leads to ∂πi / ∂q j = αij and that analogously we have ∂q j / ∂πk = β jk , we obtain ∂q j ∂q j ∂π i ∂π = i = αij , = = β jk , i , j , k = 1,2,..., s . ∂q j ∂q j ∂π k ∂πk

(22.2.18)

Thus, for any function f = f (q1 , q2 ,..., qs ; t ) or g = g (q1 , q2 ,..., qs ; t ) , the relations ∂f ∂f ∂q j ∂f ∂g ∂g = = β jk , = β jk , k = 1, 2,..., s , ∂πk ∂q j ∂πk ∂q j ∂π k ∂q j

(22.2.19)

take place; as well, for any function f = f ( π1 , π2 ,..., πs ; t ) or g = g ( π1 , π 2 ,..., π s ; t ) we have ∂f ∂f ∂g ∂f = αij , = αij , j = 1, 2,..., s . ∂q j ∂π j ∂q j ∂π j

(22.2.19')

Taking into account (22.1.22'), we can write the relation (18.2.27'), corresponding to the application of the principle of the virtual work (which led us to Lagrange’s equations (18.2.29) for holonomic constraints), in the form

⎡ d ⎛ ∂T ⎞ ∂T ⎢ dt ⎜ ∂q ⎟ − ∂q − Q j j ⎣ ⎝ j ⎠

⎤ ⎥ β jk δπk = 0 ; ⎦

because the variations δπk of the quasi-co-ordinates are arbitrary, it results

⎡ d ⎛ ∂T ⎞ ∂T ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q − Q j ⎥ β jk = 0, k = 1, 2,..., s . j ⎣ ⎝ j ⎠ ⎦

(22.2.20)

If we denote by T * the function which is obtained by replacing in T the generalized velocities q j by the quasi-velocities π k , we have


439

T (q1 , q2 ,..., qs ; q1 , q2 ,..., qs ) = T (q1 , q2 ,..., qs ; β1k π k , β2 k π k ,..., βsk π k )

= T * (q1 , q2 ,..., qs ; π1 , π 2 ,..., πs ) = T * (q1 , q2 ,..., qs ; α1 j q j , α2 j q j ,..., αsj q j ).

Starting from here and using the relations (22.2.17), (22.2.18), (22.2.18’), and (22.2.19'), we can write β jk

d ⎛ ∂T ⎞ d⎛ ∂T ⎞ ∂T ∂β jk d ⎛ ∂T * ⎞ ∂T * ∂αij β jk qi = β β π , = − + ⎜ ⎟ ⎜ ⎟ ∂q j ⎠ ∂q j ∂qi dt ⎝ ∂q j ⎠ dt ⎝ dt ⎝⎜ ∂π k ⎠⎟ ∂π i ∂qm jk ml l ⎛ ∂T * ∂T * ∂αim ⎞ ∂T * ∂T * ∂αim ∂T β jk qm ⎟ = β β π ; = β jk ⎜ + + ∂q j ∂π i ∂q j ∂πk ∂π i ∂q j jk ml l ⎝ ∂q j ⎠

replacing in (22.2.20), it results

∂α ⎞ ∂T * d ⎛ ∂T * ⎞ ∂T * ⎛ ∂αij π = Q j β jk . − + β jk βml ⎜ − im ⎟ ⎜ ⎟ dt ⎝ ∂π k ⎠ ∂πk ∂q j ⎠ ∂π i l ⎝ ∂qm Introducing the generalized forces Qk* = Q j β jk , k = 1, 2,..., s ,

(22.2.21)

corresponding to the virtual variations δπk , and using the notations (22.1.23), we get the equations * d ⎛ ∂T * ⎞ ∂T * j ∂T − + γ π = Qk* , lk dt ⎜⎝ ∂π k ⎟⎠ ∂πk ∂π i l

which we rewrite in the form d ⎛ ∂T * ⎞ ∂T * ∂T * π = Qk* , j = 1, 2,..., s ; − + γkjj ⎜ ⎟ dt ⎝ ∂π j ⎠ ∂π j ∂π i k

(22.2.22)

by analogy with Lagrange’s equations (18.2.29), we have thus obtained Lagrange’s equations in quasi-co-ordinates, in catastatic, hence scleronomic case (Hamel’s equations). One can show that, in the general non-holonomic and rheonomous case, the equations (22.2.22) are completed in the form d ⎛ ∂T * ⎞ ∂T * ∂T * ∂T * π k + γ ij − + γkji = Qk* + Rj* , j = 1,2,.., s , ⎜ ⎟ dt ⎝ ∂π j ⎠ ∂π j ∂π i ∂π i

(22.2.22')

where, starting from the constraint generalized forces Rj (see Sect. 22.2.2.1), we introduce also the generalized forces Rk* = Rj β jk , k = 1, 2,..., s ,

(22.2.21')


440

the coefficients γ ij being given by (22.2.23); these are the Boltzmann–Hamel equations. As well, the quantities p *j =

∂T * , j = 1, 2,..., s , ∂π j

(22.2.23)

are the quasi-momenta corresponding to the quasi-co-ordinates. If a virtual variation δπ1 = δπ2 = ... = δπ j −1 = 0, δπ j ≠ 0, δπ j +1 = ... = δπs = 0 corresponds to a rotation about an instantaneous axis of rotation, then the generalized forces Q j* and the quasi-momenta p *j

are, as usual, the moment of the force and the moment of

momentum, respectively, with respect to the axis. If π1 , π2 ,..., πs are generalized co-ordinates (true co-ordinates), then the relations of momentum (22.2.17) are integrable, so that the coefficients γkji vanish, while the equations (22.2.22) become the equations (18.2.29) of Lagrange. The expression (18.2.15), (18.2.15'), (18.2.15'') and (18.2.15''') of the kinetic energy allows to write T * in the form (in holonomic and catastatic, hence scleronomic case) T* =

1 * g π π , 2 ij i j

(22.2.24)

where we have introduced the notations gij* = gkl βki βlj = g *ji ,

(22.2.25)

by means of the relations (22.2.17). We calculate thus ∂T * / ∂π j = gij* π i , and then ∂g *jk d ⎛ ∂T * ⎞ = g *jk π k + π π , ⎜ ⎟ ∂πl k l dt ⎝ ∂π j ⎠ ∂T * 1 ∂gkl* = π π ; ∂π j 2 ∂π j k l

observing that the product π k πl is symmetric with respect to the indices k and l , it results * ∂g *jk ∂g *jl 1 ⎛ ∂g jk + π k πl = ⎜ ∂πl ∂πk 2 ⎜⎝ ∂πl

⎞ ⎟⎟ π k πl , ⎠

so that we can write the equations (22.2.22) in the form (we introduce the generalized forces Rj* , corresponding to a scleronomic and non-holonomic mechanical system) g *jk πk + ( [ kl , j ] + γkji gli* ) π k πl = Q j* + Rj* , j = 1, 2,..., s ,

(22.2.26)


441

where we have introduced Christoffel’s symbols of first kind [ kl , j ] =

* ∂g *jl ∂g * 1 ⎛ ∂g jk + − kl ⎜⎜ ∂πk ∂π j 2 ⎝ ∂πl

⎞ ⎟⎟ , j , k , l = 1, 2,..., s . ⎠

(22.2.27)

We have thus obtained a system of s linear equations in the quasi-velocities π k , k = 1, 2,..., s . Let g * jk = g * kj be the algebraic complement of the element g * jk in the determinant

g * = det ⎣⎡ g *jk ⎦⎤ ; the normalized algebraic complement will be g * jk = g * kj = g * jk / g * . We notice that the relations det ⎡⎣ g * jk ⎤⎦ =

1 g*

, g *jk g * mj = δm k ,

(22.2.28)

where δm k is Kronecker’s symbol, take place. Multiplying the system of equations (22.2.26) by g *mj and summing with respect to j , it results (we replace then the free index m by j ) ⎛ ⎧⎪ j ⎫⎪ j ⎞ * * πj + ⎜⎜ ⎨ ⎬ + γkl ⎟⎟ π k πl = Q j + Rj , j = 1, 2,..., s , k l ⎪ ⎪ ⎝⎩ ⎭ ⎠

(22.2.29)

where we have introduced Christoffel’s symbols of second kind, definite by ⎧⎪ j ⎫⎪ * mj ⎨ ⎬ = [ kl , m ] g , j , k , l = 1, 2,..., s , k l ⎪⎭ ⎩⎪

(22.2.27')

as well as the notations γklm = γkji gli* g * mj , m , k , l = 1, 2,..., s ,

(22.2.30)

Qm* = Q j* g * mj , Rm* = Rj* g * mj , m = 1, 2,..., s .

(22.2.30')

The system of equations (22.2.29) represents thus the normal form of Hamel’s equations, in which the quasi-accelerations are expressed as functions of quasi-velocities and quasi-co-ordinates; this form of the equations (analogue to the form (18.2.47') of Lagrange’s equations) can be always obtained, because g * ≠ 0 . In case of a non-holonomic mechanical system with m < s catastatic constraints of the form as − m + k , j q j = 0, k = 2,..., m ,

(22.2.31)

hence with s − m degrees of freedom, it is often convenient to use quasi-co-ordinates, so that m quasi-co-ordinates (in case of the notations used above, these


442

quasi-co-ordinates are the latter ones) do vanish, if we take into account the relations (22.2.31); we assume thus that these last m quasi-co-ordinates are linked to the generalized velocities by the relations πs − m + k = as − m + k , j q j , k = 1, 2,..., m ,

(22.2.32)

where the matrix ⎡⎣as − m + k , j ⎦⎤ is of rank m , the first s − m quasi-co-ordinates (which, in particular, can be generalized co-ordinates) being connected to the generalized velocities by the arbitrary linear relations π k = αkj q j , k = 1, 2,..., s − m ,

(22.2.33)

where the matrix ⎡⎣ αkj ⎦⎤ is of rank s − m . In concordance with the relations (22.2.32), we can write δπs − m + k = as − m + k , j δq j , k = 1, 2,..., m ,

(22.2.32')

and, taking into account the constraint relations (22.2.31), it results that these variations vanish δπs − m + k = 0, k = 1, 2,..., m .

(22.2.32'')

In this case, the last m terms vanish in Lagrange’s equations in quasi-co-ordinates (22.2.22), written in the form of the principle of virtual work * ⎡ d ⎛ ∂T * ⎞ ∂T * i ∂T * ⎤ ⎢ dt ⎜ ∂π ⎟ − ∂π + γkj ∂π π k − Q j ⎥ δπ j = 0 ; j ⎠ j i ⎣ ⎝ ⎦

(22.2.34)

taking into account that the first variations δπ j are independent, the equation (22.2.34) leads to the system of equations (which does not contain the constraint generalized forces, as in case of Lagrange’s equations with multipliers, e.g., in case of the equations (22.2.22'))

d ⎛ ∂T * ⎞ ∂T * s − m i ∂T * − + ∑ γkj π = Q j* , j = 1, 2,..., s − m , dt ⎜⎝ ∂π j ⎟⎠ ∂π j ∂π i k k =1

(22.2.35)

with the notations ∂α ⎞ ⎛ ∂α γkji = ⎜ il − im ⎟ βml βlj , i = 1, 2,..., s , j , k = 1, 2,..., s − m . q ∂ ∂ql ⎠ ⎝ m

We notice that the matrix ⎡⎣ β jk ⎤⎦ is the inverse of the matrix [ αik ] .

(22.2.35')


443

The system of s − m equations (22.2.35), together with the m constraint relations (22.2.31) and with the s − m linear relations (22.2.33), form the 2s − m equations of motion for the non-holonomic mechanical system in quasi-co-ordinates. Associating a corresponding number of initial conditions, we can determine the 2s − m unknown functions of time q1 , q2 ,..., qs , π 1 , π 2 ,..., πs − m . We mention that the kinetic energy T * is, in general, a function of all the s quasi-velocities π i , the coefficients γkji depending on the generalized co-ordinates q1 , q2 ,..., qs .

22.2.3 Applications In what follows, we give firstly some simple applications: the motion of the skate and the motion of a circular disc on a plane; we study then the motion of a sphere on a horizontal plane and the motion of a carriage with two or four wheels

22.2.3.1 Motion of a Skate The motion of a rigid skate on the ice plane has been considered in Sec. 13.2.2.6 (see Fig. 3.16). Let x1 , x 2 be the co-ordinates of the middle of the skate (modelled as a segment of a line of length 2l ) with respect to a system of axes Ox1x 2 in the ice plane and let θ be the angle made by the skate with the Ox1 -axis. The constraint relation is of the form tan θdx1 − dx 2 = 0 ;

(22.2.36)

it is non-holonomic (and scleronomic), as it has been shown in Chap. 3, Subsec. 2.2.6, by means of the Theorem 3.2.2 of Frobenius. As a matter of fact, this property can be put in evidence on another way too. Indeed, assuming that the constraint is a geometric one, we can find a relation of the form f (x1 , x 2 , θ ) = 0 , the differential consequence of which is ( ∂f / ∂x1 )dx1 + ( ∂f / ∂x 2 )dx 2 + ( ∂f / ∂θ )dθ = 0 ; taking into account (22.2.36), it results ⎛ ∂f + ∂f tan θ ⎞ dx + ∂f dθ = 0 . ⎜ ∂x ⎟ 1 ∂θ ⎝ 1 ∂x 2 ⎠

The displacements dx1 (as well as dx 2 ) and dθ are arbitrary, so that ∂f ∂f ∂f + = 0; tan θ = 0, ∂x1 ∂x 2 ∂θ

hence, f does not depend on θ , while the first relation, where θ is arbitrary, leads to ∂f / ∂x1 = 0 and ∂f / ∂x 2 = 0 . In this case, the function f depends of no one of the three arguments, so that it cannot represent a finite connection between x1 , x 2 and θ . The number of the generalized co-ordinates cannot be made less than three, but these ones cannot have a free variation, the mechanical system being thus non-holonomic


444

with two degrees of freedom. Another demonstration of the non-holonomy of the considered mechanical system starts from the fact that, for a position ( x1 , x 2 ) of the middle of the skate, the relation f (x1 , x 2 , θ ) = 0 leads to a certain value θ1 of the angle θ , which is not true, as we will show. Thus, if we start from the position ( x10 , x 20 ) with the value θ0 and if we reach the position ( x1 , x 2 ) on two different arcs of curve,

i.e. x 2 = f1 (x1 ) and x 2 = f\2 (x1 ), x1 ∈ [ x10 , x1 ] , respectively, then one sees that we

reach the final position with two different angles θ1 and θ2 , θ1 ≠ θ2 , arbitrarily taken; the functions f1 and f2 must satisfy the conditions f1 ( x10 ) = x 20 , f1 ( x1 ) = x 2 , f1′(x10 ) = tan θ0 , f1′(x1 ) = tan θ1 , f2 ( x10 ) = x 20 , f2 ( x1 ) = x 2 , f2′(x10 ) = tan θ0 , f2′( x1 ) = tan θ2 .

The angle θ0 can be equal to θ1 or to θ2 ; but we have supposed that θ1 ≠ θ2 . One can find easily pairs of functions f1 and f2 which satisfy the above conditions, e.g. f2 ( x1 ) = f1 (x1 ) + ( x1 − x10

where the functions

)2 ( x1

− x1 ) g (x1 ), g (x1 ) =

tan θ2 − tan θ1

( x1

− x10

)2

,

f1 ( x1 ), f2 (x1 ), g1 (x1 ), x 1 ∈ [ x10 , x 1 ] are continuous; the

arbitrary function g (x1 ) must satisfy the mentioned condition. The problem is thus reduced to the construction of a function f1 ( x1 ) which must satisfy the conditions imposed to it. Because the condition (22.2.36) maintains its form after a translation of vector (x1 − x10 , x 2 − x 20 ) , we can assume, for the sake of simplicity, that x10 = x 20 = 0 and we take x f1 (x1 ) = [ x1 ( tan θ1 + tan θ0 ) − 2 x 2 ] ⎛⎜ 1 ⎞⎟ ⎝ x1 ⎠

3

2

x − [ x1 ( tan θ1 + 2 tan θ0 ) − 3 x 2 ] ⎛⎜ 1 ⎞⎟ + x1 tan θ , ⎝ x1 ⎠ 1 g (x1 ) = 2 ( tan θ2 − tan θ1 ) ; x1

hence, the constraint relation (22.2.36) is a kinematic one. Using the generalized co-ordinates q1 = x1 , q2 = x 2 , q 3 = θ in the space Λ3 , the kinetic energy is given by T* =

1 1 M ( q12 + q22 ) + Iq32 , 2 2

(22.2.37)


445

where M is the mass of the skate and I = Ml 2 / 3 is the moment of inertia of it with respect to its middle. Obviously, the generalized forces vanish ( Q j = 0, j = 1, 2, 3 ); comparing the constraint relation (22.2.36) with (22.2.14), we notice that m = 1 , a10 = 0, a11 = tan θ , a12 = −1, a13 = 0 , the constraint generalized forces being R1 = λa11 , R2 = − λ , R3 = 0 (we denoted λ1 = λ ). Lagrange’s equations with multipliers (22.2.13) will be of the form Mq1 = λ tan θ , Mq2 = − λ ,

MI 2 q = 0 ; 6 3

(22.2.38)

we associate the initial conditions (at the moment t = 0 ) q1 (0) = q2 (0) = q 3 (0) = 0, q1 (0) = v0 , q2 (0) = 0, q3 (0) = ω0 .

(22.2.38')

The third equation (22.2.38) leads to θ (t ) = q 3 (t ) = ω0t .

(22.2.39)

Eliminating the generalized co-ordinate q2 and the multiplier λ between the relations (22.2.36) and the first two equations (22.2.38), we obtain q1 + ω0q1 tan ω0t = 0 , wherefrom, by integration, v0 sin ω0t ; ω0

(22.2.39')

v0 ( 1 − cos ω0t ) , ω0

(22.2.39'')

x1 (t ) = q1 (t ) =

we get then x 2 (t ) = q2 (t ) =

Lagrange’s multiplier being given by λ = − Mv0 ω0 cos ω0t .

(22.2.39''')

We notice thus that the middle of the skate has a uniform motion, of velocity v0 , on the circle x12 + ( x 2 − R )2 = R2 ,

(22.2.40)

of radius R = v0 / ω0 . An analogous study can be made for the motion of a ski. We can introduce also the quasi-co-ordinates π1 , π2 , π3 by the relations dπ1 = cos θdx1 + sin θ dx 2 = vdt , δπ1 = cos θδx1 + sin θδx 2 , dπ2 = − sin θdx1 + cos θ dx 2 , δπ2 = − sin θδx1 + cos θδx 2 , dπ3 = dθ , δπ3 = δθ ,

(22.2.41)


446 wherefrom

dx1 = cos θdπ1 − sin θ dπ2 , δx1 = cos θδπ1 − sin θδπ2 , dx 2 = sin θ dπ1 + cos θ dπ2 , δx 2 = sin θ δπ1 + cos θ δπ2 ,

(22.2.41')

dθ = dπ3 , δθ = δπ3 .

If we observe that dπi = αij dq j , δπi = αij δq j , dq j = β jk dπk , δq j = β jk δπk , i , j , k = 1, 2, 3 , then we obtain the matrices

[ αij ]

⎡ cos θ sin θ 0 ⎤ ⎡ cos θ − sin θ 0 ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ − sin θ cos θ 0 ⎥ , ⎡⎣ β jk ⎤⎦ = ⎢ sin θ cos θ 0 ⎥ ; ⎢ ⎥ ⎢ ⎥ 0 1 ⎥⎦ 0 1 ⎥⎦ ⎢⎣ 0 ⎢⎣ 0

1 1 2 2 = − γ 32 = −1, γ31 = − γ13 = −1 , the thus, using the formulae (22.1.23), it results γ23

other 23 components γkji vanishing. The transposition relations (22.1.23') become dδπ1 − δdπ1 = cos θ (dδx1 − δdx1 ) + sin θ (dδx 2 − δdx 2 ) − (dπ2 δπ3 − dπ3 δπ2 ), dδπ2 − δdπ2 = − sin θ (dδx1 − δdx1 ) + cos θ (dδx 2 − δdx 2 ) − (dπ3 δπ1 − dπ1 δπ3 ), dδπ3 − δdπ3 = dδθ − δdθ .

(22.2.42) Lagrange’s equations in quasi-co-ordinates (22.2.22') become (with γ ij = 0 ) d ⎛ ∂T * ⎞ ∂T * ∂T * π = Q1* + R1* , − − dt ⎜⎝ ∂π1 ⎟⎠ ∂π1 ∂π 2 3 d ⎛ ∂T * ⎞ ∂T * ∂T * π = Q2* + R2* , − + dt ⎜⎝ ∂π 2 ⎟⎠ ∂π2 ∂π1 3 d ⎛ ∂T * ⎞ ∂T * ∂T * ∂T * π π = Q3* + R3* . − + − 1 dt ⎜⎝ ∂π 3 ⎟⎠ ∂π3 ∂π 2 ∂π 1 2

Taking into account (22.2.41'), the kinetic energy (22.2.37) will be given by T* =

1 1 M ( π12 + π 22 ) + I π 32 . 2 2

(22.2.37')

The generalized forces being equal to zero ( Q1 = Q2 = Q3 = 0 ), from (22.2.21) it results

Q 1* = Q 2* = Q 3* = 0 ; starting from the constraint generalized forces

R1 = λ tan θ , R2 = − λ , R3 = 0 , the formulae (22.2.21') lead to R1* = R3* = 0 ,


R2* = − λ / cos θ .

Observing

that

447

∂T * / ∂π i = M π i , i = 1, 2, 3, ∂T * / ∂π j = 0 ,

j = 1, 2, 3 , one obtains the Boltzmann–Hamel equations

π1 − π 2 π 3 = 0, M ( π2 + π1 π 3 ) = −

λ , π = 0 , cos θ 3

(22.2.43)

which lead to the same solution (22.2.39) – (22.2.39'''), with the initial conditions (22.2.38'). Indeed, we get π1 = v0t , π2 = 0, π3 = ω0t ;

(22.2.43')

as well, we notice that π3 = 0 is a true co-ordinate.

22.2.3.2 Motion of a Heavy Circular Disc on a Plane Let be a heavy homogeneous circular disc of mass M and of radius R , which rolls on the plane Π , in which are situated the axes O ′x1′ , O ′x 2′ of the fixed frame of reference O ′x1′x 2′ x 3′ , considered in Sect. 22.1.1.1; the position of the disc is specified by five parameters, i.e.: x1′ , x 2′ , ψ, θ and ϕ (see Fig. 22.1), linked by the non-holonomic and scleronomic constraint relations (22.1.3) or (22.1.3') (relations between four of the five parameters), in case of the slidingless rolling of the disc on the rough plane Π (case in which the relative velocity of the contact point I vanishes). Hence, the considered system has only three degrees of kinematic freedom. If we denote by O the centre of the disc (the centre of the movable frame Ox1x 2 x 3 , non-rigidly linked to the disc, of axes Ox1 along OI , Ox 2 in the plane of the disc, normal to OI and parallel to the plane Π and Ox 3 normal at O to the disc, so that the frame of reference be right-handed), we can write xO′ 1 = x1′ − R cos θ sin ψ, xO′ 2 = x 2′ + R cos θ cos ψ, xO′ 3 = R sin θ ,

introducing thus five generalized co-ordinates xO′ 1 , xO′ 2 , ψ, θ , ϕ ; differentiating with respect to time, we get xO′ 1 = x1′ + Rθ sin θ sin ψ − Rψ cos θ cos ψ , xO′ 2 = x2′ − Rθ sin θ cos ψ − Rψ cos θ sin ψ, xO′ 3 = Rθ cos θ ,

so that the constraint relations (22.1.3') read xO′ 1 + R ( cos θ cos ψψ − sin θ sin ψθ + cos ψϕ ) = 0, xO′ 2 + R ( cos θ sin ψψ − sin θ cos ψθ + sin ψϕ ) = 0.

(22.2.44)


448

The kinetic energy of the mass centre O , where we assume that the whole mass M is concentrated, with respect to the pole O ′ , is given by (1/ 2)M (xO′21 + xO′22 + xO′23 ) and the kinetic energy with respect to the pole O is given by T 0 = (1/ 2)[J ( ω 12 + ω 22 ) + I 3 ω 23 ] , where I 1 = I 2 = J and I 3 are central principal moments of inertia (J = MR2 / 4, I 3 = MR2 / 2) ; thus, taking into account the relations (5.2.35) between the vector ω and Euler’s angles, one obtains T′ =

1 1 M ( x O′21 + x O′22 ) + MR2 ⎡⎣ ( 1 + cos2 θ ) ψ 2 2 8 ⎤, + ( 1 + 4 cos2 θ ) θ2 + 2ϕ 2 +4 cos θψϕ ⎦

(22.2.45)

the potential energy being given by V = −U = MgR sin θ .

(22.2.45')

If we take the generalized co-ordinates in the order q1 = xO′ 1 , q2 = xO′ 2 , q 3 = ψ , q 4 = θ , q 5 = ϕ , by comparing the constraint relations (22.2.44) with (22.2.14), we get a10 = 0, a11 = 1, a12 = 0, a13 = R cos ψ cos θ , a14 = − R sin ψ sin θ , a15 = R cos ψ , a20 = 0, a21 = 0, a22 = 1, a23 = R sin ψ cos θ , a24 = R cos ψ sin θ , a25 = R sin ψ ;

starting from the given force F = F3 i 3′ , F3 = − Mg , and from the position vector rO′ = xO′ 1 i1′ + xO′ 2 i2′ + xO′ 3 i 3′ , we can calculate the given generalized forces Q1 = Q2 = Q3 = Q5 = 0,Q4 = − MgR cos θ . Lagrange’s equations with multipliers (22.2.13) are written in the form (for j = 1, 2, 3, 5 ) MxO′ 1 = λ1 , MxO′ 2 = λ2 ,

(22.2.46)

1 d MR ⎣⎡ ψ sin2 θ + 2 ( ψ cos θ + ϕ ) cos θ ⎦⎤ = ( λ1 cos ψ + λ2 sin ψ ) cos θ , 4 dt 1 d MR ( ψ cos θ + ϕ ) = λ1 cos θ + λ2 sin ψ ; 2 dt

(22.2.46')

instead of the equation corresponding to the index j = 4 , we use the first integral of the mechanical energy T ′ = − MgR sin θ + h , h = const .

(22.2.46'')

Multiplying the equations (22.2.46) by cos ψ and sin ψ , respectively, summing and taking into account the constraint relations (22.2.44), we get the relation


449

sin θ − d ( ψ cos θ + ϕ ) ⎤ ; λ1 cos ψ + λ2 sin ψ = MR ⎡ θϕ dt ⎣⎢ ⎦⎥

replacing the left member of this relation in the equations (22.2.46'), we obtain , 3 d ( ψ cos θ + ϕ ) = 2 ψθ sin θ , ψ sin θ = 2 θϕ dt

wherefrom sin θ

dψ d = 2ϕ , 3 ( ψ cos θ + ϕ ) = 2 ψ sin θ . dθ dθ

(22.2.47)

Eliminating ϕ between these last two equations, we obtain the non-linear differential equation d2 ψ dψ 10 + 3cot − ψ, 2 dθ 3 dθ

(22.2.48)

the unknown function being ψ = ψ ( θ ) ; by the change of independent variable u = cos2 θ , we get the differential equation of second order u (1 − u )

d2 ψ 1 dψ 5 + (1 − 5u ) − ψ = 0. 2 2 du 6 du

(22.2.48')

This equation is of the type of Gauss’s hypergeometric differential equation x (1 − x )y ′′ + [ γ − ( α + β + 1)x ] y ′ − αβy = 0 ,

(22.2.49)

with the general integral of the form y (x ) = AF ( α, β , γ ; x ) + Bx 1 − γ F ( α + 1 − γ , β + 1 − γ , 2 − γ ; x ) ,

(22.2.49')

where A and B are constant and F ( α, β , γ ; x ) is Gauss’s hypergeometric series F ( α, β , γ ; x ) = 1 +

α ( α + 1)...( α + n − 1)β ( β + 1)...( β + n − 1) n x , | x |< 1, n ! γ ( γ + 1)...( γ + n − 1) n =1 (22.2.49'') ∞

∑

In case of the equation (22.2.48') we have α + β = 3 / 2, αβ = 5 / 6 and γ = 1/ 2 , so that the general integral of this equation is given by

(

)

(

)

1 1 1 3 ψ = AF α, β , ; cos2 θ + BF α + , β + , ; cos2 θ cos θ , 2 2 2 2

(22.2.48'')


450

A and B being arbitrary constants. To study the equation (22.2.48), one can use also the change of independent variable (1 − 2v )2 = cos2 θ , which leads to the equation ( γ = 2 ) v (1 − v )

d2 ψ dψ 10 + 2(1 − 2v ) − ψ = 0, 2 dv 3 dv

(22.2.50)

the general integral of which is given by

(

)

(

)

(22.2.50')

)

(22.2.50'')

θ θ θ + B ′F α − 1, β − 1, 0; sin2 ψ = A′F α, β , 2; sin2 cos2 , 2 2 2

or by

(

)

(

θ θ θ + B ′′F α − 1, β − 1, 0; cos2 ψ = A′′F α, β , 2; cos2 cosec2 , 2 2 2

where

α + β = 3, αβ = 10 / 3 , while

A′, B ′, A′′, B ′′

are arbitrary constants.

Analogously, by means of the substitution u = sin θ one obtains an equation of Gauss with γ = 2 , the general integral of which is 2

ψ = A′′′F ( α, β , 2; sin2 θ ) + B ′′′F ( α − 1, β − 1, 0; sin2 θ ) cosec2 θ ,

(22.2.51)

with α + β = 3 / 2, αβ = 5 / 6, A′′′, B ′′′ being arbitrary constants. The first equation (22.2.47) allows now to express ϕ = ϕ ( θ ) by hypergeometric series too. Taking into account (22.2.44), we can express the kinetic energy (22.2.45) in the form T′ =

1 cos θ ⎤ ; MR2 ⎣⎡ ( 1 + 5 cos2 θ ) ψ 2 + 5 θ2 + 6ϕ 2 + 12 ψϕ ⎦ 8

(22.2.45'')

the conservation theorem (22.2.46'') leads to the equation

( 1 + 5 cos2 θ ) ψ 2

cos θ = 8 g (C − sin θ ), C = h , + 5 θ2 + 6ϕ 2 + 12 ψϕ R MgR (22.2.52)

which, after replacing ψ and ϕ , allows to obtain, by a quadrature, the time t as a function of the angle θ . One can thus calculate θ = θ (t ) and then determine the generalized co-ordinates xO′ 1 = xO′ 1 ( θ ), xO′ 2 = xO′ 2 ( θ ), ψ = ψ ( θ ) and ϕ = ϕ ( θ ) , by quadratures too; one obtains thus xO′ 1 = xO′ 1 (t ), xO′ 2 = xO′ 2 (t ), ψ = ψ (t ) and ϕ = ϕ (t ) . The kinetic energy and the potential energy being given by (22.2.45'') and (22.2.45'), respectively, one can write Lagrange’s kinetic potential L = T ′ + U in the form


451

L = MR2 ⎡⎢ 6 ( ψ cos θ + ϕ )2 + 5 θ2 + ψ 2 sin2 θ − sin θ ⎤⎥ , 8 R ⎣ ⎦ 8g

1

(22.2.53)

where the generalized co-ordinates are ψ = ψ (t ), θ = θ (t ), ϕ = ϕ (t ) . We obtain thus Lagrange’s equations d ⎡12 ( ψ cos θ + ϕ ) cos θ + 2 ψ sin2 θ ⎦⎤ = 0, dt ⎣ 8g 10 θ + 12 ( ψ cos θ + ϕ ) ψ sin θ − 2 ψ 2 sin θ cos θ + cos θ = 0, R d 12 ( ψ cos θ + ϕ ) = 0, dt

which lead to the first integrals 6 ( ψ cos θ + ϕ ) cos θ + ψ sin2 θ = a , a = const,

(22.2.54)

ψ cos θ + ϕ = b , b = const;

the first of these relations can be written also in the form 6b cos θ + ψ sin2 θ = a ,

(22.2.54')

while the second equation of Lagrange becomes 4g 5 θ + 6b ψ sin θ − ψ sin θ cos θ + cos θ = 0 . R

(22.2.54'')

Eliminating ψ between the relations (22.2.54') and (22.2.54''), we get 5 θ sin 3 θ + 6ab (1 + cos2 θ ) − (a 2 + 36b 2 ) cos θ +

4g sin 3 θ cos θ = 0 ; R

multiplying by 2θ and integrating, it result 5 θ2 − 6ab

( sincos θθ − ln tan θ2 ) + a sin+ 36θb 2

2

2

2

− 6ab

( sincos θθ + ln tan θ2 ) + 8Rg sin θ = c 2

or 5 θ2 + ( a 2 + 36b 2 − 12ab cos θ )

1 8g 1 + = c , c = const , 2 sin θ R sin θ

(22.2.55)

hence a first integral of the previous equation. This equation with separate variables can be integrated by a quadrature, obtaining the time t as a function of θ , and then θ = θ (t ) . Returning to the relation (22.2.54') and then to the second relation (22.2.54),


452

we get, successively, ψ = ψ (t ) and ϕ = ϕ (t ) . By means of the constraint relations (22.1.3'), we can calculate the co-ordinates x1′ and x 2′ too. The problem can be formulated also in the quasi-co-ordinates π1 , π2 , π3 , π4 , π5 defined by the relations

dπ1 = cos ψdx1 + sin ψdx 2 , dπ2 = dψ, dπ3 = dθ , dπ4 = dx1 + R cos ψdϕ, dπ5 = dx 2 + R sin ψdϕ ;

(22.2.56)

one can thus write the corresponding equations of Lagrange.

22.2.3.3 Motion of a Sphere on a Horizontal Plane Let be a heavy rigid sphere of radius R and mass M , which can have a rolling and pivoting motion on a fixed plane O ′x1′x 2′ , linked to the fixed frame of reference R ′ , the co-ordinates x1′ , x 2′ corresponding to the contact point I with this plane; we take the origin O of the movable frame R at the centre of the sphere, the positions of its axes with respect to the fixed frame being specified by Euler’s angles ψ , θ , ϕ . The motion of the sphere can be thus studied in the space Λ5 , the generalized co-ordinates being x1′ , x 2′ , ψ, θ and ϕ (Fig. 22.13).

Fig. 22.13 Motion of a sphere on a horizontal plane

Assuming that the sphere is rolling and pivoting without sliding on the fixed plane, we can equate to zero the velocity of the contact point JJG vI′ = vO′ + ω × OI = 0 ;

(22.2.57)

JJG ′ i ′j , xO′ 1 = x1′ , xO′ 2 = x 2′ , xO′ 3 = R , OI = − Ri 3′ , we can observing that rO′ = xOj

write the above condition in the form drO′ − R ( ω2′ i1′ − ω1′i2′ ) dt = 0 .

(22.2.57')


453

We project on the axes of the movable frame and find the constraint relations (by means of the formulae (5.2.35')) dx1′ = R ( sin ψdθ − sin θ cos ψdϕ ) , dx 2′ = − R ( cos ψdθ + sin θ sin ψdϕ ) .

(22.2.57'')

Let us assume that the equations (22.2.57'') admit an integrable combination; in this case, by integration, we get f ( x1′ , x 2′ , ψ, θ , ϕ ) = 0 . Taking into account the conditions (22.2.57''), the differential consequence ∂f ∂f ∂f ∂f ∂f dx ′ + dx ′ + dψ + dθ + dϕ = 0 ∂x1′ 1 ∂x 2′ 2 ∂ψ ∂θ ∂ϕ

must be identically satisfied. Hence, besides the condition ∂f / ∂ψ = 0 we will have ∂f ∂f ∂f + R ⎛⎜ sin ψ − cos ψ ⎞⎟ = 0, ′ ′ ∂θ ∂ ∂ x x ⎝ 1 ⎠ 2 ∂f ∂f ∂f − R sin θ ⎛⎜ cos ψ + sin ψ ⎞⎟ = 0. ′ ∂ϕ ∂x 2′ ⎝ ∂x1 ⎠

Differentiating the above relations with respect to ψ and taking into account ∂f / ∂ψ = 0 , we see that the brackets in these relations vanish, so that ∂f / ∂θ = ∂f / ∂ϕ = 0 ; we obtain, from (22.2.57''), also ∂f / ∂x1′ = ∂f / ∂x 2′ = 0 . It results that the function f does depend on no one of the five generalized co-ordinates. We can thus state that the considered rigid sphere represents a non-holonomic and scleronomic mechanical system with three degrees of freedom. Such a problem is put, e.g., in the study of motion of a billiard ball. We can write the constraint relation (22.2.57'') in the form x1′ − R ( θ sin ψ − ϕ sin θ cos ψ ) = 0, x2′ + R ( θ cos ψ + ϕ sin θ sin ψ ) = 0,

(22.2.57''')

the generalized co-ordinates being q1 = x1′ , q2 = x 2′ , q 3 = ψ , q 4 = θ , q 5 = ϕ ; by comparing the constraint relations (22.2.14) with (22.2.57'''), we get a10 = 0, a11 = 1, a12 = 0, a13 = 0, a14 = − R sin ψ, a15 = R sin θ cos ψ , a20 = 0, a21 = 0, a22 = 1, a23 = 0, a24 = R cos ψ, a25 = R sin θ sin ψ .

The kinetic energy is given by ( I 1 = I 2 = I 3 = (2 / 5)MR2 , formula (3.1.27))


454

1 1 M ( x1′2 + x2′2 ) + MR2 ( ω1′2 + ω2′2 + ω3′2 ) 2 5 1 2 cos θ ) ⎤ , = M ⎡ x1′2 + x2′2 + R2 ( ψ 2 + θ2 + ϕ 2 + 2 ψϕ 2 ⎣⎢ 5 ⎦⎥ T′ =

(22.2.58)

where we took into account the formula (5.2.35'). Assuming that the fixed plane is horizontal, we notice that the work of the own weight vanishes ( Q j = 0, j = 1, 2, 3, 4, 5 ); Lagrange’s equations with multipliers (22.2.13) will be Mx1′ = λ1 , Mx2′ = λ2 ,

d ( ψ + ϕ cos θ ) = 0 , dt

2 cos θ ) = − ( λ1 sin ψ − λ2 cos ψ ) , MR ( θ + ψϕ 5 2 d MR ( ϕ + ψ cos θ ) = ( λ1 cos ψ + λ2 sin ψ ) sin θ . 5 dt

(22.2.59) (22.2.59') (22.2.59'')

The first two equations (22.2.59) specify the motion of the centre O of the sphere in the plane x 3′ = R = const , the trajectory being rectilinear and the motion uniform; the third equation (22.2.59) leads to a first integral of the moment of momentum in the form (we take into account the formula (5.2.35')) ψ + ϕ cos θ = A, A +

5 KO′ 3 ′ = const , 2 MR2

(22.2.60)

where KO′ 3 ′ is the constant component of the moment of momentum along the O ′x 3′ -axis. Because x 3′ = const , it results a potential U = const , so that the conservation theorem of mechanical energy is reduced to T ′ = h , h = const , hence to x1′2 + x2′2 +

2 2 2 cos θ ) = 2h . R ( ψ + θ2 + ϕ 2 + 2 ψϕ 5 M

Taking into account the constraint relations (22.2.58), one can write the above first integral in the form 7 2 2 cos θ ) = 2h ; ( θ + ϕ 2 sin2 θ ) = 5 ( ψ 2 + ϕ 2 cos2 θ + 2 ψϕ 5 MR2

finally, by means of the first integral (22.2.60), the first integral of mechanical energy becomes θ2 + ϕ 2 sin2 θ = B 2 , B 2 =

5 ( 2h − AKO′ 3 ′ ) . 7 MR2

(22.2.61)

From the first two equations (22.2.59) and from the relation (22.2.57'''), it results


455

sin θ ) , λ1 sin ψ − λ2 cos ψ = MR ( θ + ψϕ − d ( ϕ sin θ ) ⎤ , λ1 cos ψ + λ2 sin ψ = MR ⎡ ψθ ⎢⎣ ⎥⎦ dt

so that we can eliminate Lagrange’s multipliers. Thus, the relation (22.2.59') leads to

sin θ = 0 , θ + ψϕ

(22.2.62)

while the relation (22.2.59'') leads to the relation 2 d − d ( ϕ sin θ ) ⎤ sin θ , ( ϕ + ψ cos θ ) = ⎡⎢⎣ ψθ ⎥⎦ 5 dt dt

which can be written in the form 2 d d ( ϕ + ψ cos θ ) = ⎡⎢⎣ ψ − dθ ( ϕ sin θ ) ⎤⎦⎥ sin θ 5 dθ

too, after simplifying by θ ≠ 0 . Replacing ϕ given by (22.2.60) and effecting the computations, we get dψ 3 + cos θ 2A , + ψ = dθ sin 2 θ sin 2 θ

wherefrom, by integration, it results ψ sin2 θ = A + C cos θ , C = const .

(22.2.63)

We have thus obtained three first integrals (22.2.60), (22.2.61) and (22.2.63), sufficient to determine the angular velocities ψ , θ and ϕ ; the integration constants A, B ,C are determined by imposing initial conditions for these velocities. Eliminating ψ and ϕ between (22.2.60), (22.2.61) and (22.2.63), it results

( C + A cos θ )2 θ2 = B 2 − , sin2 θ

(22.2.64)

wherefrom, by a quadrature, we obtain θ = θ (t ) ; the equations (22.2.63) and (2.2.60) lead then, successively, to ψ = ψ (t ) and ϕ = ϕ (t ) , by two other quadratures. Replacing ψ and ϕ in (22.2.62), one observes easily that (22.2.64) is a first integral of this differential equation. Let us assume now that the horizontal plane is rotating with a constant angular velocity Ω about the O ′x 3′ -axis. In this case, the constraint relations (22.2.58) are completed in the form


456

x1′ − R ( θ sin ψ − ϕ sin θ cos ψ ) + Ωx 2′ = 0, x2′ + R ( θ cos ψ + ϕ sin θ sin ψ ) − Ωx1′ = 0;

(22.2.65)

hence, the mechanical system remains non-holonomic and scleronomic, with three degrees of freedom. Besides the generalized co-ordinates q1 , q2 , q 3 , q 4 , q 5 introduced above, we denote q 6 = t too; the quasi-co-ordinates πi , i = 1, 2,..., 6 , will be specified by the differential relations dπ1 = − dx1′ + R ( sin ψdθ − sin θ cos ψdϕ ) − Ωx 2′ dt , dπ2 = dx 2′ + R ( cos ψdθ + sin θ sin ψdϕ ) − Ωx1′dt , dπ3 = ω3′ dt = dψ + cos θ dϕ , dπ4 = ω1′dt = cos ψdθ + sin θ sin ψdϕ ,

(22.2.66)

dπ5 = ω2′ dt = sin ψdθ − sin θ cos ψdϕ, dπ6 = dt ,

while the corresponding virtual variations will be given by δπ1 = −δx1′ + R ( sin ψδθ − sin θ cos ψδϕ ) , δπ2 = δx 2′ + R ( cos ψδθ + sin θ sin ψδϕ ) , δπ3 = δψ + cos θδϕ , δπ4 = cos ψδθ + sin θ sin ψδϕ,

(22.2.67)

δπ5 = sin ψδθ − sin θ cos ψδϕ , δπ6 = 0.

Solving the systems (22.2.66) and (22.2.67) with respect to the real and virtual generalized displacements, respectively, we obtain dx1′ = − dπ1 + Rdπ5 − Ωx 2′ dπ6 , dx 2′ = dπ2 − Rdπ4 + Ωx1′dπ6 , dψ = dπ3 − sin ψcotθdπ4 + cos ψcotθdπ5 , dθ = cos ψdπ4 + sin ψdπ5 , dϕ = sin ψcosecθdπ4 − cos ψcosecθdπ5 , dt = dπ6 ,

as well as

(22.2.66')


457

δx1′ = −δπ1 + Rδπ5 , δx 2′ = δπ2 − Rδπ4 , δψ = δπ3 − sin ψcotθδπ4 + cos ψcotθδπ5 ,

(22.2.67')

δθ = cos ψδπ4 + sin ψδπ5 , δϕ = sin ψcosecθδπ4 − cos ψcosecθδπ5 , δt = 0.

The relations (22.2.17) lead to the matrices

[ αij ]

[ βij ]

⎡ −1 ⎢ ⎢0 ⎢0 = ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢⎣ 0

⎡ −1 ⎢ ⎢0 ⎢0 = ⎢ ⎢0 ⎢ ⎢0 ⎢ ⎢⎣ 0

− R sin θ cos ψ − Ωx 2′ ⎤ ⎥ R cos ψ R sin θ sin ψ − Ωx1′ ⎥ 0 cos θ 0 ⎥ ⎥, cos ψ sin θ sin ψ 0 ⎥ ⎥ − sin θ cos ψ sin ψ 0 ⎥ ⎥ 0 0 1 ⎥⎦

0 0 R sin ψ 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0

− Ωx 2′ ⎤ ⎥ 0 −R − Ωx1′ ⎥ cos ψcotθ 0 ⎥ − sin ψcotθ ⎥; cos ψ sin ψ 0 ⎥ ⎥ sin ψcosecθ − cos ψcosecθ 0 ⎥ ⎥ 0 0 1 ⎥⎦ 0

R

the formulae (22.1.23) allow to calculate 1 1 1 1 1 1 = − γ 43 = R , γ26 = − γ62 = − Ω , γ 46 = − γ64 = RΩ , γ34 2 2 2 2 2 2 = − γ53 = − R, γ16 = − γ61 = Ω , γ56 = − γ65 = − RΩ , γ35 3 3 4 4 5 5 = − γ54 = 1, γ35 = − γ53 = −1, γ 34 = − γ 43 = 1. γ 45

The transposition relations (22.1.24') are given by dδπ1 − δdπ1 = R ( dπ3 δπ4 − dπ4 δπ3 ) − Ω dπ6 ( Rδπ4 − δπ2 ) , dδπ2 − δdπ2 = − R ( dπ3 δπ5 − dπ5 δπ3 ) + Ω dπ6 ( Rδπ5 − δπ1 ) , dδπ3 − δdπ3 = dπ4 δπ5 − dπ5 δπ4 , dδπ4 − δdπ4 = dπ5 δπ3 − dπ3 δπ5 , dδπ5 − δdπ5 = dπ3 δπ4 − dπ4 δπ3 , dδπ6 − δdπ6 = 0.

.

(22.2.68)


458 Lagrange’s Q j*

equations

in

quasi-co-ordinates

(22.2.22)

become

(we

have

= 0, j = 1, 2,..., 6 ) d ⎛ ∂T * ⎞ ∂T * ∂T * − − = 0, Ω ∂π 2 dt ⎜⎝ ∂π1 ⎟⎠ ∂π1

∂T * d ⎛ ∂T * ⎞ ∂T * − +Ω = 0, ⎜ ⎟ ∂π1 dt ⎝ ∂π 2 ⎠ ∂π2 ∂T * ∂T * ⎛ ∂T * ⎞ ∂T * d ⎛ ∂T * ⎞ ∂T * − − R⎜ π 4 − π 5 ⎟ + π 5 − π = 0, ⎜ ⎟ ∂π 2 ∂π 5 4 dt ⎝ ∂π 3 ⎠ ∂π3 ⎝ ∂π1 ⎠ ∂π 4 d ⎛ ∂T * ⎞ ∂T * ∂T * ( π 3 − Ω ) − − +R ⎜ ⎟ ∂π 1 dt ⎝ ∂π 4 ⎠ ∂π4 ∂T * d ⎛ ∂T * ⎞ ∂T * ( π 3 − Ω ) + − +R ⎜ ⎟ ∂π 2 dt ⎝ ∂π 5 ⎠ ∂π5

(22.2.69)

∂T * ∂T * π 5 + π = 0, ∂π 3 ∂π 5 3 ∂T * ∂T * π 4 + π = 0, ∂π 3 ∂π 4 3

(22.2.69')

∂T * ⎛ ∂T * ⎞ d ⎛ ∂T * ⎞ ∂T * − + − R Ω π π 5 ⎟ = 0, 4 ⎜ ⎟ ⎜ ∂π 2 dt ⎝ ∂π 6 ⎠ ∂π6 ⎝ ∂π 1 ⎠

where we noticed that π1 = π 2 = 0 (the constraint relations (22.2.65)) and π 6 = 1 . Taking into account (22.2.65), (22.2.66), we can express the kinetic energy in the form T* =

1 M ⎡ ( π − Rπ 5 + Ωx 2′ π 6 )2 + ( π 2 − Rπ 4 + Ωx1′π 6 )2 + i 2 ( π 32 + π 42 + π 52 ) ⎤⎦, 2 ⎣ 1 (22.2.58')

where we have introduced the gyration radius i given by i 2 = I / M . We can calculate ∂T * ∂T * = − M ( Rπ 5 − Ωx 2′ ) , = M ⎣⎡ ( R2 + i 2 ) π 4 − RΩx1′ ⎦⎤ , ∂π1 ∂π 4 ∂T * ∂T * = − M ( Rπ 4 − Ωx1′ ) , = M ⎡⎣ ( R2 + i 2 ) π 5 − RΩx 2′ ⎤⎦ , ∂π 2 ∂π 5 ∂T * ∂T * = Mi 2 π 3 , = M Ω ⎡⎣Ω ( x1′2 + x2′2 ) − R ( x1′π 4 + x1′π 5 ) ⎤⎦; ∂π 3 ∂π 6

as well, taking into account (22.2.19), we get ∂T * ∂T * = M ( Rπ 5 + Ωx 2′ ) , = MRΩ ( Rπ 5 − Ωx 2′ ) , ∂π1 ∂π4 ∂T * ∂T * = − M ( Rπ 4 − Ωx1′ ) , = − MRΩ ( Rπ 4 − Ωx1′ ) , ∂π2 ∂π5 ∂T * ∂T * = 0, = MRΩ 2 ( x 2′ π 4 − x1′π 5 ) . ∂π3 ∂π6


459

The third equation (22.2.69) leads to π3 = 0 , hence to π 3 = ω3′ = const ,

(22.2.70)

and the fourth and the fifth equation of this system lead to ( π 4 = ω1′ , π 5 = ω2′′ )

( R2 ( R2

+ i 2 ) ω 1′ − RΩ ( Rω2′ − Ωx 2′ ) = 0, + i 2 ) ω 2′ + RΩ ( Rω1′ − Ωx1′ ) = 0,

(22.2.70')

where we took into account the constraint relations (22.2.65); using, further, these relations and integrate them, we obtain

( R2 + i 2 ) ω1′ − RΩx1′ ( R2 + i 2 ) ω2′ − RΩx 2′

= C 1 ( R2 + i 2 ) , = C 2 ( R2 + i 2 ) ,

(22.2.71)

C 1 and C 2 being arbitrary constants. The same relations (22.2.65) allow to eliminate the components of the rotation vector, so that x1′ + Ωx 2′ = C 2 R , x2′ − Ωx1′ = C 1R ,

(22.2.72)

with the notation Ω = [ i 2 /(R2 + i 2 )]Ω ; there result the equations x1′ + Ω 2 x1′ = C 3 , C 3 = R ( C 2 − C 1Ω ) , x2′ − Ω 2 x 2′ = C 4 , C 4 = R ( C 1 + C 2 Ω ) ,

(22.2.72')

the integration constants being determined by initial conditions. Hence, in general, the centre of the sphere which moves on a rough horizontal plane, which is rotating with a constant angular velocity about a fixed axis, describes an ellipse, the parameters of which depend on the initial conditions and which is fixed with respect to the inertial frame of reference.

22.2.3.4 Motion of the Two-Wheeled Carriage Let us consider a Roman carriage, formed by a rigid solid S of mass M , supported at the points A1 and A2 on a horizontal plane O ′x1′x 2′ by two equal wheels, of radii R and masses m , which are independently rotating with the velocities ϕ 1 and ϕ 2 , about the centres C 1 and C 2 , respectively; the movable frame of reference has the pole at the middle of the distance between the two centres ( C 1C 2 = 2a ), the Ox1 -axis being along the line of the centres, which makes the angle θ with the O ′x1′ -axis, the Ox 2 -axis being horizontal, while the Ox 3 -axis is vertical and parallel to the O ′x 3′ -axis. The mass centre C of the car is on the Ox 2 -axis, so that OC = l (Fig. 22.14).

460


Let O be the projection of the pole O on the plane O ′x1′x 2′ of co-ordinates x1′ , x 2′ ; in this case, the variables q1 = x1′ , q2 = x 2′ , q 3 = θ , q 4 = ϕ1 , q 5 = ϕ2 can form a system of generalized co-ordinates.

Fig. 22.14 Motion of a two-wheeled carriage

Assuming that the wheels are rolling and pivoting without sliding on the fixed plane, we impose the condition that the velocities of the points of both wheels, which – at the considered moment – coincide with the points of contact A1 and A2 , respectively, do vanish. Thus, for the point A2 we can write JJJJJG JJJJJG JJJJG vA′ 2 = vC′ 2 + ωC 2 × C 2 A1 = vO′ + ω0 × OC 2 + ωC 2 × C 2 A2 = 0 ,

where JJJJJG JJJJG vO′ = x1′i1′ + x2′ i2′ ,OC 2 = a ( cos θi1′ + sin θi2′ ) ,C 2 A2 = − Ri 3′ , ω0 = θi 3′ , ωC 2 = ϕ 2 ( cos θi1′ + sin θi2′ ) , ik′ , k = 1, 2, 3 , being the unit vectors of the axes of the fixed frame of reference. Projecting on the axes of this frame, we obtain (the projection on the O ′x 3′ -axis is identically equal to zero) x1′ − ( a θ + Rϕ 2 ) sin θ = 0, x2′ + ( a θ + Rϕ 2 ) cos θ = 0 .

Analogously, equating to zero the absolute velocity of the point A , it results

(22.2.73)


x1′ + ( a θ − Rϕ 1 ) sin θ = 0, x2′ − ( a θ − Rϕ 1 ) cos θ = 0 .

461 (22.2.73')

These four constraint relations are not independent (the rank of the matrix of the unknowns x1′ , x2′ , θ, ϕ 1 and ϕ 2 is not four). From the relations (22.2.73) and (22.2.73') one obtains the relations

x1′ sin θ − x2′ cos θ = −a θ + Rϕ 1 = a θ + Rϕ 2 ,

(22.2.74)

wherefrom results the holonomic constraint 2a θ + R ( ϕ 2 − ϕ 1 ) = 0 ;

(22.2.74')

indeed, by integration we have 2a θ + R ( ϕ2 − ϕ1 ) = 2ak ,

(22.2.75)

so that one can determine the non-dimensional constant k if the initial values of the three angles are given. As well, from each system of relations (22.2.73) and (22.2.73'), respectively, one obtains the differential consequence x1′ cos θ + x2′ sin θ = 0 ;

(22.2.76)

one can thus see that the velocity vO′ has no component along the axle of the carriage (the Ox 1 -axis), hence being normal to this one, which was to be expected, because the axle cannot slide along itself (the velocities of the points A1 and A2 vanish at any moment and the bar C 1C 2 is rigid). From the relations (22.2.74) we get also the constraint relation x1′ sin θ − x2′ cos θ =

1 R ( ϕ 1 + ϕ 2 ) . 2

(22.2.76')

One can show that the system (22.2.76), (22.2.76') does not admit any integrable combinations, so that the two-wheeled carriage is a non-holonomic system with two degrees of freedom. If we denote by T0′ the kinetic energy of the carriage (the axle and the mechanical system which lies on it) with respect to the fixed frame or reference, we can write T0′ =

1 M ( x1′2 + x2′2 + i 2 θ2 ) , 2

(22.2.77)

where i is the gyration radius of the system with respect to the Ox 3 -axis, in the hypothesis in which its mass centre is at the point O . In the case in which the mass centre is at the point C ( x1′ − l sin θ , x 2′ + l cos θ , R ), one must use the velocity of


462

this point with respect to the fixed frame, given by 2 2 2 2 vC′ = (x1′ − l θ cos θ ) + (x2′ − l θ sin θ ) , replacing the moment of inertia Mi by

M (i 2 − l 2 ) , corresponding to Koenig’s theorem; one obtains thus T0′ =

1 M ⎡ x ′2 + x2′2 − 2l θ ( x1′ cos θ + x2′ sin θ ) + i 2 θ2 ⎦⎤ . 2 ⎣ 1

(22.2.77')

We denote by T1′ and T2′ the kinetic energies of the two wheels; we can write T2′ =

1 1 m vC′22 + ( I 1 ω12 + I 2 ω22 + I 3 ω32 ) , 2 2

where I k are the principal moments of inertia of the respective wheel with respect to its centre, while ωk , k = 1, 2, 3 , are the corresponding angular velocities, along the directions of the axes of the movable frame; we notice that I 2 = I 3 = I , I 1 corresponding to the axis C 2 x 1 , while ω1 = ϕ 2 , ω2 = 0, ω3 = θ . Because vC′ 2 = (x1′ − a θ sin θ )2 i1′ + (x2′ + a θ cos θ )2 i2′ , it results

1 { m ⎣⎡ x1′2 + x2′2 + a 2 θ2 − 2a θ ( x1′ sin θ − x2′ cos θ ) ⎦⎤ + I θ2 + I 1ϕ 22 } ; 2

T2′ =

(22.2.78)

analogically T1′ =

1 { m ⎡⎣ x1′2 + x2′2 + a 2 θ2 + 2a θ ( x1′ sin θ − x2′ cos θ ) ⎤⎦ + I θ2 + I1ϕ 12 } . (22.2.78') 2

Finally, the kinetic energy T ′ = T0′ + T1′ + T2′ is expressed in the form 1 ⎡ M ′ ( x1′2 + x2′2 ) + I ′θ2 + I 1 ( ϕ 12 + ϕ 22 ) ⎤⎦, 2⎣ M ′ = M + 2m , I ′ = 2 ( I + ma 2 ) + Mi 2 ,

T′ =

(22.2.79)

if C ≡ O , and in the form T′ =

1 ⎡ M ′ ( x1′2 + x2′2 ) + I ′θ2 + I 1 ( ϕ 12 + ϕ 22 ) ⎤⎦ − 2 I θ ( x1′ cos θ + x2′ sin θ ) 2⎣ (22.2.79')

in the general case. Writing Lagrange’s equations, one must not take into account the constraint relation (22.2.76) in the expression of the kinetic energy (22.2.79'); it can be used only after writing these equations. As well, we notice that the own weight (the only force which is applied upon the mechanical system) does not produce work; hence, the generalized forces vanish. We can thus write the first integral of mechanical energy in the form (valid also if the centre of mass is not at O )


463

M ′ ( x1′2 + x2′2 ) + I ′θ2 + I 1 ( ϕ 12 + ϕ 22

) = 2h ,

(22.2.80)

where h is the energy constant. Using the general form (22.2.14) of the constraint relations, the relations (22.2.73'), (22.2.76) allow to write a10 = 0, a11 = sin θ , a12 = − cos θ , a13 = a , a14 = − R, a15 = 0, a20 = 0, a21 = sin θ , a22 = − cos θ , a23 = −a , a24 = 0, a25 = − R, a 30 = 0, a 31 = cos θ , a 32 = sin θ , a 33 = 0, a 34 = 0, a 35 = 0.

Limiting ourselves to the case C ≡ O multipliers (22.2.13) are written in the form

( l = 0 ), Lagrange’s equations with

M ′x1′ = ( λ1 + λ2 ) sin θ + λ3 cos θ , M ′x2′ = − ( λ1 + λ2 ) cos θ + λ3 sin θ , I ′θ = ( λ1 − λ2 ) a , I 1ϕ1 = − λ1R , I 1ϕ2 = − λ2 R .

(22.2.81)

Eliminating Lagrange’s multipliers λ1 , λ2 between the last three equations, we get I ′θ = −

a I ( ϕ − ϕ2 ) , R 1 1

wherefrom I ′θ = −

a I ( ϕ − ϕ2 ) + I ′ ( C 1t + C 2 ) , R 1 1

(22.2.82)

C 1 ,C 2 being two integration constants. From (22.2.75), (22.2.82) it results R C1 R C2 − k + k, ,ψ = 2a K 2a K C −k 2a R a I1 + ,K = . ϕ1 − ϕ2 = ωt + 2 2a R I ′ R K

θ (t ) = ωt + ψ , ω =

(22.2.83)

Eliminating the multipliers λ3 between the first two equations (22.2.81) and then the multipliers λ1 , λ2 , by means of the last two equations of Lagrange, we can write M ′ ( x1′ sin θ − x2′ cos θ ) = λ1 + λ2 = −

I1 ( ϕ − ϕ2 ) ; R 1

we notice that x1′ sin θ − x2′ cos θ =

d ( x ′ sin θ − x2′ cos θ ) = −a θ + Rϕ1 = a θ + Rϕ2 , dt 1


464

where we took into account the constraint relations (22.2.75), (22.2.76). Observing that θ = 0 , it results ϕ1 = ϕ2 = 0 , so that ϕ1 (t ) = ω1t + ψ1 , ϕ2 (t ) = ω2t + ψ2 , a a ω , ω2 = ω0 − ω , R R a a ψ1 = ψ0 + ( ψ − k ) , ψ2 = ψ0 − ( ψ − k ) , R R ω1 = ω0 +

(22.2.83')

the angles θ , ϕ1 , ϕ2 depending thus on the angular velocities ω and ω0 and on the non-dimensional constants k , ψ , ψ0 , which are determined by initial conditions. In this case, the relation (22.2.74) can be written in the form x1′ sin θ − x2′ cos θ = Rω0 ;

taking into account the relation (22.2.76) too, we obtain also the co-ordinates of the point O x1′ = x1′0 −

ω0 ω R cos ( ωt + ψ ) , x 2′ = x 2′0 − 0 R sin ( ωt + ψ ) , ω ω

(22.2.83'')

where the constants x 1′0 , x 2′0 are specified by the initial conditions too. Lagrange’s multipliers λ1 = λ2 = 0, λ3 = Rωω0 ,

(22.2.84)

and the generalized constraint forces R1 = Rωω0 cos ( ωt + ψ ) , R2 = Rωω0 sin ( ωt + ψ ) , R3 = R4 = R5 = 0

(22.2.84')

are determined by the equations (22.2.81). 22.2.3.5 Motion of the Four-Wheeled Carriage Let be a carriage with four wheels, of centres C1 ,C 2 ,C 3 and C 4 , which are rotating independently by the angles ϕ1 , ϕ2 , ϕ3 and ϕ4 , respectively; we denote by C ′ and C ′′ the middles of the two axles and take the pole O of the movable frame of reference at the mass centre, considered to be situated on the straight line C ′C ′′ , which is taken as Ox1 -axis and is parallel to the plane O ′x1′x 2′ . The Ox1 -axis makes the angle θ with the O ′x1′ -axis, while the Ox 3 -axis is parallel to the O ′x 3′ -axis. The two axes of the wheels are of lengths 2a and 2b , respectively, and make the angle ψ between them (Fig. 22.15). The wheels of centres C1 and C 2 are of radii R ′ and masses m ′ , while the wheels C 3 and C 4 are of radii R ′′ and masses m ′′ ; the mass of the carriage without wheels is M . Let be x 1′ , x 2′ the co-ordinates of the point C ′′ ; in this case, the co-ordinates of the points C ′ and


465

O will be x1′ − l cos θ , x 2′ − l sin θ and x1′ − c cos θ , x 2′ − c sin θ , respectively, where l = C ′C ′′ and c = OC ′′ . We can use thus eight generalized co-ordinates x 1′ , x 2′ , θ , ψ , ϕ1 , ϕ2 , ϕ3 and ϕ4 .

Fig. 22.15 Motion of a four-wheeled carriage

As in the case of the two-wheeled carriage, we can write the constraint relation (we notice that the angle θ has another significance that in the preceding case) x1′ sin ( θ + ψ ) − x2′ cos ( θ + ψ ) = 0

(22.2.85)

for the front wheels. Passing from the co-ordinates of the point C ′′ to the co-ordinates of the point C ′ , one obtains the constraint relation x1′ sin θ − x2′ cos θ + l θ = 0

(22.2.85')

for the backwheels. Analogically, equating to zero the velocities of the contact points of the wheels of centres C 1 and C 2 , respectively, we obtain x1′ cos θ + x2′ sin θ + a θ + R ′ϕ 1 = 0, x1′ cos θ + x2′ sin θ − a θ − R ′ϕ 1 = 0;

(22.2.86)

by equating to zero the velocities of the contact points of the wheels of centres C 3 and C 4 , respectively, we can also write

x1′ cos ( θ + ψ ) + x2′ sin ( θ + ψ ) + b ( θ + ψ ) + R ′′ϕ 3 = 0, x1′ cos ( θ + ψ ) + x2′ sin ( θ + ψ ) − b ( θ + ψ ) − R ′′ϕ 4 = 0.

(22.2.86')

We have obtained six non-holonomic constraint relations, the mechanical system being thus non-holonomic and scleronomic with two degrees of freedom.


466

The kinetic energy of the axles and of the carriage which is supported by them is of the form T ′ = T0′ + T1′ + T2′ + T3′ + T4′ , where, by a calculation analogous to that of the preceding subsection, we have 1 1 M ⎡⎣ x1′2 + x2′2 + c 2 θ2 + 2cθ ( x1′ sin θ − x2′ cos θ ) ⎤⎦ + I 3 θ2 , 2 2 1 ⎡ 2 2 2 2 T1′ = m ′ ⎣ ( x1′ + l θ sin θ ) + ( x2′ − l θ cos θ ) + a θ 2 1 1 +2a θ ( x1′ cos θ + x2′ sin θ ) ⎤⎦ + J ′θ2 + I ′ϕ 12 , 2 2 1 2 2 T2′ = m ′ ⎡⎣ ( x1′ + l θ sin θ ) + ( x2′ − l θ cos θ ) + a 2 θ2 2 1 1 −2a θ ( x1′ cos θ + x2′ sin θ ) ⎤⎦ + J ′θ2 + I ′ϕ 22 , 2 2 1 2 2 2 2 T3′ = m ′′ { x1′ + x2′ + b ( θ + ψ ) 2 1 1 2 +2b ( θ + ψ ) [ x1′ cos ( θ + ψ ) + x2′ sin ( θ + ψ ) ] } + J ′′ ( θ + ψ ) + I ′′ϕ 32 , 2 2 1 2 T4′ = m ′′ { x1′2 + x2′2 + b 2 ( θ + ψ ) 2 1 1 2 −2b ( θ + ψ ) [ x1′ cos ( θ + ψ ) + x2′ sin ( θ + ψ ) ] } + J ′′ ( θ + ψ ) + I ′′ϕ 42 ; 2 2 T0′ =

we have denoted by I 3 the moment of inertia of the carriage without wheels with respect to the Ox 3 -axis, by I ′ and I ′′ the principal moments of inertia of the wheels C 1 and C 3 , respectively, with respect to their axes and by J ′ and J ′′ the other principal moments of inertia of the same wheels. Thus, we obtain 1 M ′ ( x1′2 + x2′2 ) + ( Mc + 2m ′l ) θ ( x1′ sin θ + x2′ cos θ ) + J θ2 2 1 2 + ( m ′′l 2 + J ′′ ) ( θ + ψ ) + ⎣⎡ I ′ ( ϕ 12 + ϕ 22 ) + I ′′ ( ϕ 32 + ϕ 42 ) ⎦⎤, 2

T′ =

(22.2.87)

where M ′ = M + 2 ( m ′ + m ′′ ) , J =

1 ( Mc 2 + I 3 ) + m ′ ( l 2 + a 2 ) + J ′ 2

(22.2.87')

represent the total mass of the carriage and a quantity of the nature of a moment of inertia, respectively. One can thus obtain the eight Lagrange’s equations with multipliers, to which one associates the six constraint relations; hence, the eight generalized co-ordinates and the six Lagrange’s multipliers can be determined.


467

22.2.4 Other Equations of Motion In the following, we put in evidence other equations of motion, corresponding to the non-holonomic mechanical system, i.e.: Chaplygin’s equations, Voronets’s equations, Volterra’s equations and Maggi’s equations; as well, we obtain the canonical form of the equations of motion (Hamilton’s equations in quasi-co-ordinates) too (Chaplygin, S.A, 1954). 22.2.4.1 Chaplygin’s Systems. Chaplygin’s Equations S. A. Chaplygin noticed that, in the case of many non-holonomic conservative mechanical systems, the generalized co-ordinates q1 , q2 ,..., qh , qh + 1 ,..., qs can be chosen so that the generalized velocities corresponding to the first h co-ordinates be considered independent; the other s − h co-ordinates do not intervene, neither in the coefficients ch + k , j of the catastatic constraint relations (22.1.1') (where we make ak 0 = 0 ), written in the form (the matrix ⎡⎣akj ⎤⎦ is of rank h ) qh + k =

h

∑ ch +k , j q j , k j =1

= 1, 2,..., s − h ,

(22.2.88)

nor in the expression of the Lagrangian L , written without taking into account these constraint relations (so that ∂T / ∂qh + k = 0, Qh + k = 0, k = 1, 2,..., s − h ); in these systems, called Chaplygin systems, the equations of motion can be separated from the non-integrable constraint equations. We preserve the denomination of Chaplygin systems also for the non-conservative systems with general non-holonomic constraints (non-catastatic, for which ak 0 ≠ 0 ), if the generalized forces and the coefficients ckj do not depend explicitly on the co-ordinates qh +1 , qh + 2 ,..., qs . The motion of the skate, the motion of a circular disc on a plane and the motion of the two-wheeled carriage, considered in the previous section, correspond just to such systems. Obviously, the constraint relations (22.2.88) allow to write the relations

δqh + k =

h

∑ ch +k , j δq j , k j =1

= 1, 2,..., s − h ,

(22.2.88')

for the virtual generalized displacements, the virtual generalized displacements δq1 , δq2 ,..., δqh being independent too. Starting from the d’Alembert-Lagrange theorem (18.2.27'), written in the form s −h ∂T ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ⎡ d ⎛ ∂T ⎞ ⎤ − Qk + h ⎥ δqh + k = 0, ⎟ − ∂q ⎟ − ∂q − Q j ⎥ δq j + ∑ ⎢ dt ⎜ ∂q ⎝ h +k ⎠ ⎦ j h +k ⎠ ⎦ j =1 k =1 ⎣ h

∑ ⎢⎣ dt ⎝⎜ ∂q j

using the relations (22.2.88') and observing that in a double sum one can invert the order of summation, we obtain s −h d ⎛ ∂T ⎞ ⎤ ⎡ d ⎛ ∂T ⎞ ∂T ⎟ ⎥ δq j = 0 ; ⎟ − ∂q − Q j + ∑ ch + k , j dt ⎜ ∂q ⎝ h +k ⎠ ⎦ j ⎠ j =1 k =1 h

∑ ⎢⎣ dt ⎜⎝ ∂q j


468

but the virtual generalized displacements which intervene are independent so that one can write

d ⎛ ∂T ⎞ ∂T s − h d ⎛ ∂T ⎞ − + ∑ ch + k , j = Q j , j = 1, 2,..., h . ⎜ ⎟ dt ⎝ ∂q j ⎠ ∂q j k =1 dt ⎜⎝ ∂qh + k ⎟⎠

(22.2.89)

We denote by an asterisk an expression in which the generalized velocities qh + 1 , qh + 2 ,..., qs , hence the generalized velocities which are considered as dependent, have been eliminated by means of the constraint relations (22.2.88). We can thus write ∂T * ∂T s − h ∂T = + ∑ ch + k , j , ∂q j ∂q j k =1 ∂qh + k ∂T * ∂T s − h ∂T = + ∂q j ∂q j k∑ =1 ∂qh + k

∂ch + k ,l , l =1 ∂q j h

∑

where we took into account ∂qh + k ∂q = ch + k , j , h + k = ∂q j ∂q j

∂ch + k , j ql ; l = 1 ∂ql h

∑

observing that d c = dt h + k , j

∂ch + k , j ql , l = 1 ∂qk h

∑

we can also write s − h ∂T d ⎛ ∂T * ⎞ d ⎛ ∂T ⎞ s − h d ∂T = + ∑ ch + k , j + ∑ ⎜ ⎟ ⎜ ⎟ dt ⎝ ∂q j ⎠ dt ⎝ ∂q j ⎠ k =1 dt ∂qh + k k =1 ∂qh + k

h

∑

l =1

∂ch + k ,l ∂ql

ql .

The equations (22.2.89) can be rewritten in the form

d ⎛ ∂T * ⎞ ∂T * s − h ∂T − + ∑ dt ⎜⎝ ∂q j ⎟⎠ ∂q j k =1 ∂qh + k

∂ch + k , j ⎞ ⎛ ∂ch + k ,l − q = Q j , j = 1, 2,..., h , ∂ q ∂ql ⎟⎠ l j l =1 (22.2.89') h

∑ ⎜⎝

obtaining thus Chaplygin’s equations presented in 1895 at a conference and published in 1897. Eliminating the generalized velocities qh + 1 , qh + 2 ,..., qs from the expressions ∂T / ∂qh + k by means of the constraint relations (22.2.88), the system (22.2.89') becomes of h differential equations and with h unknown functions q1 (t ), q2 (t ),..., qh (t ) , a number equal to the number of degrees of freedom of the mechanical system. These unknown functions are thus determined independently of the


469

dependent generalized co-ordinates qh + 1 (t ), qh + 2 (t ),..., qs (t ) , which can be subsequently obtained, by s quadratures, from the constraint relations (22.2.88). If the constraint relations are integrable (holonomic), hence if ∂ch + k ,l ∂q j

=

∂ch + k , j ∂ql

, j , l = 1, 2,..., h , k = 1, 2,..., s − h ,

(22.2.90)

then Chaplygin’s equations are reduced to Lagrange’s classical equations. Putting in evidence the conservative part ∂U / ∂q j and the non-conservative part Q j of the generalized forces (see formula (18.2.24)), one can introduce the kinetic potential L = T + U , so that Chaplygin’s equations read d ⎛ ∂L * ⎞ ∂L * s − h ⎛ ∂L ⎞ − + ∑⎜ ⎟ dt ⎝⎜ ∂q j ⎠⎟ ∂q j k = 1 ⎝ ∂qh + k ⎠

∂ch + k , j ⎞ ⎛ ∂ch + k ,l − q = Q j , j = 1, 2,..., h . ∂q j ∂ql ⎠⎟ l l =1 (22.2.89'')

* h

∑ ⎝⎜

Let us write now the equations of motion of the skate (see Sect. 22.2.3.1). We choose as independent generalized co-ordinates the co-ordinates q1 = x1 and q 3 = θ , the dependent co-ordinate q2 = x 2 being specified by the constraint relation x2 = x1 tan θ .

Observing that c21 = tan θ , c23 = 0 , we can write the equations (22.2.89'') ]n the form (only the derivative ∂c21 / ∂θ = 1/ cos2 θ is non-zero) * θ ⎞ d ⎛ ∂L * ⎞ ∂L * ⎛ ∂L ⎞ ⎛ − + − ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = 0, dt ⎝ ∂x1 ⎠ ∂x1 ⎝ ∂x2 ⎠ ⎝ cos2 θ ⎠

(

)

* x1 d ⎛ ∂L * ⎞ ∂L * ⎛ ∂L ⎞ − +⎜ = 0. ⎟ ⎜ ⎟ 2 ∂θ ∂ dt ⎝ ∂θ ⎠ x θ cos ⎝ 2 ⎠

In this case, the kinetic potential is given by (we use the formula (22.2.37)) 1 2

L* = M

( cosx θ ) 1

2

+

1 2 Iθ 2

and one obtains x1 + x1 θ tan θ = 0, θ = 0 ,

being led to the same results as in Sect. 22.2.3.1. We can consider also a somewhat more general case in which the skate, of weight M g , slides on the plane Ox 1x 2 , inclined with the angle α with respect to the horizontal line (Fig. 22.16). Chaplygin’s equations are the same, Lagrange’s function being given by


470 1 2

L* = M

x12 1 + I θ2 + mgx1 sin α ; 2 2 cos θ

besides the equation θ = 0 , we find

x1 + x1 θ tan θ = g sin α cos2 θ . Assuming that v 0 = 0 , we get x1 (t ) = a 2 sin2 ω0t , a 2 = x 2 (t ) =

g sin α, 2 ω02

1 2 a ( 2 ω0t − sin 2 ω0t ) ; 2

(22.2.91)

hence, the skate AB has a uniform motion of rotation about the contact point C , which – in its displacement – describes a cycloid.

Fig. 22.16 Motion of a skate on an inclined plane

We can associate to the independent generalized co-ordinates q1 , q2 ,..., qh other independent parameters, that is quasi-co-ordinates π1 , π2 ,..., πh , by means of the relations (the Greek indices are summed from 1 to h , according to the summation convention of the dummy indices) π γ = αγεqε ,

(22.2.92)

of the form (22.2.17), where αγε = αγε (q1 , q2 ,..., qh ), γ , ε = 1, 2,..., h . Using these relations and the s − h constraint relations (22.2.88), we can express all the generalized velocities by means of the h quasi-velocities in the form q j = β j σ π σ , j = 1, 2..., s .

(22.2.92')

To the virtual generalized displacements correspond relations of the form δq j = β j σ π σ , j = 1, 2..., s ,

(22.2.92'')


471

where δπσ are the virtual variations of the quasi-co-ordinates. Taking into account these relations, the d’Alembert–Lagrange theorem ⎡ d ⎛ ∂L ⎞ ∂L ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q ⎥ δq j = 0 j ⎦ ⎣ ⎝ j ⎠

leads to

⎡ d ⎛ ∂L ⎞ ∂L ⎤ ⎢ dt ⎜ ∂q ⎟ − ∂q ⎥ β j σ = 0, σ = 1, 2,..., h , j ⎦ ⎣ ⎝ j ⎠ where the Latin indices are summation indices from 1 to s , corresponding to Einstein’s convention of summation, while L = L (q1 , q2 ,..., qh , q1 , q2 ,..., qs ) is Lagrange’s kinetic potential. Taking into account (22.2.92''), we get

∂L * ∂L ∂L ∂β j σ ∂L * ∂L = + π σ , = β , ∂q γ ∂q γ ∂q j ∂q γ ∂π σ ∂q j j σ wherefrom ∂L ∂L * ∂L ∂β j σ βjε = βγε − βγε π σ , ∂q j ∂q γ ∂q j ∂q γ d ⎛ ∂L ⎞ d ⎛ ∂L * ⎞ ∂L ∂β j ε − βjε = βγσ π σ , ⎜ ⎟ dt ⎝ ∂q j ⎠ dt ⎝⎜ ∂π ε ⎠⎟ ∂q j ∂q γ

with j = 1, 2,..., s , γ , ε, σ = 1, 2,..., h . We notice ( ∂β j ε / ∂q j )βγε π σ = ( ∂β j ε / ∂πσ )π σ , to which we associate two other analogous relations; we can thus write ∂β j ε ∂β j ε ∂β j σ ∂β j σ βγσ = βγε = , , ∂q γ ∂πσ ∂q γ ∂πε ∂L * ∂L * βγσ = . ∂q γ ∂πσ

Replacing in the equations of motion, it results * d ⎛ ∂L * ⎞ ∂L * ⎛ ∂L ⎞ ⎛ ∂β j σ ∂β j γ ⎞ π σ = 0 , − +⎜ − ⎜ ⎟ ⎟ ⎜ ∂πσ ⎟⎠ dt ⎝ ∂π γ ⎠ ∂πγ ⎝ ∂q j ⎠ ⎝ ∂πγ

(22.2.93)

with j = 1, 2,..., s , γ , σ = 1, 2,..., h ; we obtain thus Chaplygin’s equations in quasi-co-ordinates. If the quasi-co-ordinates are true co-ordinates βγε = δ γε , γ , ε = 1, 2,..., h (we introduce Kronecker’s symbol), so that the equations


472

(22.2.93) are reduced to Chaplygin’s classical equations (22.2.89''). The equations (22.2.93) are different from the Boltzmann-Hamel equations (22.2.22), (22.2.22'), because there are involved only the coefficients β jσ , nor the coefficients αγε . 22.2.4.2 Voronets’s Equations P. V. Voronets considers, in 1901, non-holonomic mechanical systems acted upon by conservative forces, which derive from the potential U . The constraint non-holonomic relations are written in the form (22.2.88), but the coefficients ch + k depend on all the generalized co-ordinate q1 , q2 ,..., qs . The quantities βijk =

∂ch + k ,i ∂q j

−

∂ch + k , j ∂qi

+

∂ch + k , j ⎛ ∂ch + k ,i ⎞ ch +l , j − ch +l ,i ⎟ ∂qh +l ∂qh + l ⎠ l =1

s −h

∑ ⎜⎝

(22.2.94)

cannot vanish simultaneously, because the system of relations is non-integrable. Lagrange’s equations with multipliers can be written in the form d ⎛ ∂T ⎞ − dt ⎜⎝ ∂q j ⎟⎠ d ⎛ ∂T dt ⎜⎝ ∂qh + k

∂ (T + U ) = ∂q j

s −h

∑ λk ch + k , j , j

k =1

= 1, 2,..., h ,

⎞ ∂ (T + U ) = λk , k = 1,2,..., s − h . ⎟ − ∂q ⎠ h +k

Taking into account the constraint relations (22.2.88), we can replace the function T by a function Θ , so that T ( q1 , q2 ,..., qs , q1 , q2 ,..., qs ; t ) = Θ ( q1 , q2 ,..., qs , q1 , q2 ,..., qh ; t ) ;

observing that ∂Θ ∂T s − h ∂T = + ch + k , j , ∂q j ∂q j k∑ = 1 ∂qh + k

we can write

d ⎛ ∂Θ ⎞ d ⎛ ∂T ⎞ s −h ⎡ d ⎛ ∂T ⎞ dch + k , j ∂T ⎤ . = + ∑ ⎢ ch + k , j + ⎜ ⎟ ⎜ ⎟ dt ⎝ ∂q j ⎠ dt ⎝ ∂q j ⎠ k =1 ⎣ dt ⎝⎜ ∂qh + k ⎠⎟ dt ∂qh + k ⎥⎦ Using the above Lagrange’s equations with multipliers, we obtain

∂ (T + U ) dch + k , j ∂T d ⎛ ∂Θ ⎞ ∂ (T + U ) s − h ⎡ = + ∑ ⎢ ch + k , j + ⎜ ⎟ ∂q j ∂qh + k dt ⎝ ∂q j ⎠ dt ∂qh + k k =1 ⎣

⎤ ⎥, j = 1,2,..., s . ⎦

By means of the relations ∂Θ ∂T s − h ∂T = + ∂qi ∂qi k∑ = 1 ∂qh + k

h

∑

j =1

∂ch + k , j ∂qi

q j , i = 1, 2,..., s ,


we can eliminate the derivatives

∂T / ∂q j

473 and

∂T / ∂qh + k ,

j = 1, 2,..., h ,

k = 1, 2,..., s − h ; taking into account the notations (22.2.94), we get Voronets’s equations in the form

∂ (Θ + U ) s − h h k ∂T d ⎛ ∂Θ ⎞ ∂ (Θ + U ) s − h q , j = 1,2,..., h . = + ∑ ch + k , j + ∑ ∑ β ji ⎜ ⎟ ∂q j ∂qh + k ∂qh + k i dt ⎝ ∂qi ⎠ k =1 k =1 i =1 (22.2.95)

If the co-ordinates qh + 1 , qh + 2 ,..., qs do not appear explicitly in the expression of the kinetic energy, of the potential U and of the constraint relations, then we find again Chaplygin’s equations (22.2.89'), Voronets’s equations having thus a more general character. The equations (22.2.95) read d ⎛ ∂Θ ⎞ ∂Θ s − h ∂Θ − − ∑ ch + k , j ⎜ ⎟ ∂qh + k dt ⎝ ∂q j ⎠ ∂q j k =1 s −h h s h − ∂T − ∑ ∑ β jik q = Q j + ∑ ch + k , j Qh + k , j = 1, 2,..., h , h + k i ∂ q k =1 i = 1 k =1

(22.2.95')

in the case in which the mechanical system is non-conservative, Voronets deduced these equations by means of Hamilton’s variational principle too; as well, he found the form of these equations also in quasi-co-ordinates. 22.2.4.3 Volterra’s Equations V. Volterra introduced, in 1898, s parameters p1 , p2 ,..., ps , s ≤ 3n , which he called characteristics of a discrete mechanical system of n particles of masses M k , in the representative space E 3n (see Sect. 18.1.1.2); these characteristics are linked to the generalized velocities X k , k = 1, 2,..., 3n , by the relations (Volterra, V., 1893, 1899) X k = ξkj p j , k = 1, 2,..., 3n ,

(22.2.96)

where ξkj = ξkj ( X1 , X 2 ,..., X 3 n ) and where we have used the summation convention of the dummy indices from 1 to s . If s = 3n , the relations (22.2.96) being non-integrable, then the parameters p j become quasi-co-ordinates. We denote by w j , j = 1, 2,..., s , the characteristics of the virtual displacements, so that δXk = ξkj δw j , k = 1, 2,..., 3n ;

hence, it results

δXk =

∂X k δw j . ∂p j

(22.2.96')


474

The kinetic energy of the mechanical system can be expressed in the form T =

1 3n 1 M k X k2 = Eij pi p j , ∑ 2 k =1 2

(22.2.97)

where Eij = E ji =

3n

∑ M k ξki ξkj

k =1

(22.2.97')

are known functions of the co-ordinates X1 , X 2 ,..., X 3 n . We introduce the notations (l ) bqr =

∂ξkq

3n 3n

∑ ∑ M k ∂X p ξkl ξpr ,

k =1 p =1

=

(m ) aqr

(l ) elm ⎡⎣ bqr

−

(l ) brq

(22.2.98)

⎤ , elm Emr = δlr , ⎦

where δlr is Kronecker’s symbol and q , r , l , m = 1, 2,..., s . Let Pj =

3n

∑ Xk ξkj , j

k =1

= 1, 2,..., s ,

(22.2.98')

be the corresponding given generalized forces; we introduce the notation Tj =

∂T = ∂w j

3n

∂T

∑ ∂Xk ξkj , j

= 1, 2,..., s .

k =1

(22.2.98'')

By a non accurate demonstration, which he corrected afterwards (incorrect too!), Volterra obtained the equations d ⎛ ∂T ⎞ ∂T pr + Tj + Pj , j = 1, 2,..., s , = a (jrm ) ⎜ ⎟ dt ⎝ ∂p j ⎠ ∂pm

(22.2.99)

which we call Volterra’s equations; but these equations are correct as it has been shown by Yu. I. Neĭmark and N. A. Fufaev. We can write these equations also in the form d ⎛ ∂T ⎞ = b(jrl ) pr pl + Pj , j = 1, 2,..., s . dt ⎜⎝ ∂p j ⎟⎠

(22.2.99')

22.2.4.4 Maggi’ Equations G. A. Maggi admitted, in 1896, that the virtual generalized displacements δq j , j = 1, 2,..., s , which verify the m constraint relations (22.1.2), can be expressed in the linear form


δq j =

s −m

∑ Eij εi , j

i =1

475

= 1, 2,..., s ,

(22.2.100)

where ε1 , ε2 ,..., εs − m are independent parameters. The d’Alembert–Lagrange equation, written in the form (18.2.27'), leads to s −m

⎧

⎫ ⎡ d ⎛ ∂T ⎞ ∂T ⎤ ⎟ − ∂q ⎥Eij − Ei ⎬εi = 0 , j ⎦ ⎠ ⎩ j =1 ⎭ s

∑ ⎨ ∑ ⎣⎢ dt ⎝⎜ ∂q j

i =1

with the notation Ei =

s

∑ Q j Eij , i j =1

= 1,2,..., s − m .

(22.2.101)

Taking into account that the parameters ε1 , ε2 ,..., εs − m are independent, we obtain Maggi’s equations ⎡ d ⎛ ∂T ⎞ ∂T ⎟− ⎠ ∂q j j =1 s

∑ ⎣⎢ dt ⎝⎜ ∂q j

⎤ ⎥Eij = Ei , i = 1, 2,..., s − m . ⎦

(22.2.102)

Maggi showed in 1901 that, in case of a discrete mechanical system of s particles, these equations are reduced, in particular, to Volterra’s equations if conditions of the form (22.2.90) are fulfilled. 22.2.4.5 Canonical Form of the Equations of Motion in Quasi-co-ordinates S. A. Chaplygin and J. Quangel have shown in 1906 how the equations of motion of a non-holonomic mechanical system can be obtained in quasi-co-ordinates, in a form analogue to that of Hamilton’s equations for holonomic mechanical system; one can thus extend known results obtained in case of holonomic systems to non-holonomic ones. We assume that the considered mechanical system are conservative, so that we introduce the kinetic potential L = T + U ; using the constraint relations (22.1.1'), we can write Lagrange’s equations with multipliers d ⎛ ∂L ⎞ ∂L − = dt ⎜⎝ ∂q j ⎟⎠ ∂q j

m

∑ λk akj , j

k =1

= 1, 2,..., s .

(22.2.103)

Introducing Hamilton’s function (19.1.11), we get Hamilton’s equations q j =

∂H ∂H + , p = − ∂p j j ∂q j

m

∑ λk akj , j

k =1

= 1,2,..., s ,

(22.2.104)

corresponding to conservative non-holonomic mechanical systems. Together with the constraint relations (22.2.1'), one obtains a complete system of 2s + m differential q j , p j , j = 1,2,..., s , k = 1, 2,..., m , and equations for the unknowns λk , k = 1, 2,..., m , as functions of the time t .


476

Having to do with conservative forces, we can introduce the kinetic potential L * ; hence, the equations (22.2.35) read d ⎛ ∂L * ⎞ ∂L * ∂L * − + γkji π = 0, j = 1, 2,..., s − m . ⎜ ⎟ dt ⎝ ∂π j ⎠ ∂π j ∂π j k

(22.2.105)

In general, L * depends on all quasi-velocities π i , i = 1, 2,..., s , which are linked to the generalized velocities q j , j = 1,2,..., s , by the relations (22.2.17). We assume that m non-holonomic and catastatic constraint relations (22.2.31) take place; quasi-co-ordinates πi will be chosen so that the last m quasi-velocities do vanish πs − m + k = 0, k = 1,2,..., m ,

the

(22.2.106)

if the constraint relations (22.2.31) hold. As. well, corresponding to the relations (22.2.33), we assume that

q j =

s −m

∑ β jk πk , j

k =1

= 1,2,..., s .

(22.2.107)

Thus, the equations (22.2.105), (22.2.107), in which we take into account the conditions (22.2.106), form a complete system of 2s − m differential equations for the unknowns q j (t ), j = 1, 2,..., s and π k (t ), k = 1, 2,..., s − m . Introducing the quasi-momenta (22.2.23), we define the function K = pi* π i − L * ,

(22.2.108)

analogue to Hamilton’s functions. Taking into account

δq j =

s −m

∑ βjk δπk ,

k =1

∂T * = ∂π j

s −m

∑ β jk

k =1

∂T * , j = 1,2,..., s , ∂qk

we can write δK = π j δp *j + p *j δπ j − = π j δp *j −

s −m

∑

k =1

∂L * ∂L * δq j − δπ j ∂q j ∂π j

∂L * ∂L * β jk δπk = π j δp *j − δπ j , ∂q j ∂π j

the canonical quasi-co-ordinates being thus π j and p *j , j = 1,2,..., s . After eliminating the quasi-velocities, the variation of the function K = K (q1 , q2 ,..., qs , p1* , p2* ,..., ps* ; t ) is of the form


δK =

477

∂K ∂K * ∂K ∂K δq + δp j = δπ j + * δp *j , ∂q j j ∂p *j ∂π j ∂p j

so that it results ∂K ∂L * ∂K − = , ; * ∂ ∂ π πj ∂p j j

π j =

the equations of motion (22.2.105) are thus rewritten in the form π j =

∂K * ∂K ∂K − γkji p *j , p j = − , j = 1,2,..., s − m , * ∂ π ∂p j ∂p *j j

(22.2.109)

obtaining the canonical equations of motion in a non-holonomic case. The conditions (22.2.106) lead to ∂K ∂ps*− m + k

= 0, k = 1, 2,..., m ,

(22.2.106')

and the relations (22.2.107) to q j =

s −m

∂K

k =1

k

∑ β jk ∂p*

= 0, j = 1, 2,..., s .

(22.2.107')

The equations (22.2.106'), (22.2.107') and (22.2.209) form a complete system of 3s − m differential equations to determine the unknown functions q j (t ), p *j (t ) ,

j = 1,2,..., s , and π k (t ), k = 1, 2,..., s − m .

Let us start now from Chaplygin’s equations (22.2.89''), where the function

L = L (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) and the coefficients ch + k , j do not depend on the s − h last generalized co-ordinates qh + 1 , qh + 2 ,..., qs , having to do with a system of h

differential equations with the unknown functions q j (t ), j = 1, 2,..., h . We introduce the generalized momenta p j (t ), j = 1,2,..., h , by the relations pj =

∂L * , j = 1,2,..., h ; ∂q j

(22.2.110)

Hamilton’s function is given by H* =

k

∑ p j q j j =1

− L* .

(22.2.111)


478

By calculations analogue to these above, we get

q j =

∂H * ∂L * ∂H * , = ,− ∂p j ∂q j ∂q j

and the system (22.2.89'') can be replaced by q j = p j = −

∂H * , j = 1, 2,..., h , ∂p j

* ∂ch + k , j ⎞ ∂H * ∂H * s − h ⎛ ∂L ⎞ ⎛ ∂ch + k ,l − ∑⎜ − . ⎟ ⎜ ∂q j ∂ql ⎟⎠ ∂pl k =1 ⎝ ∂qh + k ⎠ ⎝ ∂q j

(22.2.112)

We obtained thus Chaplygin’s equations in a canonical form; they form a complete system of 2h equations for the unknown functions q j (t ) and p j (t ), j = 1,2,..., h .

22.3 Gibbs–Appell equations Starting from the acceleration energy as “central function”, P. Appell studies in detail, in 1899, the equations of motion given by W. Gibbs in 1879; we present these equations and apply them to various particular cases (Appell, P., 1941–1953).

22.3.1 Gibbs–Appell Equations of Motion We introduce, in what follows, the acceleration energy, making a thorough study of it; using this mechanical quantity, we deduce then the Gibbs–Appell equations, valid for holonomic mechanical systems, as well as for non-holonomic ones. These equations represent the simplest and the most extensive form of the equations of motion. 22.3.1.1 Acceleration Energy Searching a form of the equations of motion which be unique either for the holonomic mechanical systems or for the non-holonomic ones, the rôle of the kinetic energy will be taken by the acceleration energy, called Appell’s function too and defined by S =

1 n 1 n mi ai2 = ∑ mi ri2 ; ∑ 2 i =1 2 i =1

(22.3.1)

expressing the accelerations ai by means of the generalized co-ordinates, the generalized velocities and the generalized accelerations in the form (18.2.14), the accelerations energy is given by S = S 2 + S1 + S 0 ,

(22.3.2)

where S2 =

1 γ q q , γ = γkj = 2 jk j k jk

n

∂r

∑ mi ∂qij

i =1

⋅

∂ri , j , k = 1, 2,..., s , ∂qk

(22.3.2')


479

is a quadratic form in the generalized accelerations, S1 = γ j qj , γj =

n

⎛ ∂r

∑ mi ⎜⎝ ∂qij

i =1

⋅

∂2 ri ∂r ∂r ∂r qk ql +2 i ⋅ i + i ⋅ ri ∂qk ∂ql ∂q j ∂qk ∂q j

⎞ ⎟ , j = 1, 2,..., s , ⎠

(22.3.2'')

is a linear form in the generalized accelerations and ∂2 ri ⎡ ∂2 ri ⎢ ∂q ∂q ⋅ ∂q ∂qm q j qk ql qm l ⎣ j k ∂2 ri ∂ri ∂2 ri ∂r ⎛ ∂ri ∂ri ⎞ ⎤ +4 ⋅ ⋅ + ⋅ ri ⎟ q j qk +4 i ⋅ ri q j + ri2 ⎥ q j qk ql + 2 ⎜ 2 ∂q j ∂qk ∂ql ∂q j ∂qk ∂q j ⎝ ∂q j ∂qk ⎠ ⎦ (22.3.2''') S 0 = γ0 , γ0 =

1 n mi 2 i∑ =1

is a constant with respect to these accelerations. We notice that γ jk = g jk , the coefficients being the same as these which appear in case of the kinetic energy, which are given by the formula (18.2.15'). We can write 2

S2 =

1 n ⎛ ∂r ⎞ mi ⎜ i qj ⎟ ≥ 0 . ∑ 2 i =1 ⎝ ∂q j ⎠

(22.3.3)

This quadratic form vanishes only if each bracket vanishes, hence for ( ∂ri / ∂q j )qj = 0, i = 1, 2,..., n . Writing these linear equations in components, we notice that the matrix [ ∂Xk / ∂q j ] of the coefficients is of rank s (the co-ordinates Xk are specified in Sect. 18.1.1.2); it results qj = 0, j = 1, 2,..., s , the quadratic form S 2 being thus positive definite. All the properties mentioned in Sect. 18.2.1.3 for the coefficients g jk remain valid. We notice as well, that S 2 is a non-singular quadratic form (eventually, excepting the (isolated) points in which the correspondence (18.2.1) (or (18.2.2)) is not one-to-one). In case of catastatic constraints, we have ri = 0, i = 1, 2,..., n , so that γ 0 = γ j = γ jk = 0, j , k = 1, 2,..., s ; hence, S2 = S1 = S0 = 0 , so that S = 0 , (the acceleration energy does not depend explicitly on time). As well,

γj =

∂r

n

∑ mi ∂qij

i =1

γ0 =

⋅

∂2 ri q q , j = 1,2,..., s , ∂qk ∂ql k l

∂2 ri ∂2 ri 1 n ⋅ mi q q q qm , ∑ ∂q j ∂qk ∂qk ∂ql j k l 2 i =1

(22.3.4) (22.3.4')

the respective coefficients being homogeneous forms in generalized velocities. In general, the coefficients of the quadratic form are of the form


480

γ jk = g jk = g jk (q1 , q2 ,..., qs ; t ), j , k = 1, 2,..., s , and the other coefficients are of the

form γ j = γ j (q1 , q2 ,..., qs ; t ), j = 0,1,2,..., s ; we have γ jk = γ jk (q1 , q2 ,..., qs ), j , k = 1, 2,..., s , γ j = γ j (q1 , q2 ,..., qs ), j = 0,1, 2,..., s ,

in case of catastatic constraints. We notice also that, in fact, the conditions ri = 0, i = 1, 2,..., n , are only necessary conditions of catastaticity, so that the above properties can take place even if the constraints are not catastatic (if they are non-holonomic). If the mechanical system S is studied with respect to an inertial frame of reference R ′ and with respect to a non-inertial frame of Koenig type R (which does not rotate with respect to the fixed inertial frame), then we notice that ai′ = aO′ + ai , i = 1, 2,..., n , so that S′ =

n 1 n 1 n 1 n mi a ′i 2 = ∑ mi aO′2 + ∑ mi aO′ ⋅ ai + ∑ mi a ′i 2 , ∑ 2 i =1 2 i =1 2 i =1 i =1

wherefrom it results S′ =

n 1 M aO′2 + aO′ ⋅ ∑ mi ai + SO ; 2 i =1

(22.3.5)

1 n mi a2i 2 i∑ =1

(22.3.5')

we have denoted by SO =

the acceleration energy with respect to the frame of Koenig type. In case of a Koenig frame of reference, the pole of which coincides with the mass centre ( O ≡ C ), the static moment with respect to C vanishes, so that n

∑ mi a

i =1

i

= 0;

it results

S ′ = SO +

1 n M aO′2 2 i∑ =1

(22.3.5'')

and we can state Theorem 22.3.1 (of Koenig type). The acceleration energy of a discrete mechanical system with respect to a given frame of reference is equal to the sum of the acceleration energy of the same system with respect to a Koenig frame of reference and the


481

acceleration energy of the mass centre at which we have considered that the whole mass of the mechanical system is concentrated, with respect to the given frame.

22.3.1.2 Equations of Motion Starting from the relations (22.2.18), which link the kinematic characteristics (and the quasi-co-ordinates) to the generalized co-ordinates, and assuming the existence of m non-holonomic constraint relations (22.1.19), (22.1.20), we eliminate the corresponding kinematic characteristics, remaining with ωi , i = 1, 2,..., h , h = s − m , non-zero kinematic characteristics. In this case, the generalized velocities will be expressed in the form

qk =

h

∑ βjk π j j =1

+ βk 0 , k = 1, 2,..., s ,

(22.3.6)

as functions of quasi-velocities, where βkj = βkj (q1 , q2 ,..., qs ; t ), j = 0,1, 2,..., h , it results also that

δqk =

h

∑ βjk δπj j =1

+ βk 0 , k = 1, 2,..., s .

(22.3.6')

Taking into account the expression (18.2.4') of the velocity, we get

vi =

h

∑ Bij π j j =1

δri =

h

+ Bi 0 , i = 1, 2,..., n ,

∑ Bij δπj , i j =1

= 1, 2,..., n ,

(22.3.7) (22.3.7')

where Bij =

∂ri ∂r β , B = i βk 0 + ri , ∂qk kj i 0 ∂qk

(22.3.7'')

with Bij = Bij (q1 , q2 ,..., qs ; t ), i = 1, 2,..., n , j = 0,1, 2,..., h . One can thus calculate the acceleration ai =

h

∑ Bij πj j =1

+ ..., i = 1, 2,..., n ,

(22.3.8)

where we have put in evidence only the terms which contain the quasi-accelerations; it results ∂ai = Bij , i = 1, 2,..., n , j = 1, 2,..., h . ∂πj

(22.3.8')


482 We can thus write n

∑ mi ai

i =1

n

h

∑ ∑ mi ai

⋅ δri =

i =1 j =1

⋅ Bij δπ j =

h

n

∑ ∑ mi ai

⋅

j =1 i = 1

∂ai δπ = ∂πj

h

∂S

∑ ∂πj δπj , j =1

where we have introduced the acceleration energy (22.3.1). As well, n

∑ Fi

i =1

⋅ δri =

n

h

∑ ∑ Fi

i =1 j = 1

⋅ Bij δπ j =

h

∑ Qj* δπj

,

j =1

with the quasi-forces Q j* =

n

∑ Fi

i =1

⋅ Bij , j = 1, 2,..., h .

(22.3.9)

The theorem of virtual work, written in the form (18.2.27), leads to h

⎛ ∂S

∑ ⎜⎝ ∂πj j =1

⎞ − Q j* ⎟ δπ j = 0 ; ⎠

the variations δπ j being independent, there result the Gibbs–Appell equations ∂S = Q j* , j = 1, 2,..., h , ∂πj

(22.3.10)

the number of which is equal to the number of kinetic degrees of freedom. We have thus h differential equations of second order, to which we associate m non-holonomic constraint relations (22.1.19), (22.1.20), for the s unknowns qk , k = 1, 2,..., s ; thus, the number h + m = s of equations will be equal to the number of unknowns. In particular, we can take as quasi-co-ordinates just the independent generalized co-ordinates. If the mechanical system is holonomic ( m = 0, h = s ), then the Gibbs-Appell equations become ∂S = Q j , j = 1, 2,..., s , ∂qj

(22.3.11)

obtaining a system of equations equivalent to Lagrange’s equations.

22.3.2 Applications We present in what follows some applications of the Gibbs–Appell equations to the motion of a rigid solid with a fixed point, as well as to the motion of a disc on a horizontal plane or to the motion of a sphere on an inclined plane or on a horizontal plane in uniform motion.


483

22.3.2.1 Motion of a Rigid Solid with a Fixed Point Let be a rigid solid S , which is rotating with the angular velocity ω about the fixed point O ′ ≡ O , with respect to an inertial frame of reference R ′ and to a non-inertial frame R , which is rotating about the fixed frame with the angular velocity Ω . The velocity of a point P of the rigid solid with respect to the inertial frame is given by v ′ = ω × r , where r is the position vector with respect to both frames of reference. In this case, the acceleration a ′ is obtained in the form a′ =

∂ × (ω × r) (ω × r) + Ω × (ω × r) = ω × r + ω × [( ω − Ω ) × r ] + Ω ∂t = ω × ( ω × r ) − r × ω + ω × ( r × Ω ) + Ω × ( ω × r ) ,

where we have taken into account the relation (A.2.37), which is linking the derivative with respect to the fixed frame to the derivative with respect to the movable one; using the relations (2.1.49), (2.1.50') too, we obtain, finally, a ′ = ( ω ⋅ r ) ω − ω2 r − r × χ, χ = ω + Ω × ω .

Projecting on the axes of the frame of reference

(22.3.12)

R , it results

a1′ = − ( ω22 + ω32 ) x1 + ( ω1 ω2 − χ 3 ) x 2 + ( ω1 ω3 + χ 2 ) x 3 , a2′ = ( ω1 ω2 + χ 3 ) x1 − ( ω32 + ω12 ) x 2 + ( ω2 ω3 − χ 1 ) x 3 , a 3′ = ( ω3 ω1 − χ 2 ) x1 + ( ω3 ω2 + χ 1 ) x 2 − (

ω12

+

ω22

(22.3.12')

) x3 ,

with χ 1 = ω 1 + Ω2 ω3 − Ω3 ω2 , χ 2 = ω 2 + Ω3 ω1 − Ω1 ω3 ,

(22.3.12'')

χ 3 = ω 3 + Ω1 ω2 − Ω2 ω1 ;

The acceleration energy is thus given by S′ =

1 ( a ′2 + a2′2 + a3′2 )μ( r )dV , 2 ∫ ∫ ∫V 1

where μ( r ) is the density and V is the volume of the solid. We can choose the quasi-co-ordinates so that π k = ωk , k = 1,2, 3 . Because we are interested only in the dependence on πk , hence on ω k , we hold back in the expression of a1′2 only the terms a1′2 = ( χ 3 − 2 ω1 ω2 ) χ 3x 22 + ( χ 2 − 2 ω1 ω3 ) χ 2x 32 +2 ( ω22 + ω32 ) ( χ 3x1x 2 − χ 2x1x 3 ) + 2 [ ω1 ( ω2 χ 2 − ω3 χ 3 ) − χ 2 χ 3 ] x 2 x 3 ;


484

we proceed analogically for a2′2 , a 3′2 . Taking into account the definitions given in Sect. 3.1.2.1 for the moments of inertia, we can calculate the part SO of S ′ , which is of interest in what follows. We can assume, without any loss of generality, that the axes of the frame of reference are rigidly linked to the rigid solid S ; in this case, Ω = ω , so that χ k = ω k = πk , k = 1, 2, 3 . If, for the sake of simplicity, we choose the principal axes of inertia of the rigid solid at the fixed point O as axes Ox k , k = 1, 2, 3 , we obtain S′ =

1 ( I π2 + I 2 π22 + I 3 π32 ) + ( I 3 − I 2 ) π1 π 2 π 3 2 1 1 + ( I 1 − I 3 ) π2 π 3 π1 + ( I 2 − I 1 ) π3 π1 π 2 ,

(22.3.13)

where we have put in evidence the principal moments of inertia. Observing that the work of the moments of the given forces with respect to the given pole is expressed in the form MO ⋅ ωdt =

3

3

j =1

j =1

∑ M j ω j dt = ∑ M j dπ j ,

it results Q j = MOj , j = 1, 2, 3 . In this case, the Gibb–Appell equations (22.3.11) lead to Euler’s equations (15.1.11''). In the particular case in which the ellipsoid of inertia of the rigid solid is of rotation, we can choose as axis of inertia just the Ox 3 -axis. In this case Ω1 = ω1 , Ω2 = ω2 , but Ω3 ≠ ω3 ; all the centrifugal moments of inertia vanish and I 1 = I 2 = J ≠ I 3 . We obtain χ 1 = π1 + π 2 ( π 3 − Ω3 ) , χ 2 = π2 − π 1 ( π 3 − Ω3 ) , χ 2 = π3 ,

so that S′ =

1 [ J ( π1 + π2 ) + I 3 π3 ] + ( I 3 π 3 − J Ω3 ) ( π1 π 2 − π2 π 1 ) , 2

(22.3.14)

where we have neglected the terms which do not contain quasi-accelerations. The Gibbs-Appell equations will be of the form J π1 + ( I 3 π 3 − J ω3 ) π 2 = MO 1 , J π2 − ( I 3 π 3 − J ω3 ) π 1 = MO 2 ,

(22.3.15)

I π3 = MO 3 ,

where MOj , j = 1,2, 3 , are the components of the resultant moment MO with respect to the fixed pole.


485

22.3.2.2 Motion of a Heavy Circular Disc on a Horizontal Plane Using the Gibbs–Appell equations, we take again the problem of motion of a heavy circular disc, of radius R and mass M , on a horizontal plane, problem considered in Sect. 22.2.3.2 in quasi-co-ordinates (see Fig. 22.1). To calculate the acceleration energy, we use the Theorem 22.3.1. We notice that the acceleration energy with respect to the pole O is given by (22.3.14); to calculate the acceleration energy of the mass centre O . we calculate firstly the acceleration aO′ of this point. JJG The velocity of the contact point I must vanish; hence, vI′ = vO′ + ω × OI = 0 , wherefrom we get the components of the velocity vO′ in the form vO′ 1 = 0, vO′ 2 = − Rω3 , vO′ 3 = Rω2 .

The acceleration will ( Ω1 = ω1 , Ω2 = ω2 )

be

given

by

aO′ = ∂vO′ / ∂t + Ω × vO′ ,

so

that

aO′ 1 = R ( ω22 + Ω3 ω3 ) , aO′ 2 = − R ( ω 3 + ω1 ω2 ) , aO′ 3 = R ( ω 2 − ω1 ω3 ) ;

taking into account ω j = π j , j = 1, 2, 3 , we obtain aO′2 = R2 ⎣⎡ π22 + π32 + 2 π1 ( π 2 π3 − π 3 π2 ) ⎦⎤ ,

(22.3.16)

where we have neglected the terms which do not contain the characteristics, The acceleration energy of the disc is thus given by 1 ⎡J π2 + ( J + MR2 ) π2 + ( I 3 + MR2 ) π23 ⎤⎦ 2⎣ 1 + MR2 π1 ( π 2 π3 − π 3 π2 ) + ( I 3 π 3 − J Ω3 ) ( π1 π 2 − π2 π 1 ) . S′ =

(22.3.17)

The work of the weight will be Q4 dq 4 = − MgR cos θdθ , where we have used the notations in Sect. 22.2.3.2. In the considered case we have s = 5 generalized co-ordinates, linked by m = 2 non-holonomic relations; hence, h = s − m = 5 − 2 = 3 . The Gibbs–Appell equations read J π1 + ( I 3 π 3 − J Ω3 ) π 2 = 0,

(J

+ MR2 ) π2 − ( I 3 + MR2 ) π 1 π 3 + J Ω3 π 1 = − MgR cos θ ,

( I3

+ MR

2

) π3

(22.3.18)

+ MR π 1 π 2 = 0. 2

Observing that ω = ψ i 3′ + θi2 + ϕ i 3 , Ω = ψ i 3′ + θi2 , of components (along the axes of the frame of reference R ), ω1 = − ψ sin θ , ω2 = θ, ω3 = ψ cos θ + ϕ ,


486

the system of differential equations (22.3.18) becomes ( π k = ωk , k = 1, 2, 3 ) cos θ ) = I 3 ω3 θ, J ( ψ sin θ + 2 ψθ

(J

+ MR2 ) θ + ( I 3 + MR2 ) ω3 ψ sin θ − J ψ 2 sin θ cos θ = − MgR cos θ ,

( I3

+ MR

2

) ω 3

(22.3.18')

sin θ , = MR ψθ 2

with the unknown functions ψ = ψ (t ), θ = θ (t ) and ω3 = ω3 (t ) . The first equation (22.3.18') takes the form

J d ( ψ sin2 θ ) = ω3 θ sin θ . I 3 dt

(22.3.18'')

To obtain the differential equations which define ω3 and ψ as functions of θ , we write the third equation (22.3.18') and the equation (22.3.18'') in the form ⎛ 1 + I 3 ⎞ dω3 + ψ = 0, J d ⎡ 1 − u 2 ψ ⎤ + ω = 0 , ( ) ⎦ 3 ⎜ ⎟ I 3 du ⎣ ⎝ MR2 ⎠ du

where we have used the change of variable u = cos θ . Eliminating the unknown functions ω3 and ψ , respectively, we obtain dω d ⎡ d2 ζ 1 − u 2 ) 3 ⎤ − λω3 = 0, ( 1 − u 2 ) 2 − λζ = 0 , ( du ⎢⎣ du ⎦⎥ du

(22.3.19)

with the notation

I MR2 . ζ = ( 1 − u 2 ) ψ , λ = 3 J I 3 + MR2

(22.3.19')

The disc being homogeneous, we have I 3 = 2J , so that

λ =

2MR2 . I 3 + MR2

(22.3.19'')

If the disc is whole, then I 3 = MR2 / 2 and it results λ = 4 / 3 , while if the disc is reduced to its circumference we have I 3 = MR2 and λ = 1 ; analogically, one can consider also the case of a circular annulus. We mention that the first equation (22.3.19) is of Legendre type, which defines the hypergeometric functions (see Sect. 22.2.3.2 too). The theorem of uniqueness allows to state that the system (22.3.18') has the solution π ψ = 0, θ = , ω3 = ω , ω = const , 2

(22.3.20)


487

if this one corresponds to the initial conditions; in this case, the point of contact describes a straight line ( ψ = const ), the plane of the disc remaining normal to the fixed plane ( θ = π / 2 ), while the rotation of the disc is uniform ( ϕ = ωt + ω0 , ω0 = const ). If we put θ = π / 2 + ε , ψ = ξ and ω3 = ω + η in the equations (22.3.18') and if we linearize the system thus obtained, then we can write

J ξ = I 3 ωε , ( J + MR2 ) ε + ( I 3 + MR2 ) ωξ = MgRε, η = 0 , wherefrom J ( J + MR2 ) ε + ⎡⎣ I 3 ( I 3 + MR 2 ) ω 2 − JMgR ⎤⎦ ε = 0 ;

if I 3 ( I 3 + MR2 ) ω 2 > JMgR , then the solution is stable, in a first approximation.

22.3.2.3 Motion of a Heavy Sphere on an Inclined Plane Let be now the case of a heavy homogeneous sphere, of radius R and mass M , which has a motion of rolling without sliding on a plane inclined by an angle α with respect to a horizontal plane. We choose, in the inclined plane, the O ′x 2′ -axis along the direction of maximal inclination and the O ′x1′ -axis parallel to the horizontal plane; the

Fig. 22.17 Motion of a heavy sphere on an inclined plane

O ′x 3′ -axis is normal to this plane (Fig. 22.17). A point P of the sphere will be

specified by the co-ordinates x1′ , x 2′ , x 3′ = R of the centre C of the sphere and by Euler’s angles ψ , θ and ϕ . Because the sphere rolls without sliding, it results that the velocity of the contact point I must vanish, so that


488

JJG vI′ = vC′ + ω × CI = 0 ,

where ω is the rotation velocity of the movable frame, rigidly linked to the sphere. Projecting on the axes of the fixed frame of reference, we obtain the non-holonomic constraint relations x1′ − Rω2′ = 0, x2′ + Rω1′ = 0 ;

(22.3.21)

we can thus define the quasi-co-ordinates π1 , π2 , π3 by the relations (5.2.35') in the form π1 = ω1′ = ϕ sin θ sin ψ + θ cos ψ , π 2 = ω2′ = − ϕ sin θ cos ψ + θ sin ψ ,

(22.3.22)

π 3 = ω3′ = ϕ cos θ + ψ .

The acceleration energy is given by S =

1 ⎡ M ( x1′2 + x2′2 ) + I ( ω 1′2 + ω 2′2 + ω 3′2 ) ⎤⎦ 2⎣ 1 = ⎣⎡ ( I + MR2 )( ω 1′2 + ω 2′2 ) + I ω 3′2 ⎦⎤ 2 1 = ⎡⎣ ( I + MR2 )( π12 + π22 ) + I π32 ⎤⎦, 2

(22.3.23)

where we took into account (22.2.21), (22.2.22) and where I is the moment of inertia of the sphere with respect to one of the diameters. The work of the own weight is given by Mg sin αdx 2′ = − Mg ω1′ sin αdt = − MgR sin αdπ1 = Q1 dπ1 ,

because dx 3′ = 0 . The Gibbs–Appell equations read

(I

+ MR2 ) π1 = − MgR sin α, π2 = 0, π3 = 0 ;

(22.3.24)

it results, by integration, π1 = − Kt + C 1 , π 2 = C 2 , π 3 = C 3 , K =

MgR sin α , I + MR2

(22.3.24')

where C 1 ,C 2 ,C 3 are arbitrary constants. The co-ordinates of the point P of the sphere will be thus given by the equations x1′ = C 2 Rt + C 2′ , x 2′ =

1 KRt 2 + C 1Rt + C 1′ , 2

(22.3.25)


489

ψ = [ ( Kt + C 1 ) sin ψ + C 2 cos ψ ] cotθ + C 3 θ = − ( Kt + C 1 ) cos ψ + C 2 sin ψ,

(22.3.25')

ϕ = − [ ( Kt + C 1 ) sin ψ + C 2 cos ψ ] cosecθ .

If the sphere moves on a horizontal plane ( α = 0 ), then it results K = 0 and the mass centre has a uniform and rectilinear motion; in the case in which α ≠ 0 , the mass centre is moving along a parabola of axis parallel to the O ′x 2′ -axis or along the direction of this axis if the initial velocity (for t = 0 ) vanishes ( C 1 = C 2 = 0 ) or is along a parallel to it ( C 2 = 0 ).

22.3.2.4 Motion of a Sphere on a Horizontal Plane in Uniform Rotation Let us consider the motion of a homogeneous sphere, of radius R and mass M , on a horizontal plane O ′x1′x 2′ , which has a uniform rotation of angular velocity Ω about the vertical axis O ′x 3′ ; the axes O ′x1′ and O ′x 2′ are fixed in the plane of rotation. The problem has been previously studied in Sect. 22.2.3.3 (see Fig. 22.13). The constraint relations (22.2.65) can be written in the form x1′ = Rω2′ − Ωx 2′ , x2′ = − Rω1′ + Ωx1′ ,

(22.3.26)

x1′ , x 2′ and x 3′ = R being the co-ordinates of the centre C of the sphere; there result the acceleration x1′ = Rω 2′ + Ω ( Rω1′ − Ωx1′ ) , x2′ = − Rω 1′ + Ω ( Rω2′ − Ωx 2′ ) ,

the acceleration of the centre C being given by aC′2 = x1′2 + x2′2 = R 2 ( ω 1′2 + ω 2′2 ) + 2 RΩ [ ω 2′ ( Rω1′ − Ωx 1′ ) − ω 1′ ( Rω2′ − Ωx 2′ ) ] ,

where we have ignored the terms which do not contain derivatives of second order of the quasi-co-ordinates, specified by ω1′ = π1 , ω2′ = π 2 . The acceleration energy is given by ( I = Mi 2 , where I is the central axial moment of inertia, i is the corresponding gyration radius and ω3′ = π 3 ) S =

1 1 M ⎡a ′2 + i 2 ( π12 + π22 + π32 ) ⎦⎤ = M { ( R2 + i 2 )( π12 + π22 2 ⎣ C 2 + i 2 π32 + 2 RΩ [ π2 ( Rπ 1 − Ωx1′ ) − π1 ( Rπ 2 − Ωx 2′ ) ] } ;

) (22.3.27)

observing that Q j* = 0, j = 1, 2, 3 , the Gibbs–Appell equations relative to the quasi-accelerations πk lead to the equations (22.2.70), (22.2.70'), studied in Sect. 22.2.3.3. If we use the quasi-accelerations x1′ , x2′ , ω 3′ = π3 and notice that


490 ω 1′ = −

1 1 ( x′ − Ωx1′ ) , ω 2′ = − ( x1′ + Ωx2′ ) , R 2 R

then the acceleration energy is written in the form S =

1 M 2

( )

i ⎧⎡ ⎨ ⎢1 + R ⎩⎣

2

( ) Ω ( x′x ′ − x′x ′ ) ⎫⎭⎬ ,

i ⎤ 2 2 2 2 ⎥ ( x1′ + x2′ ) + i π3 + 2 R ⎦

2

1 2

2 1

(22.3.27')

where, as well, we maintained only the terms which contain quasi-accelerations. Assuming that upon the sphere acts a system of arbitrary given forces, the torsor of which at the centre C is given by R and M , of the components Rj , M j , j = 1, 2, 3 , and writing the constraint relations (22.3.26) by means of the virtual displacements in the form ( ω j′ = π j , j = 1, 2, 3 ) δx1′ = Rδπ2 , δx 2′ = − Rδπ1 ,

(22.3.26')

we obtain the virtual work δW = R ⋅ δr ′ + M ⋅ δπ = R1 δx1′ + R2 δx 2′ + M 1 δπ1 + M 2 δπ2 + M 3 δπ3 M = ⎛⎜ R1 + 2 R ⎝

⎞ δx ′ + ⎛ R − M 1 ⎟ 1 ⎜ 2 R ⎠ ⎝

⎞ δx ′ + M δπ ; ⎟ 2 3 3 ⎠

hence, Q1* = R1 + M 2 / R , Q2* = R2 − M 1 / R , Q3* = M 3 . The Gibbs–Appell equations (22.3.10) lead to M ⎡⎣ ( R2 + i 2 ) x1′ + i 2 Ωx2′ ⎤⎦ = R ( RR1 + M 2 ) , M ⎡⎣ ( R2 + i 2 ) x2′ − i 2 Ωx1′ ⎤⎦ = R ( RR2 − M 1 ) ,

(22.3.28)

Mi 2 π3 = M 3 .

If the sphere is subjected only to the action of the own weight, then Rj = M j = 0, j = 1, 2, 3 , and we find again the equations (22.2.70), (22.2.72). Assuming now that the plane which is rotating is inclined by the angle α with respect to the horizontal plane, we have R2 = Mg sin α , the other components of the torsor vanishing. Denoting z = x1 + ix 2 , i = −1 , we can write the system (22.3.28) in the form z − ikz = ik , K =

i 2Ω R2 g sin α = , K , R2 + i 2 R2 + i 2

(22.3.29)

K t; k

(22.3.29')

the solution being given by z = C 1 + C 2e ikt −


491

the complex constants C 1 ,C 2 are determined by initial conditions.

22.4 Other Problems on the Dynamics of Non-holonomic Mechanical Systems We will consider also other aspects of interest concerning non-holonomic problems; let us thus mention the collisions – phenomena with discontinuity – as well as the obtaining of first integrals for the equations of motion.

22.4.1 Collisions The phenomenon of collision is due to the apparition of some percussive forces or to the sudden application of some non-holonomic constraints; in what follows, we present the discontinuous phenomena which put in evidence both cases.

22.4.1.1 Basic Equations From the very beginning, we notice that the non-holonomy does not introduce something essential from the point of view of the percussions, defined by the same formula (10.1.40) in the form P =

lim

∫

t ′′

t ′′ − t ′ → 0 + 0 t ′

F (t )dt ,

(22.4.1)

where F is a percussive force, the limit being considered in the sense of the theory of distributions in the collision interval [t ′, t ′′ ], | t ′′ − t ′ |< ε, ε > 0 arbitrary, which contains only one moment of discontinuity t0 . The corresponding general theorems are given in Sect. 10.1.2.3, and the general study for a single particle is presented in Sect. 13.1.1.2. Corresponding to the theorem of momentum 13.1.2, stated in the space E 3 , we can pass to the Theorem 18.3.1, stated in the space of configurations Λs , according to which the jump of the generalized momentum of a discrete mechanical system subjected to constraints, corresponding to a generalized co-ordinate, at a moment of discontinuity is equal to the sum of the given and constraint generalized percussions, corresponding to the same co-ordinate and which act upon this system at that moment. We assume that, in the interval of percussion, both the generalized co-ordinates q1 , q2 ,..., qh , qh + 1 ,..., qs of a representative point P and the kinematic characteristics akj , ak 0 , k = 1, 2,..., m , j = 1, 2,..., h of the constraint relations (22.1.1) or (22.1.1') remain constant. As in case of the Chaplygin system (see Sect. 22.2.4.1), we assume that the generalized velocities corresponding to the first h co-ordinates can be considered to be independent and that the other s − h co-ordinates do not intervene in the coefficients ch + k , j , ch + k ,0 , k = 1, 2,..., s − h , j = 1, 2,..., h , of the non-holonomic constraint relations, written in the form (the matrix [akj ] is of rank h ) qh + k =

h

∑ ch + k , j q j j =1

+ ch + k ,0 , k = 1, 2,..., s − h ;

(22.4.2)


492

introducing the virtual generalized displacements, the relations (22.4.2) take the form (22.2.88'). Supposing that at least a percussive generalized force intervenes, the generalized velocities q j′ at the initial moment t ′ become q j′′ at the final moment t ′′ . The d’Alembert–Lagrange theorem becomes ⎡ d ⎛ ∂T * ⎞ ∂T * ⎤ ⎟ − ∂q − Q − Rj ⎥ δq j = 0 , q ∂ j ⎝ j ⎠ ⎦ j =1 ⎣ h

∑ ⎢ dt ⎜

(22.4.3)

where T * is the kinetic energy, which is written taking into account the constraint relations (22.4.2) (eventually, the quasi-velocities); Q j and Rj , j = 1, 2,..., h , are the given and the constraint (which appear only in the non-holonomic case) generalized forces, respectively. By introducing the constraint generalized forces, the constraint relations (22.2.88') have been eliminated, so that, in (22.4.3), the virtual generalized displacements are independent; we are thus led to the system of h differential equations d ⎛ ∂T * ⎞ ∂T * − = Q j + Rj , j = 1, 2,..., h , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(22.4.3')

which describe the motion of the non-holonomic system (the motion of the representative point P in the space Λs ), at any moment t (including the moments t ∈ [t ′, t ′′ ] ). Integrating on the collision interval, we can write ∂T * ∂q j

t ′′

− t′

t ′′ ∂T *

∫t ′

∂q j

dt =

t ′′

∫t ′ (Qj

+ Rj )dt , j = 1, 2,..., h .

A mean value theorem applied to an integral the integrand of which is a finite quantity allows to neglect the integrals which result from the non-percussive part of the given generalized forces and from the constraint generalized forces (which are, as well, finite, as it has been shown by Beghin and Rousseau in 1903 for a non-holonomic mechanical system with ideal constraints), as well as from the partial derivatives ∂T * / ∂q j , j = 1, 2,..., h . Defining the generalized percussion in the form (see Sect. 18.3.1.8 too)

Pj =

lim

∫

t ′′

t ′′ − t ′ → 0 + 0 t ′

Q j dt , j = 1,2,.., h ,

(22.4.4)

we get, finally, ⎛ ∂T * ⎞ ⎛ ∂T * Δ⎜ ⎟=⎜ ⎝ ∂q j ⎠ ⎝ ∂q j

⎞′′ ⎛ ∂T * ⎟ − ⎜ ∂q ⎠ ⎝ j

⎞′ ⎟ = Pj , j = 1, 2,..., h . ⎠

(22.4.5)


493

Hence, in the collision phenomenon, the equations (22.4.5) of the non-holonomic mechanical systems have the same form as those used for the holonomic ones (see Sect. 18.3.1.8); as a difference, in case of the non-holonomic mechanical systems, the system is reduced with s − h constraint relations, while T * contains only independent parameters, the functions being expressed by means of the relations (22.4.2). Writing the constraint relations (22.4.2) for the ends of the collision interval, we obtain, by subtraction (the kinematic coefficients are constants), Δqh + k =

h

∑ ch +k , j Δqj , k j =1

= 1, 2,..., s − h ,

Δq j = q ′′ − q j′ , j = 1, 2,..., h .

(22.4.6) (22.4.6')

The generalized velocities at the start of the collision phenomenon qi′ , i = 1, 2,..., s , and the generalized percussions

Pj , j = 1, 2,..., h , are known; the unknown

generalized velocities q j′′, j = 1, 2,..., h , at the end of the collision phenomenon, will be given by the relations (22.4.5), while the other generalized velocities qh′′+ k , k = 1, 2,..., s − h , will be given by the relations (22.4.6), (22.4.6'). The collision phenomenon considered above corresponds to the intervention of some percussive forces. We consider now the case of a sudden application of some non-holonomic constraints. Let thus be a mechanical system subjected to no one percussive force, which after eliminating the holonomic constraints, has s degrees of freedom and is represented in the space Λs by the representative point P , specified by the generalized co-ordinates q1 , q2 ,..., qs . If, at the moment t ′ , we introduce s − h ideal non-holonomic relations of the form (22.4.2), then the virtual generalized displacements will be linked by the relations (22.2.88'), so that h of them can be considered to be independent. Taking into account these relations, the d’Alembert–Lagrange theorem (22.4.3) reads

∑ { dt ⎜ h

j =1

∂T * ⎞ ∂T * d ⎛ ∂T * s − h + ∑ ch + k , j − − Q j − Rj ∂qh + k ⎟⎠ ∂q j ⎝ ∂q j k =1

s −h

⎡ dch + k , j ∂T * ⎛ ∂T * ⎞⎤⎫ −∑ ⎢ + ch + k , j ⎜ + Qh + k + Rh + k ⎟ ⎥ ⎬ δq j = 0; ∂ ∂ t q q d h +k ⎝ h +k ⎠⎦⎭ k =1 ⎣

(22.4.7)

integrating with respect to time on the collision interval [t ′, t ′′ ] and making the same considerations as above (given generalized forces and finite constraints, finite partial derivatives ∂T * / ∂qi , i = 1, 2,..., s , constant kinematic coefficients, independent generalized displacements), we obtain the system of algebraic equations

⎛ ∂T * ⎞ s −h ⎛ ∂T * ⎞ Δ⎜ + ∑ ch + k Δ ⎜ ⎟ ⎟ = 0, j = 1, 2,..., h , ⎝ ∂q j ⎠ k =1 ⎝ ∂qh + k ⎠

(22.4.8)


494 ⎛ ∂T * ⎞ ⎛ ∂T * Δ⎜ ⎟=⎜ ⎝ ∂q j ⎠ ⎝ ∂q j

⎞′′ ⎛ ∂T * ⎟ − ⎜ ∂q ⎠ ⎝ j

⎞′ ⎟ , j = 1, 2,..., h , ⎠

(22.4.8')

for the determination of the s generalized velocities after collision. Let us denote by T* the function T * considered as depending explicitly on the generalized velocities q1 , q2 ,..., qh and implicitly on them by the generalized velocities qh +1 , qh + 2 ,..., qs , by means of the constraint relations (22.4.2); in this case, the equations (22.4.8), (22.4.8') can be written, formally, in a compact form ⎛ ∂T* ⎞ ⎛ ∂T* Δ⎜ ⎟=⎜ ⎝ ∂q j ⎠ ⎝ ∂q j

⎞′′ ⎛ ∂T* ⎟ − ⎜ ∂q ⎠ ⎝ j

⎞′ ⎟ = 0, j = 1, 2,..., h . ⎠

(22.4.9)

One must give a special attention to these equations, because one cannot eliminate the generalized velocities qh + 1 , qh + 2 ,..., qs , effecting then the differentiation with respect to the other generalized velocities; indeed, the constraints (22.4.2) take not place for t < t ′ , so that these ones cannot be used to eliminate the generalized velocities mentioned above from the expression ( ∂T* / ∂q j )′ , j = 1, 2,..., h . In what concerns the elimination of these generalized velocities from the expressions ( ∂T* / ∂q j )′′ ,

j = 1, 2,..., h , corresponding to the end of the collision phenomenon, there can arise two situations: the non-holonomic constraints suddenly applied (i) continue to act or (ii) disappear for t > t ′′ . The case (i) corresponds to a plastic collision, for which k = 0 , while the case (ii) corresponds to a perfect elastic collision, for which k = 1 ; k is the restitution coefficient (see Sect. 10.1.1.1). In general, we can have 0 < k < 1 . Obviously, the generalized velocities at the beginning of the collision phenomenon q1 , q2 ,..., qs are known; to determine the generalized velocities which result at the end of the phenomenon of collision we proceed in a different way, as k = 0 or 0 < k < 1 . So, in the first case, the constraints (22.4.2) are maintained for t > t ′′ too and we can eliminate the generalized velocities from ( ∂T* / ∂q j )′′ , j = 1, 2,..., h , the problem being thus completely solved. In the second case, one cannot make the elimination mentioned above, having to our disposal h algebraic equations q1′′, q2′′,..., qh′′ ; the other generalized velocities qh′′+ 1 , qh′′+ 2 ,..., qs′′ will be specified by the relations (22.4.2), the problem being thus completely solved. In the second case, one cannot make the elimination mentioned above, having to our disposal h equations (22.4.9) for s unknowns q1′′, q2′′,..., qs′′ ; it is necessary to associate other s − h relations of physical nature, obtained – eventually – on an experimental way, to can solve the problem. The problem becomes more complicated if, beside the non-holonomic constraints – which can be also non-ideal – appear percussive forces too. In the case k = 0 we have s unknown percussions and s − (s − h ) = h generalized velocities at the end of the collision phenomenon, as well unknown; hence, there are s + h unknowns to be determined. For this, we have at our disposal h equations of dynamical equilibrium


495

(22.4.5) and s − h constraint relations (22.4.2); we need other h relations, which can be obtained assuming that the constraints are ideal ( s − (s − h ) = h relations) or imposing other supplementary conditions of physical nature. In the case 0 < k ≤ 1 we have also h + (s − h ) = s equations at our disposal; but the number of the unknowns is 2s ( s percussions and s generalized velocities at the end of the collision). The condition of ideal constraints involves only h relations, so that there are still necessary s − h relations of physical nature to solve the problem. The above results are valid for discrete mechanical systems (systems of particles or systems of rigid solids).

22.4.1.2 Applications We consider, in what follows, some simple applications, which present a certain interest in themselves. Let be thus a rigid solid which slides on a horizontal plane and is acted upon by a percussive force contained in the plane and normal to the direction of advance. We idealize the rigid solid by a skate of mass M which slides on the plane Ox1x 2 , A being the contact point, C the mass centre ( AC = a ), the percussion force P acting at the point B ( AB = b ) (Fig. 22.18); the position of the skate will be specified by the point A( x1 , x 2 ) and by the angle θ made with the Ox1 -axis. As we have seen in Sects. 3.2.2.6 and in 22.2.3.1, the constraint relation is of the form x2 = x1 tan θ .

(22.4.10)

Fig. 22.18 Motion of a skate on a horizontal plane, acted upon by a percussive force

The co-ordinates of the mass centre are given by

xC1 = x1 + a cos θ ,

xC 2 = x 2 + a sin θ ; in this case, corresponding to the formula (22.2.37), we can

express the kinetic energy in the form T =

1 ⎡ 2 2 M ( x − a θ sin θ ) + ( x2 + a θ cos θ ) + i 2 θ2 ⎤⎦ , 2 ⎣ 1

(22.4.11)

where i is the gyration radius with respect to the mass centre. Taking into account the constraint relation (22.4.10), it results


496 T* =

1 ⎡ x12 ⎤ M + ( a 2 + i 2 ) θ2 ⎥ . 2 ⎢⎣ cos2 θ ⎦

(22.4.11')

The work effected by the percussion P is given by δW = − P sin θδx1 + P cos θδx 2 + Pb δθ = Pb δθ ,

so that the generalized percussions will be

P1 = P = 0, Pθ = Pb . Assuming that the

skate is at rest at the beginning of the collision phenomenon, the equations (22.2.5) with (22.4.11') lead to x1′′ = x2′′ = 0, M ( a 2 + i 2 ) θ ′′ = Pb .

(22.4.12)

Fig. 22.19 Motion of a homogeneous circular disc, which is rotating about one of its diameters

Let us consider now a homogeneous circular disc of mass M and radius a , which is rotating with an angular velocity ω about a diameter D of it. At a certain moment, the point P of the circumference, specified by the angle α and the distance a sin α with respect to the axis D (Fig. 22.19), is suddenly stopped; the velocity of this point at the respective moment is aω sin α , while the moment of momentum with respect to the axis D ′ which passes through P and is parallel to D is given by K D ′ = (1/ 4)Ma 2 ω (the moment of inertia with respect to the axis D is I D = (1/ 4)Ma 2 ), remaining unchanged after stopping the point P . In this case, the

moment of momentum with respect to the axis Δ , tangent at P to the circumference is K Δ = (1/ 4)Ma 2 ω sin α . Starting from the moment of inertia I D and applying the Huygens–Steiner theorem (see Sect. 3.1.2.6), we obtain the axial moment of inertia I Δ = (5 / 4)Ma 2 and then the angular velocity K Δ / I Δ = (1/ 5)ω sin α , hence a velocity which is the fifth part of the initial velocity of the point P . Let be, as well, a homogeneous sphere of radius a , which moves freely in the space and which, at the initial moment t ′ strikes a rough horizontal plane O ′x1′x 2′ (the O ′x 3′ -axis is directed towards the part from which comes the sphere, the considered

frame of reference being fixed). After collision, the sphere remains on the plane and can


497

have a motion of pivoting or of rolling without sliding; there are thus imposed the constraint relations (see Sect. 22.2.3.3, formula (22.2.57')) x1′ − a ω2′ = 0, x2′ − a ω1′ = 0, x 3′ = 0 ,

(22.4.13)

according to which the velocity of the contact point vanishes. Introducing Euler’s angles ψ , θ , ϕ and using the relations (5.2.35') which are linking the components of the rotation velocity vector ω along the axes of the fixed frame of reference to these angles, we can write three constraint relations in the form (see the formulae (22.2.57'') too) x1′ = a ( θ sin ψ − ϕ sin θ cos ψ ) , x2′ = −a ( θ cos ψ − ϕ sin θ sin ψ ) ,

(22.4.13')

x 3′ = 0,

the first two relations being non-holonomic. The kinetic energy is given by T =

1 M ⎡ x ′2 + x2′2 + x 3′2 + i 2 ( ω1′2 + ω2′2 + ω3′2 ) ⎤⎦ , 2 ⎣ 1

(22.4.14)

where i is the gyration radius with respect to a diameter; taking into account the relations (5.2.35'), it results T =

1 cos θ ) ⎤ . M ⎡ x 2 + x22 + x 32 + i 2 ( ψ 2 + θ2 + ϕ 2 + 2 ψϕ ⎦ 2 ⎣ 1

(22.4.14')

By means of the relations (22.4.13'), we can write ∂T * ∂T ∂T ∂x1′ ∂T ∂x2′ = + + = Mi 2 ( ψ + ϕ cos θ ) , ∂x2′ ∂ψ ∂ψ ∂ψ ∂x1′ ∂ψ ∂T * ∂T ∂T ∂x1′ ∂T ∂x2′ = + + = M ⎡⎣ i 2 θ + a ( x1′ sin ψ − x2′ cos ψ ) ⎦⎤, ∂x2′ ∂θ ∂θ ∂θ ∂x1′ ∂θ ∂T * ∂T ∂T ∂x1′ ∂T ∂x2′ = + + ∂ϕ ∂ϕ ∂x1′ ∂ϕ ∂x2′ ∂ϕ = M ⎣⎡i 2 ( ϕ + ψ cos θ ) − a sin θ ( x1′ cos ψ + x2′ sin ψ ) ⎦⎤;

in this case, the equations (22.4.8') read ψ ′′ + ϕ ′′ cos θ = ψ ′ + ϕ ′ cos θ ,

( i2

+ a 2 ) θ ′′ = i 2 θ ′ + a ⎡ ( x1′ )′ sin ψ − ( x2′ )′ cos ψ ⎤ , ⎣ ⎦ ( i 2 + a 2 sin2 θ ) ϕ ′′ + i 2 ψ ′′ cos θ

= i 2 ( ϕ ′ + ψ ′ cos θ ) − a ⎡ ( x1′ )′ cos ψ − ( x2′ )′ sin ψ ⎤ sin θ , ⎣ ⎦


498

where we took into account the constraint relations (22.4.13') which take place at the end of the collision phenomenon and remain still valid. Hence, one obtains ψ ′′ =

1 { ⎡ ( i 2 + a 2 ) ψ ′ + a 2ϕ ′2 cos θ ⎤⎦ sin θ ( i + a 2 ) sin θ ⎣ 2

+ a ( x1′ cos ψ + x2′ sin ψ ) cos θ } , θ ′′ = ϕ ′′ =

1 ⎡ i 2 θ ′ + a ( x1′ sin ψ − x2′ cos ψ ) ⎦⎤, ( i + a2 ) ⎣

(22.4.15)

2

1 ⎡ i 2 ϕ ′ sin θ − a ( x1′ cos ψ + x2′ sin ψ ) ⎤⎦; ( i + a 2 ) sin θ ⎣ 2

taking into account (22.4.13') and (5.2.35'), we obtain the velocities of the centre of the sphere, after collision, in the form a ( i 2 ω2′ + ax1′ ) , i2 + a2 a = 2 ( i 2 x2′ + a ω1′ ) , i + a2 xC′′ 3 = 0.

xC′′ 1 = xC′′ 2

(22.4.16)

22.4.2 First Integrals of the Equations of Motion We consider, in what follows, the forms taken by the first integrals of the systems of equations of motion in case of non-holonomic constraints, as well as the possibility to obtain them; some applications put in evidence the results thus obtained.

22.4.2.1 Determination of First Integrals Starting from Lagrange’s equations for a holonomic mechanical system, we find, in certain conditions, the first integral of Painlevé (18.2.61), the first integral of Jacobi (18.2.63) or a first integral of Jacobi type (18.2.66), the last two ones being particular cases of Painlevé’s first integral (see Sect. 18.2.3.4); the mechanical systems which admit one of the two integrals are generalized conservative systems, for which the generalized mechanical energy (18.2.68) is introduced. Jacobi’s first integral allows the reduction of Lagrange’s system of equations (18.2.38) for a natural mechanical system, to a system of equations corresponding to a mechanical system with a number of degrees of freedom less with a unity (see Sect. 18.2.3.5). The existence of hidden co-ordinates (the kinetic energy T does not depend explicitly on these co-ordinates) or of ignorable co-ordinates (Lagrange’s function L does not depend explicitly on these co-ordinates) allows, as well, the determination of first integrals; in this case, corresponding to the Routh–Helmholtz theorem, the problem is reduced to the integration of a system of equation of Lagrange for a mechanical system with a smaller number of degrees of freedom (see Sects. 18.2.3.6 and 18.2.3.7). In the case of a non-holonomic mechanical system for which one of the generalized co-ordinates, let be qk , is ignorable ( ∂L / ∂qk = 0 ) one can write a first integral starting from Lagrange’s equations only if some conditions are fulfilled. We introduce thus Chaplygin’s equations (22.2.89''), in which we assume that the non-conservative


499

forces Q j , j = 1, 2,..., h , vanish, the constraint relations being of the form (22.2.88). These equations take the form of Lagrange’s equations for j = 1 if the relations ∂ch + k ,l ∂ch + k ,1 , k = 1, 2,..., s − h , l = 1, 2,..., h , = ∂q1 ∂ql

(22.4.17)

are verified. Introducing the function uh + k = uh + k (q1 , q2 ,..., qs ) by the relations uh + k =

q1

∫q

10

ch + k ,1 dq1 , k = 1, 2,..., s − h ,

(22.4.18)

where q10 is an arbitrary constant, and using the relations (22.4.17), we can write ∂uh + k = ∂ql

q1

∫q

10

∂ch + k ,1 d q1 = ∂ql

q1

∫q

10

∂ch + k ,l dq1 , ∂q1

which leads to ch + k ,l =

∂uh + k + ch + k ,l ( q10 , q2 ,..., qs ) ; ∂ql

replacing in (22.2.88), we obtain, finally, dqh + k = duh + k +

h

∑ ch + k , j ( q10 , q2 ,..., qs ) dq j .

j =2

(22.4.19)

Hence, in case of a non-holonomic mechanical system, we can write an equation of motion corresponding to a certain generalized co-ordinate, in the form of an equation of Lagrange type, if, in the non-holonomic constraint relations, it is possible to separate a total differential, so that in the other constraint relations does no more appear the respective generalized co-ordinate. Let us consider now the system of equations of motion in quasi-co-ordinates (22.2.35), with the notations (22.2.35'), written in the hypothesis in which the constraint relation are catastatic, of the form (22.2.31); assuming that the generalized forces Q j* , j = 1, 2,..., s , are conservative, we can introduce the kinetic potential L * = T + U , and the equations of motion take the form d ⎛ ∂L * ⎞ ∂L * s − m i ∂L * π = 0, j = 1, 2,..., s − m , − + ∑ γkj ∂π j k dt ⎜⎝ ∂π j ⎟⎠ ∂π j k =1

(22.4.20)

where L * is a function of the generalized co-ordinates q1 , q2 ,..., qs and, in general, of the generalized quasi-velocities π1 , π 2 ,..., πs . If the condition s −m

∂α

∑ βmk βl 1 ⎛⎜⎝ ∂qmil

k =1

−

∂αim ⎞ ∂L * π = 0 ∂ql ⎟⎠ ∂π j k

(22.4.21)


500

is fulfilled, then the equation (22.4.20), corresponding to the quasi-co-ordinate π1 takes the form of an equation of Lagrange; this condition is fulfilled if, e.g., γki 1 = 0, i = 1, 2,..., s , k = 1, 2,..., s − m . If, moreover, the quasi-co-ordinate π1 is ignorable (which takes place, e.g., if the kinetic energy T * does not depend explicitly on any generalized co-ordinate), then one obtain a finite integral of the form ∂L * = const . ∂π1

(22.4.22)

Let us return to the equations (22.2.35), (22.2.35'), where T * is a function of the generalized co-ordinates q1 , q2 ,..., qs and, in general, on the generalized velocities π1 , π 2 ,..., π s ; we notice that, due to non-holonomic constraints, we can equate to zero the generalized quasi-velocities πs − m + k = 0, k = 1, 2,..., m , in the expressions ∂T * / ∂π i , i = 1, 2,..., s , but only after effecting the differentiation. Multiplying each member of the equation (22.2.35) by π j and summing, it results s −m

⎡

j =1

⎣

∂L * ⎤ d ⎛ ∂L * ⎞ − π j = ⎟ ∂π j ⎦⎥ ⎝ ∂π j ⎠

∑ ⎢ π j dt ⎜

s −m

∑ Qj* π j , j =1

where we took into account that s −m s −m

∑ ∑ γkji πk π j

k =1 j =1

= 0, i = 1, 2,..., s ,

according to the relations (22.2.35) and to the fact that one can equate to zero the generalized quasi-velocities πs − m + k = 0, k = 1, 2,..., m . We notice that dT * = dt

s −m ⎛

∑ ⎜ πj j =1

⎝

∂T * ∂T * + π j ∂π j ∂π j

⎞ ⎟, ⎠

because T * does not depend explicitly on time, having catastatic constraints. In general, T * = T2* + T1* + T0* ,

(22.4.23)

where T2* is a quadratic form in the generalized quasi-velocities π j , j = 1, 2,..., s − m , T1* is a linear form in the same generalized quasi-velocities, while T0* does not depend on these generalized quasi-velocities; as well, Euler’s theorem for homogeneous functions leads to s −m

∑ π j j =1

∂T * = 2T2* +T1* . ∂π j


501

Finally, we can write d ⎡ 2T * + T1* − (T2* + T1* + T0* ) ⎤⎦ = dt ⎣ 2

s −m

∑ Q j* π j

,

j =1

so that d (T2* − T0*

s −m

) = ∑ Q j* dπj

.

j =1

(22.4.24)

Assuming that the generalized forces Q j* , j = 1, 2,..., s − m , derive from a simple potential U or from a generalized potential with the scalar part U 0 , we introduce the potential energy V = −U or V = −U 0 , respectively, so that Q j* = −

∂V , j = 1, 2,..., s − m . ∂π j

(22.4.25)

In this case, the relation (22.4.24) becomes

d (T2* − T0* + V

)=0

(22.4.24')

and we obtain the first integral of the generalized mechanical energy

E * = T2* − T0* + V = h ,

(22.4.26)

where E is the generalized mechanical energy, analogue to the first integral (18.2.68) obtained in case of holonomic mechanical systems. If T0* = 0 , the kinetic energy T * being, e.g., a quadratic form in the generalized quasi-velocities π1 , π 2 ,..., π s − m , then the first integral (22.4.26) is reduced to the first integral of the mechanical energy E* = T* +V = h .

(22.4.27)

22.4.2.2 Applications Let be a circular disc, of radius R and mass M , which rolls without sliding on the fixed horizontal plane O ′x1′x 2′ ; the O ′x 3′ -axis is directed towards the part in which is the disc. A movable system of axes has the pole at the centre O of the disc, the Ox 3 -axis being normal to its plane and the Ox1 -axis being horizontal; the tangent to the disc at the contact point I makes the angle ψ with the O ′x1′ -axis, the axes O ′x 3′ and Ox 2 make the angle θ , while the angle ϕ specifies the rotation of the disc (Fig. 22.20). The conditions at the contact point are dx1′ = R cos ψdϕ , dx 2′ = R sin ψdϕ .

(22.4.28)


502

We introduce the generalized co-ordinates q1 = θ , q2 = ϕ , q 4 = ψ , q 4 = x1′ , q 5 = x 2′ , the mechanical system having 5 − 2 = 3 degrees of freedom. We choose the angles θ , ψ and ϕ as independent co-ordinates and notice that the constraint relations do not contain the generalized co-ordinate θ ; we can thus write an equation of Lagrange corresponding to this generalized co-ordinate.

Fig. 22.20 Motion of a circular disc, which rolls without sliding on a fixed horizontal plane

Taking into account (22.4.28), we find Lagrange’s functions in the form 1

L * = ⎡⎣ ( J + MR2 ) θ2 + ( I 3 + MR2 ) ( ϕ − ψ sin θ ) 2

2

+ J ψ 2 cos2 θ ⎤⎦ − MgR cos θ ,

(22.4.29)

where I 1 = I 2 = J and I 3 are the central principal moments of inertia of the disc, while M g is its weight.

Comparing the constraint relations (22.4.28) to the relations (22.2.88), we get c42 = R cos ψ, c52 = R sin ψ ,

the other coefficients being equal to zero; we notice that ch + k ,1 ∂c42 ∂c ∂ ( R cos ψ ) ∂ ( R sin ψ ) = = 0, 52 = = 0, = 0, ∂q1 ∂θ ∂q1 ∂θ ∂ql

so that the relations (22.4.17) are verified. In this case, Chaplygin’s equation is reduced to Lagrange’s equations d ⎛ ∂L * ⎞ ∂L * − = 0, ∂θ dt ⎜⎝ ∂θ ⎟⎠


503

which takes the form

(J

+ MR2 ) θ + ( I 3 + MR2

) ( ϕ − ψ sin θ ) ψ cos θ

+J ψ 2 sin θ cos θ − MgR sin θ = 0,

but we cannot write analogous equations corresponding to the generalized co-ordinates ψ and ϕ . We notice also that, in this case, the generalized co-ordinate θ is not an ignorable co-ordinate, because the kinetic potential L * depends explicitly on θ . But we can write a first integral of the mechanical energy in the form

(J

+ MR2 ) θ2 + ( I 3 + MR2

) ( ϕ − ψ sin θ )

2

+ J ψ 2 cos2 θ + 2 MgR cos θ = 2h ,

(22.4.30) where h is the energy constant, which is determined by the initial conditions. Let us consider now also the free motion of a homogeneous sphere, of radius R and mass M , on a horizontal plane O ′x1′x 2′ , the O ′x 3′ -axis being directed towards the part where is the sphere, the constraint relations being given by x1′ − Rω2′ = 0, x2′ − Rω1′ = 0 .

(22.4.31)

We introduce the quasi-velocities π1 , π 2 , π 3 , π 4 , π 5 by means of the relations (see the relations (5.2.35')) π1 ≡ ω1′ = θ cos ψ + ϕ sin θ sin ψ, π 2 ≡ ω2′ = θ sin ψ − ϕ sin θ cos ψ , π 3 ≡ ω3′ = ϕ cos θ + ψ ,

(22.4.32)

π 4 = R ( θ sin ψ − ϕ sin θ cos ψ ) − x1′ , π 5 = R ( θ cos ψ + ϕ sin θ sin ψ ) − x2′ ,

where we have used Euler’s angles θ , ψ and ϕ ; hence, it results ψ = − π1 sin ψcotθ + π 2 cos ψcotθ + π 3 , θ = π1 cos ψ + π 2 sin ψ , ϕ =

π1 sin ψ π 2 cos ψ − , sin θ sin θ x1′ = Rπ 2 − π 4 ,

x2′ = − Rπ 1 − π 5 .

Taking into account these relations, the kinetic energy

(22.4.32')


504 T =

1 M ⎡ x ′2 + x2′2 + i 2 ( ω1′2 + ω2′2 + ω3′2 ) ⎤⎦ , 2 ⎣ 1

(22.4.33)

where i is the gyration radius with respect to a diameter of the sphere, takes the form T* =

1 M ⎡ ( Rπ 2 − π 4 )2 + ( Rπ 1 − π 5 )2 + i 2 ( π12 + π 22 + π 32 ) ⎤⎦ . 2 ⎣

(22.4.33')

It results thus ∂T * = M ⎣⎡ ( R2 + i 2 ) π1 − Rπ 5 ⎦⎤, ∂π1 ∂T * = M ⎡⎣ ( R2 + i 2 ) π 2 − Rπ 4 ⎤⎦, ∂π 2 ∂T * ∂T * = Mi 2 π 3 , = M ( Rπ 2 − π 4 ) , ∂π 3 ∂π 4 ∂T * = − M ( Rπ1 − π 5 ) . ∂π 5 1 1 2 2 3 3 = − γ32 = 1, γ 31 = − γ13 = 1, γ12 = − γ21 = 1, The non-zero coefficients γki j are γ23 4 4 5 5 γ13 = − γ 31 = R, γ23 = − γ 32 = R . The condition (22.4.21) is verified, as one can see

by a direct calculation. We notice also that T * does not depend explicitly on the quasi-co-ordinates, so that the quasi-co-ordinates π1 , π2 and π3 are ignorable (they are hidden too); the equations (22.4.20) lead to the first integrals ( L * = T * , the potential energy being constant in the motion on a horizontal plane) ∂T * ∂T * ∂T * = const, = const, = const . ∂π1 ∂π 2 ∂π 3

(22.4.34)

Putting π 4 = π 5 = 0 , which corresponds to the constraint relations (22.4.31), we notice that the first integrals (22.4.34) lead to π1 = ω1′ = const, π 2 = ω2′ = const, π 3 = ω3′ = const .

(22.4.35)

Taking into account the constraint relations (22.4.31), the kinetic energy (22.4.33) takes the form T* =

1 M ⎡ ( R2 + i 2 2 ⎣

)( ω1′2

+ ω2′2 ) + i 2 ω3′2 ⎦⎤ ;

(22.4.36)

one obtains thus the first integral of the mechanical energy

( R2

+ i2

)( ω1′2

+ ω2′2 ) + i 2 ω3′2 =

2h , M

the energy constant h being determined by initial conditions.

(22.4.37)

Chapter 23 Stability and Vibrations Some problems of stability of equilibrium and motion have been put in evidence, in an incipient form, in the frame of Newtonian mechanics, in several of the preceding chapters; but these problems need a more profound study, with a general character, in the frame of Lagrangian or Hamiltonian mechanics. The motion of a mechanical system S can take the form of linear or even non-linear vibrations; the corresponding mechanical phenomenon needs a thorough study in a general form too. After a study of the stability of discrete mechanical systems, one considers the vibrations with a deterministic character of these systems. The corresponding mechanical phenomena are linked between them, the vibrations around a stable position of equilibrium playing a very important rôle.

23.1 Stability of Mechanical Systems In the study of the behaviour of a mechanical system S , at rest with respect to an inertial frame of reference, appears also the notion of stability, at a small change of its configuration of equilibrium. In a qualitative definition, if the mechanical system S returns to its initial configuration, then the position of equilibrium is stable; otherwise, it is instable. A first study for a particle has been made in Sects. 4.1.1.7 and in 7.2.3. The respective problems are extended to the motion of a discrete mechanical system, case in which intervene also the velocities of the particles at an initial state, which must be maintained in certain limits to can speak about the stability of the motion. A study in the frame of Hamiltonian mechanics, in the phase space Γ 2s , is interesting in this order of ideas. We will make, in what follows, a study of the stability of equilibrium and motion, followed by various applications.

23.1.1 Stability of Equilibrium After some introductory notions, we present the Lagrange–Dirichlet theorem and the theorems of Lyapunov and Chetaev. Thus, important criteria concerning the stability of equilibrium are given. We mention a detailed study of the linear systems too (Chetaev, N.G., 1963; Lyapunov, A.M., 1949).


505

506


23.1.1.1 Introductory Notions To fix the ideas, we start from a very simple particular case, that is the case of a particle P of weight G , constrained to stay on a fixed smooth circle, situated in a vertical plane (Fig. 23.1) (see Sects. 4.1.1.5 and 4.1.1.7 too); we have to do with an ideal constraint. The positions of equilibrium are P1 and P2 . The problem is put analogously in case of the mathematical pendulum (see Sect. 7.1.3.1). By a small perturbation of the position of equilibrium P2 , that is by a displacement of the particle P in a position sufficiently close to P2 and imparting to it an initial velocity of sufficiently small intensity, one obtains a motion in which the particle P : (i) remains in an arbitrarily small neighbourhood of the position of equilibrium P2 ; (ii) the kinetic energy (hence, the modulus of the velocity of the particle) remains inferior to an as small as we wish quantity. The properties (i) and (ii) characterize the stability of the position of equilibrium P2 .

Fig. 23.1 Positions of equilibrium of a heavy particle on a smooth circle in a vertical plane

These ideas can be correspondingly generalized for a discrete mechanical system S (we will consider such systems in what follows), the position of which is specified by a representative point P in the configuration space Λs or in the phase space Γ 2s . Thus, we say that a position of equilibrium of a discrete mechanical system S is stable if, after a sufficiently small perturbation of the corresponding representative point P , with sufficiently small arbitrary velocities, so that the magnitude of the kinetic energy of the mechanical system S be sufficiently small, it results a motion in which the representative point P occupies a position in a as small as we wish neighbourhood with respect to the position of equilibrium, while the magnitude of the kinetic energy (hence, the magnitude of the velocities of all particles (or of all rigid solids which form the system) of the mechanical system S ) remains inferior to an arbitrary small limit. Otherwise, if the representative point P leaves a neighbourhood of the position of equilibrium or if the kinetic energy of the mechanical system S is greater than a positive number which does not depend on the initial conditions (hence, the magnitude of the velocities of some of the particles have this property too), then the positions of equilibrium is instable.

Stability and Vibrations

507

Returning to the problem previously considered, we see that the point P1 is an instable position of equilibrium (Fig. 23.1). As a matter of fact, this position is called labile; if the radius of the circle tends to infinity, the circle becoming a horizontal straight line, then the position of equilibrium is indifferent (also instable). In general, we say that a position of a discrete mechanical system S is a position of equilibrium if, at the initial moment, it had this position, with null velocities, and remained – further – in the same position; obviously, this is valid for the representative point P too. Let us suppose that the position of equilibrium is specified by the generalized co-ordinates q1 , q2 ,..., qs ; corresponding to the first principle of Newton, the generalized forces must vanish ( Q j = 0, j = 1,2,..., s ) in this case. The generalized velocities q1 , q2 ,..., qs must vanish too at the initial moment. If the generalized forces depend, in general, both on the generalized co-ordinates and on the generalized velocities ( Q j = Q j (q1 , q2 ,..., qs , q1 , q2 ,..., qs ), j = 1, 2,..., s ), then the generalized co-ordinates must correspond to the position of equilibrium, while the generalized velocities must vanish. Without any loss of generality, we can choose the origin of the co-ordinates just at the position of equilibrium of the representative point P (the equilibrium taking place for q1 = q2 = ... = qs = 0 ); the non-zero generalized co-ordinates of any other position of the representative point P characterize a deviation from the position of equilibrium, being called perturbations of the discrete mechanical system S . The position of equilibrium is called stable position of equilibrium if, for initial perturbations q j0 sufficiently small and for initial generalized velocities q j0 , j = 1, 2,..., s , sufficiently small too (at the moment t = t0 ), the mechanical system S does not leave an arbitrary small neighbourhood, having – as well – arbitrary small velocities, in all the period of motion. More precisely, if – for an arbitrarily given ε > 0 – one can determine δ = δ ( ε, t0 ) > 0 , so that for t ≥ t0 the inequalities q j (t ) < ε, q j (t ) < ε, j = 1, 2,..., s ,

(23.1.1)

take place, assuming that at the initial moment t = t0 hold the inequalities q j0 (t ) < δ , q j0 (t ) < δ , j = 1, 2,..., s ,

(23.1.1')

then the position of equilibrium q j = 0, q j = 0, j = 1, 2,..., s , is stable; obviously, without any loss of generality, we can take t0 = 0 . It is convenient to give a geometric representation to the inequalities (23.1.1), (23.1.1') in a 2s -dimensional space q1 , q2 ,..., qs , q1 , q2 ,..., qs . In the particular case s = 1 , let be two vicinities of the origin O , corresponding to the mentioned inequalities, in the plane ( q1 , q1 ) (Fig. 23.2). If the point O corresponds to a stable position of equilibrium and if for a given ε > 0 one chooses a corresponding δ > 0 , then any motion which starts at the moment


508

t0 from the interior of a square of centre O and side 2δ remains in the interior of a square which has the same centre and is of side 2ε .

Fig. 23.2 Motion around a stable position of equilibrium in a phase plane

Let be, for instance, the case of a non-damped linear oscillator, which leads to a linear differential equation of the form (see Sect. 8.2.2.1 too) q + ω 2q = 0 ;

(23.1.2)

putting the initial conditions q (t0 ) = q 0 , q (t0 ) = q0 , it results q0 sin ω (t − t0 ), ω q (t ) = q0 cos ω (t − t0 ) − q 0 sin ω (t − t0 ).

q (t ) = q 0 cos ω (t − t0 ) +

(23.1.2')

In this case q (t ) ≤ q 0 +

1 q < ε, q (t ) ≤ q0 + ω q 0 < ε ω 0

(23.1.2'')

only if | q 0 |< δ and | q0 |< δ , where we choose, e.g., δ = min( ε / 2 ω , ωε / 2) . We notice, in the above example, that, for finding a stable position of equilibrium, it has been necessary to solve completely the initial value problem, hence to determine the co-ordinates of the representative point P ,

q j = q j (t ; q10 , q20 ,..., qs0 , q10 , q20 ,..., qs0 ), j = 1, 2,..., s ,

(23.1.3)

as functions of the initial conditions and of time (in the particular case s = 1 ); it is possible (the most times) to be not very simple to do it. Thus, it is worth to find a stability criterion for the position of the equilibrium which does not need the integration of the differential equation (or of the system of differential equations) of motion of the discrete mechanical system.


509

It is necessary to make some observations before dealing with such a criterion. If only the trajectory of the representative point P remains in the neighbourhood of a position of equilibrium, its velocities, in a perturbed motion, differing very much from those in the non-perturbed motion, we say that the motion is orbitally stable. Obviously, a stable motion is orbitally stable too; but an orbital stable motion is not – in general – stable. The notion of stability, introduced for a position of equilibrium, has been extended also to the motion of a discrete mechanical system S . If, to very small perturbations of the initial conditions corresponds a motion which – from the point of view of the trajectory of the representative point, as well as of its generalized velocities – remains always in the neighbourhood of the non-perturbed motion, then the considered motion is stable; otherwise, it is instable. More precisely, let us denote by xi (t ), i = 1, 2,..., 2s , the state of the representative point P in the phase space Γ 2s (the generalized co-ordinates and the generalized momenta, hence the canonical co-ordinates), in a non-perturbed motion the discrete mechanical system S ; let be, as well, x i (t ), i = 1, 2,..., 2s , the corresponding variables in a perturbed motion. If, an arbitrary ε > 0 being given, one can determine δ = δ ( ε, t0 ) > 0 so that, for any t ≥ t0 , the inequalities x i (t ) − xi (t ) < ε, i = 1, 2,..., 2s ,

(23.1.4)

take place, assuming that at the initial moment t = t0 hold the inequalities x i (t0 ) − xi (t0 ) < δ , i = 1, 2,..., 2s ,

(23.1.4')

then the motion of the representative point P (hence, of the discrete mechanical system S ) is stable. We can express the above condition of stability also in the form 2s

∑ [ xi (t ) − xi (t ) ]2

i =2

< r2 ,

(23.1.5)

where r is an as small as we wish real number; no one of the absolute values | x i (t ) − xi (t ) |, i = 1, 2,..., 2s , can be greater than r . If x i (t ) are continuous functions and if r is sufficiently small, then the representative point P will be as close as we wish from the representative point P , in the interior of the hypersphere Sr , 2s

∑ [ xi (t ) − xi (t ) ]2

i =1

= r2 ,

(23.1.5')

of centre P and radius r . Indeed, we can pass from one definition to another one, observing that the point P is at the moment t0 in the interior of a hypersphere S δ , of radius δ , and that, in the perturbed state, at the moment t , the point P is in the interior of a hypersphere S ε , of radius ε , concentric with S δ .


510

A representative point P , which verifies the inequality 2s

∑ [ xi (t ) − xi (t ) ]2

i =1

> r2 ,

(23.1.5'')

is exterior to the hypersphere Sr ; hence, there exists at least an absolute value | x i (t ) − xi (t ) |, i = 1, 2,..., 2s , which is greater than the value r . Hence, if r is sufficiently great, the condition (23.1.5'') corresponds to instable motion. Let dx i = Xi (x1 , x 2 ,..., x 2 s ; t ), i = 1, 2,..., 2s , dt

(23.1.6)

be a system of 2s differential equations of first order, e.g., Hamilton’s canonical equations; we assume that the gives functions Xi , i = 1, 2,..., 2s , satisfy the conditions of the Cauchy-Lipschitz theorem of existence and uniqueness (see Sect. 19.1.1.4). We make firstly a study of the stability of a certain solution xi (t ) = 0, i = 1, 2,..., 2s , of this system, which – obviously – verifies the relations dxi (t ) = Xi ( x1 (t ), x2 (t ),..., x2 s (t ); t ) , i = 1, 2,..., 2s . dt

(23.1.6')

Subtracting the differential equations (23.1.6), (23.1.6') one from the other and making the change of functions x i (t ) = xi (t ) + yi (t ), i = 1, 2,..., 2s , we may write d yi = Xi ( y1 + x1 (t ), y2 + x2 (t ),..., y2 s + x2 s (t ); t ) dt − [ x1 (t ), x2 (t ),..., x2 s (t ); t ] , i = 1, 2,..., 2s ,

(23.1.7)

obtaining thus the system of differential equations of the perturbed motions, called so by Alexandr Mikhailovich Lyapunov; this denomination is due to the fact the variables yi are equal to the perturbations x i (t ) − xi (t ), i = 1, 2,...,2s . This system admits – obviously – the solutions yi = 0, i = 1,2,..., 2s , which can be considered as corresponding to a configuration of equilibrium. Thus, the passing from the system (23.1.6) to the system (23.1.7) means the passing from the study of the stability of motion to the study of the stability of a configuration of equilibrium; hence, we will begin with the study of the position of equilibrium of the representative point P . Without any loss of generality, we can return to the system of equations (23.1.6), assuming that the solutions x1 = x 2 = ... = x 2 s = 0 are verified; hence, we study the stability of these solutions. If, an arbitrary ε > 0 being given, one can determine δ = δ ( ε, t0 ) > 0 , so that, for any t > t0 , the inequality ρ (t ) =

x12 (t ) + x 22 (t ) + ... + x 22s (t ) < ε

(23.1.8)


511

takes place, assuming that, at the moment t = t0 , holds the inequality ( x i (t0 ) = x i0 ,

i = 1, 2,..., 2s ) ρ0 =

( x10 )2 + ( x 20 )2

+ ... + ( x 20s

then the trivial solution x i = 0, i = 1, 2,..., 2s

)2

< δ,

(23.1.8')

is stable in Lyapunov’s sense.

Otherwise, the motion is instable. The Cauchy–Lipschitz theorem ensures the existence and the uniqueness of the solution and, as well, of the continuous dependence of it at the initial data in the vicinity or the moment t = t0 . To introduce the notion of stability in Lyapunov’s sense, it is necessary to can prolong the non-perturbed solutions, as well as all the perturbed solutions for any t > t0 . Hence, the stability problem is the problem of its continuous dependence on the initial data on an infinite interval of time. If the above solution is stable in Lyapunov’s sense and if there exists δ > 0 , so that the condition (23.1.8') be verified, the condition lim ρ (t ) = lim x12 (t ) + x 22 (t ) + ... + x 22s (t ) = 0

t →∞

t →∞

(23.1.8'')

being also involved, then we say that the solution is asymptotically stable (or completely stable); hence, the point P of the perturbed position tends to the position of stable equilibrium in an infinite time. The domain in the space Γ 2s for which any motion which starts from a point of it is asymptotically stable is called domain of attraction of the point x i = 0, i = 1, 2,...,2s ; if this domain is just the whole space, then the corresponding solution is called globally stable. If the functions Xi = Xi ( x1 , x 2 ,..., x 2 s ; t ), i = 1,2,..., 2s , depend explicitly on time, then the system of differential equations (23.1.6) is non-autonomous; otherwise, hence if Xi = Xi (x1 , x 2 ,..., x 2 s ), i = 1, 2,..., 2s , the system is called autonomous or dynamical. Let us consider now an autonomous system and let us suppose that the solutions x i (t ), i = 1, 2,..., 2s , correspond to periodical motions; the trajectory C in the phase space will be closed. Let be a point P ′ ∈ C and a point P is the neighbourhood of this curve; by definition, the distance from the point P to the curve C is given by d( P ,C ) = inf {d( P , P ′), P ′ ∈ C } . If, being given ε > 0 arbitrary, there exists δ = δ ( ε, t0 ) > 0 , so that any solution x1 (t ), x 2 (t ),..., x 2 s (t ) which passes through the point P0 (x10 , x 20 ,..., x 20s ) of Γ 2s at

the moment t = t0 and fulfils the condition d(P0 ,C ) < δ

has the property

(23.1.9)


512

d(P ,C ) < ε

(23.1.9')

for any t > t0 , the point P corresponding to the solution being an arbitrary point of the trajectory of the perturbed motion, then we say that the trajectory C is orbitally stable (or stable in Poincaré’s sense). If the trajectory C is orbitally stable and if there exists δ > 0 , so that the condition (23.1.9) holds and the condition

lim d(P ,C ) = 0

(23.1.9'')

t →∞

be also involved, then we say that the trajectory C is asymptotically orbitally stable; hence, a perturbed curve tends to the curve C in an infinite time. 23.1.1.2 Functions of a Definite Sign We begin with some considerations concerning the functions of a definite sign, which intervene in the study of stability problems. Let be a uniform and continuous function F = F ( x1 , x 2 ,..., x n ) , defined in a neighbourhood

of

the

origin,

so

that

F (0, 0,..., 0) = 0 ;

the

co-ordinates

x i = qi , i = 1,2,..., n = s , can be the co-ordinates of a representative point P in the

configuration space Λs . We say that the function F is positive definite (negative definite) if, for

x12 + x 22 + ... + x n2 < h , it takes only positive (negative) values and

vanishes only at the origin; if, without changing the sign, the function F = F (x1 , x 2 ,..., x n ) vanishes also outside the origin, then we say that it is of constant sign (positive or negative). For instance, the function F = x16 + x 24 + x 32 is positive definite, while the function F = −(x1 − x 2 )2 is of constant sign (negative). One sees immediately that the origin represents a minimum (maximum) for a positive (negative) definite function; hence, the derivative of first order vanishes at the origin too, so that, in a Maclaurin expansion into series, we can write F =

1 n n ∂2 F cij x i x j + ..., cij = , ∑ ∑ 2 i =1 j =1 ∂x i ∂x j

(23.1.10)

where we have neglected the terms of higher order. The linear terms must vanish, otherwise F cannot be of a definite sign. The quadratic form 1 n n ∑ cij xi x j , cij = c ji , i , j = 1, 2,..., n , 2 i∑ = 1 j =1

(23.1.11)

determines the sign of the function F in a sufficiently small neighbourhood of the origin, so that a study of it becomes necessary. Hence, if the quadratic form is positive (negative) definite, then the function F is positive (negative) definite too.


513

Sylvester’s criterion gives a necessary and sufficient condition for the quadratic form to be positive definite. For this, let be the determinants c11

c12

... c1k

c21 Δk = ...

c22 ...

... c2 k ... ... , k = 1, 2,...n ,

ck 1 ck 2

(23.1.11')

... ckk

introduced by Jacobi, starting from the matrix

[ cij ]

c11

c12

... c1 n

c21 = ...

c22 ...

... c2 n ... ... .

cn 1 cn 2

(23.1.11'')

... cnn

If all the determinants are positive ( Δk > 0, k = 1, 2,..., n ), then the quadratic form is positive definite (it can be written in the form of a sum of squares, after Jacobi’s formula). If the function F is negative definite, then the function − F is positive definite and all the coefficients cij change of sign; in this case, the necessary and sufficient condition that the quadratic form be negative definite is expressed by means of the determinants Δk too; the determinants of even index must be positive, while these of odd index must be negative ( ( −1)k Δk > 0, k = 1, 2,.., n ). One can show that, if the function F is of definite sign, then the surface F = F (x1 , x 2 ,..., x n ) = C , C = const , is a closed surface, the origin being in the interior. Observing that x k = x k (t ), k = 1, 2,..., n , it is useful to calculate also the total derivative of the function F with respect to time

dF = dt

n

∑ F,i

i =1

dx i = dt

n

∑ F,i Xi ,

i =1

(23.1.12)

where we have considered the equations dx i = Xi (x1 , x 2 ,..., x n ), i = 1, 2,..., n ; dt

(23.1.12')

the sign of the function dF / dt can give interesting informations concerning the motion of the representative point P ∈ Λs , n = s . Assuming that the function F is positive definite, then the vector grad F is normal to the surface F = C and is directed towards the exterior (in the increasing sense of the function F ); we can write


514

dF = v ⋅ gradF dt

(23.1.12'')

Fig. 23.3 The piercing of the surface F = C by the trajectory of the representative point for dF / dt < 0 (a) and for dF / dt > 0 (b).

too, where v is the velocity of the representative point P . It results thus that: if dF / dt < 0 , then the trajectory of the point P pierces the surface F = C from the exterior towards the interior (Fig. 23.3a); if dF / dt > 0 , then the trajectory of the point P pierces the surface F = C from the interior towards the exterior (Fig. 23.3b), while if dF / dt = 0 , dF / dt > 0 , then the trajectory of the point P is tangent to the surface F = C (eventually, it can stay on this surface). If the function F is negative definite, then one can make inverse affirmations. 23.1.1.3 Lagrange–Dirichlet Theorem In Sect. 4.1.1.7 we have stated the Theorem 4.1.2, according to which the position of equilibrium of a particle subjected only to the action of a given gravitational field is a position of stable or instable (labile or indifferent) equilibrium, as this one has a minimal or non-minimal (maximal or stationary) applicate, respectively. This result, obtained by E. Torricelli in 1644, can be extended to the case of a mechanical system, stating thus. Theorem 23.1.1 (E. Torricelli). The position of equilibrium of a discrete mechanical system S , subjected only to the action of a given gravitational field, is a position of stable, labile or indifferent equilibrium as the gravity centre of the system has a minimal, maximal or stationary applicate (with respect to a frame of reference, one of the axes of which is along the direction of action of the considered field – assuming that it is uniform – having a sense opposite to this action). This is the first important result concerning the stability of the position of equilibrium of a mechanical system, which corresponds to a particular case of given external forces (gravitational field, hence, own weight), but valid both for a discrete and continuous mechanical system. Let us try now to enlarge the frame of our study to the case of conservative forces, starting from a simple potential. For this, we will deal only with a discrete mechanical system S , the configuration of which is specified by the representative point P , of generalized co-ordinates q1 , q2 ,..., qs , in the space of configurations Λs . Let be


515

q1 = q2 = ... = qs = 0 , the considered position of equilibrium. The potential function U = U (q1 , q2 ,..., qs ) is determined neglecting an additive constant, which we choose

so as to have U = U (0, 0,..., 0) = 0 . We suppose that the origin corresponds to an isolated point of extremum for the function U , let this one be a maximum; the respective point corresponds to a position of equilibrium of the discrete mechanical system S , because Qj =

∂U = 0, j = 1, 2,..., s . ∂q j

(23.1.13)

In this case, in a Δ -neighbourhood of the origin | q j |< Δ, j = 1,2,..., s , will take place the strict inequality U = U (q1 , q2 ,..., qs ) < U (0, 0,..., 0) = 0 , assuming that not all the generalized co-ordinates vanish simultaneously. The mechanical energy will be expressed in the form E (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) = T − U =

1 g q q − U ; 2 jk j k

(23.1.14)

if at least a generalized velocity is non-zero (neither we assume that the origin is a singular point, nor the Δ -neighbourhood of the origin contains such a point), then we have T > 0 , so that it results E > 0 in the Δ -neighbourhood of the origin, if not all the canonical co-ordinates vanish. Hence, the mechanical energy has an isolated minimum (equal to zero) at O . Let be an arbitrary number ε so that 0 < ε < Δ and let be the ε -neighbourhood defined by (Fig. 23.4, for s = 1 ) q j < ε, q j < ε, j = 1, 2,..., s .

(23.1.15)

Fig. 23.4 Graphical representation of the Lagrange–Dirichlet theorem in the phase plane

The frontier of this neighbourhood being a closed set of points, the function E attains its minimum E ∗ > 0 on this frontier, on which we will have, obviously,


516

− E ≥ E ∗ > 0 ; but the continuous function E vanishes at the origin, hence it will exist always a δ -neighbourhood ( δ ≤ ε ) of the point O so that, in this neighbourhood, E < E ∗ . Therefore, at the initial moment the conditions (23.1.1') hold, then the initial mechanical energy E 0 verifies the relation E 0 < E ∗ ; but the discrete

mechanical system is conservative, so that we have – at any moment – E = E 0 , hence E < E ∗ too. Consequently, we can state that, in its motion, the representative point P cannot reach the frontier of the ε -neighbourhood on which E ≥ E ∗ , being always in its interior, We can thus state (we introduce the potential energy V = −U ) Theorem 23.1.2 (Lagrange–Dirichlet). If, for a certain position of a conservative mechanical system S , the potential energy has an isolated minimum, then this position is a position of stable equilibrium for it. This theorem has been enounced by J.-L. Lagrange in 1788, is his famous treatise “Mécanique analytique” and has been accurately proved, afterwards, by P.G. Lejeune-Dirichlet. It has been shown in Sect. 4.1.1.7 for the case of a single particle (Theorem 4.1.4) (Lagrange, J.L., 1788). If the potential energy V has an isolated minimum, then the potential function U = −V has an isolated maximum. In case of non-conservative generalized forces, we use the general study made in Sect. 18.2.1.3, adding to the conservative part of the force a non-conservative part, so that Qj =

∂U + Q j , Q j = Q j (q1 , q2 ,..., qs ), j = 1, 2,..., s . ∂q j

(23.1.13')

If the power of the non-potential generalized forces P = Q j q j vanishes ( P = 0 ), then these forces are gyroscopic, being conservative (they derive from a generalized potential, depending – obviously – on the distribution of the generalized velocities); one can use the conservation theorem of mechanical energy (indeed, dW = − dV ) , so that the above proof does not change, the Theorem 23.1.2 remaining, further, valid. If P < 0 , then the non-potential generalized forces are dissipative. The mechanical energy E = T − U decreases in time, in this case, so that E ≤ E 0 (instead of E = E 0 ); if E 0 < E ∗ , then it results E < E ∗ too during the motion and further the

Theorem 23.1.2 can be enounced. One must also show that, in the cases mentioned above (for which P ≤ 0 , | q j |< Δ, j = 1,2,..., s ), the position of equilibrium is maintained. We assume, as above, that the position of equilibrium takes place for q j = 0, j = 1, 2,..., s . We assume also that, among the functions Q j , there exists at least one for which Q j (0, 0,..., 0) ≠ 0 ; in this case, due to the continuity, Q j ≠ 0 in a Δ -neighbourhood

too. Because the canonical co-ordinates are independent, we can choose their values in this neighbourhood so that P > 0 , which contradicts the hypotheses of gyroscopic


517

forces; we have thus a contradiction. Hence, by adding gyroscopic or dissipative forces, the position of equilibrium remains the same. We can apply the Theorem 23.1.2 to the example given before (the example in Fig. 23.1 and the linear oscillator of equation (23.1.2)), the position of equilibrium being a stable one. One can show that the position of equilibrium of a conservative discrete mechanical system (or to which gyroscopic or dissipative forces have been added) is, as well, stable if the potential energy V corresponding to this position has a non-strict minimum, but is such that, in any ε -neighbourhood of the position of equilibrium, there exists a closed hypersurface f (q1 , q2 ,..., qs ) = 0 which contains this position and on which the values of the potential energy V are strictly greater than its value at the position of equilibrium. For instance, a conservative system with one degree of freedom and a potential energy V = q 4 sin2 (1/ q ) and V (0) = 0 has the position q = 0 as stable position of equilibrium. We have shown in Sect. 1.1.1 how can be expressed a condition of equilibrium by means of a hypersphere Sr of equation (23.1.5'); one can obtain thus a proof of the Theorem 23.1.2 in a synthetic but rigorous form. We introduce an isoenergetic hypersurface formed by the set of representative points P (in the phase space Γ 2s ), the canonical co-ordinates of which verify the condition E (P ) − E (Pmin ) = c , c = const ;

(23.1.16)

the mechanical energy E ( P ) has an isolated minimum at Pmin and the constant c can take arbitrary values (positive values on isoenergetic hypersurface which have points in the neighbourhood of Pmin ). If c = 0 , then the hypersurface is reduced to the point Pmin . The mechanical energy E is a continuous function so that, for increasing

positive, small values of c , the corresponding isoenergetic hypersurfaces are closed, each of them containing the point Pmin , as well as all the preceding hypersurfaces, in the order in which the constant c increases. Taking into account that the mechanical energy is conserved during the motion (the generalized forces are conservative or quasi-conservative), the representative point P describes a curve situated on the isoenergetic hypersurface which passes through P0 , corresponding to the state of the discrete mechanical system at the initial moment; the constant c has the value c = E ( P0 ) − E ( Pmin ) .

(23.1.16')

For the trajectory of the representative point P be in the interior of the hypersphere S ε it is necessary and sufficient that to the initial perturbation of the state of equilibrium does correspond a representative point P0 , which belong to an isoenergetic hypersurface situated entirely in S ε ; taking into account how the isoenergetic hypersurfaces are situated around Pmin , it results that such a choice of the initial perturbation P0 is always possible, the Theorem 23.1.2 being thus proved.


518

In the case of a discrete mechanical system of n particles, of masses mi , of weights mi g and applicates z i , i = 1, 2,...., n , with respect to a horizontal plane (in opposite sense to the gravity action), the potential energy is given by −U =

n

∑ mi gzi

i =1

=Mgz , M =

n

∑ mi ,

i =1

(23.1.17)

where z is the applicate of the gravity centre, we find again Torricelli’s theorem ( −U has an isolated minimum if the applicate z is minimal). The theorem remains valid if, besides particles, the discrete mechanical system contains rigid solids too. 23.1.1.4 Stability of the Equilibrium of an Autonomous Discrete Mechanical System with One Degree of Freedom, in Linear Approximation We begin the study of an autonomous discrete mechanical system with the most simple case, that in which there is only one degree of freedom; the phase space Γ 2 has two dimensions, the representative point P being of canonical co-ordinates q , p (the generalized co-ordinate and the generalized momentum). Hamilton’s system of differential equations can be written in the autonomous general form dx 1 dx = X1 (x1 , x 2 ), 2 = X 2 (x1 , x 2 ) . dt dt

(23.1.18)

The Cauchy–Lipschitz theorem 19.1.2 of existence and uniqueness ensures us, in sufficiently large conditions, that through any point of the phase space passes only one integral curve of the system (23.1.18) ( x1 = x1 (t ), x 2 = x 2 (t ) ), which satisfies the given initial conditions (in a Cauchy problem); hence, two integral curves cannot have common points. x1 = α1 , x 2 = α2 be a position of equilibrium so that X1 ( α1 , α2 ) Let = X 2 ( α1 , α2 ) = 0 ; one can thus obtain all the positions of equilibrium. By the translation x1 → x1 + α1 , x 2 → x 2 + α2 one obtains X1 (0, 0) = X 2 (0, 0) = 0 ; one can thus admit, without any loss of generality, that the origin represents a position of equilibrium. Assuming that, in general, X1 ( x1 , x 2 ) ≠ 0 , we can write. X (x , x ) dx 2 = 2 1 2 , X1 ( x 1 , x 2 ) dx 1

(23.1.18')

so that, at a regular point (for which x12 + x 22 ≠ 0 ), the tangent to the integral curve is well determined; if X1 = X 2 = 0 , then the tangent is non-determined, having to do with a singular point (e.g., the origin). We expand into a Maclaurin series and obtain X1 (x1 , x 2 ) = a11x1 + a12 x 2 + R1 (x1 , x 2 ), X 2 ( x1 , x 2 ) = a21x1 + a22 x 2 + R2 (x1 , x 2 ),

(23.1.19)


519

where R1 , R2 are remainders (higher powers of the variables x1 , x 2 ) and where aij , i , j = 1, 2 , are constants. In linear approximation, we consider the system (called the system of the first approximation too) dx 1 dx = a11x1 + a12 x 2 , 2 = a21x1 + a22 x 2 ; dt dt

(23.1.20)

the corresponding characteristic equation is a12 ⎤ ⎡ a11 − λ det A = det ⎢ ⎥ =0 a22 − λ ⎦⎥ ⎣⎢ a21

(23.1.20')

or, in a developed form, λ2 − (a11 + a22 )λ + a11a22 − a12a21 = 0 .

(23.1.20'')

The roots λ1 and λ2 of this equation are the eigenvalues of the matrix A and the form of the integral curves in the neighbourhood of the position of equilibrium depends on them. The solution of the system of differential equations is, in this case, of the form x1 = C 1 eλ1t + C 2 eλ2t , x 2 =

1 ⎡ ( λ − a1 )C 1 eλ1t + ( λ2 − a1 )C 2 eλ2t ⎤⎦ , a2 ⎣ 1

(23.1.21)

where C 1 ,C 2 are two integration constants; by a change of axes, passing from the orthogonal Cartesian system Ox 1x 2 to the oblique Cartesian system O ξ1 ξ2 defined by the relations x1 = ξ1 + ξ2 , x 2 =

1 [ ( λ1 − a1 ) ξ1 + ( λ2 − a1 ) ξ2 ] , a2

(23.1.21')

one obtains the parametric representation ξ1 = C 1 eλ1t , ξ2 = C 2 eλ2t .

(23.1.21'')

(i) Distinct real roots of the same sign. Eliminating the time t between the equations (23.1.21''), we get ξ2 = C ξ1n ,

(23.1.22)

where n = λ2 / λ1 , and C is a new integration constant, specified by the initial conditions; the curve are parabolas of degree n .


520

If | λ1 | 1 , and the parabolas tend to be tangent at the origin to the O ξ1 -axis (Fig. 23.5a); if | λ1 |>| λ2 | , then it results n < 1 and the parabolas tend to be tangent at the origin to the O ξ2 -axis (Fig. 23.5b). The respective singular point is called node.

Fig. 23.5 Representation of the solution for real positive distinct eigenvalues λ: (a) λ1 < λ2 ; (b) λ1 > λ2 , around an instable node

If the roots λ1 and λ2 are both positive and if t increases, then the representative point moves away from the singular point, which is an instable node (Fig. 23.5), while if the roots λ1 and λ2 are both negative and if t increases, then the representative point comes close to the singular point, which is – in this case – a stable asymptotic node (Fig. 23.6).

Fig. 23.6 Representation of the solution for real negative distinct eigenvalues λ: (a) λ1 < λ2 ; (b) λ1 > λ2 , around a stable node

(ii) Real roots of opposite signs. Assuming that λ1 > 0, λ2 < 0 and denoting n = − λ1 / λ2 , we obtain curves of the form

ξ2 =

C , ξ1n

(23.1.22')

hence a family of hyperbolas of degree n . The singular point is called saddle (Fig. 23.7). If t → ∞ , then ξ1 → ∞ and ξ2 → ∞ , the representative point moving off from the origin. A saddle point is instable. If λ1 < 0, λ2 > 0 , then the trajectories are the same, but they are traveled through in an inverse sense. We mention that, in the first


521

case, the O ξ2 -axis is a stable variety (Fig. 23.7), while, in the second case, the O ξ1 -axis is such a variety.

Fig. 23.7 Representation of the solution for real eigenvalues λ of opposite signs, around an instable saddle point

(iii) A non-zero root and a zero root. Let λ1 = λ and λ2 = 0 be the roots; we can write

ξ1 = C 1 eλt , ξ2 = C 2 .

(23.1.23)

The trajectories in the phase plane are semi-straight lines parallel to the O ξ1 -axis, which start from the vicinity of the O ξ2 -axis if λ > 0 (Fig. 23.8a) or tend to this axis for t → ∞ if λ < 0 (Fig. 23.8b). Outside the origin, there are still positions of equilibrium on the O ξ2 -axis, for C 1 = 0 and C 2 arbitrary.

Fig. 23.8 Representation of the solution for the eigenvalues λ1 = λ , λ2 = 0 : (a) λ > 0 ; (b) λ < 0 .

(iv) Double roots. In case of a double root, obvious real, we have λ1 = λ2 = λ . We distinguish two subcases: (a) The equations are uncoupled and we can use the results of case (i) with n = 1 . The trajectories of the representative point form a family of semi-straight lines, concurrent at the origin; if the roots are positive and t increases, then the representative point moves away from the origin, obtaining a unstable star (Fig. 23.9a), while if the


522

roots are negative and t increases, then the representative point moves close to the singular point, resulting a stable star (Fig. 23.9b). If λ = 0 , then the general solution is ξ1 = C 1 , ξ2 = C 2 , C 1 ,C 2 = const ,

(23.1.24)

and any point of the phase space is a point of equilibrium.

Fig. 23.9 Representation of the solution for double eigenvalues λ1 = λ2 = λ : (a) λ > 0 , instable star; (b) λ < 0 , stable star

(b) The equations cannot be uncoupled and the solutions are

ξ1 = C 1 eλt , ξ2 = C 2teλt ,

(23.1.24')

Eliminating the time, we get ξ2 = K1 ξ1 ln K 2 ξ1 ,

(23.1.24'')

where K1 , K 2 are other constants of integration, hence a family of logarithmic curves, the singular point being a node; this node is asymptotically stable if λ < 0 and instable if λ > 0 . If λ = 0 , then the general solution is ξ1 = C 1 + C 2t , ξ2 = C 2 ,

(23.1.24''')

the phase trajectories being straight lines λ2 = const , parallel to the O ξ1 -avis, the velocities being constant, in direct proportion to ξ2 (Fig. 23.8); there exist points of equilibrium (C 1 , 0) , different from the origin, on the straight line ξ2 = 0 . (v) Complex roots. If complex roots are α ± β i , then the solutions will be ξ1 = C 1 eαt cos βt , ξ2 = C 2 eαt sin βt ,

(23.1.25)

where we used Euler’s formula. The trajectories of the representative point P form a family of spirals, the singular point being called focus; if α > 0 , then the point P moves away from the origin, which is an instable focus (Fig. 23.10a), while if α < 0 ,


523

then the point P tends to the singular point for t → ∞ , having – in this case – an asymptotically stable focus (Fig. 23.10b).

Fig. 23.10 Representation of the solution for complex eigenvalues of real part α: (a) α > 0 , instable focus; (b) α < 0 , stable focus

If α = 0 , then the roots are purely imaginary and the solutions are ξ1 = C 1 cos βt , ξ2 = C 2 sin βt .

(23.1.25')

Eliminating the time t between these relations, we get the trajectories of the representative point in the form

ξ12 2

C1

+

ξ 22 C 22

= 1,

(23.1.25'')

hence a family of ellipses. The singular point is a centre, being stable (Fig. 23.11).

Fig. 23.11 Representation of the solution for purely imaginary eigenvalues around a stable centre

The notions of point of centre type, point of focus type, point of saddle type etc. have been introduced by H. Poincaré in the study of the behaviour of the integral curves in the neighbourhood of singular points (Poincaré, H., 1952). Let us apply the above theory to the case of the mathematical pendulum of equation (see Sect. 7.1.3.1)


524

g θ + ω 2 sin θ = 0, ω 2 = , l

(23.1.26)

l being the length of the pendulum, and g the gravity acceleration; the position of the particle in motion is given by the angle θ = θ (t ) . With the notations x1 = θ , x 2 = dx1 / dt , we obtain the system of differential equations of first order dx 1 dx = x1 , 1 = − ω 2 sin x1 ; dt dt

(23.1.27)

retaining only the linear terms, we can write dx 1 dx = x 2 , 1 = − ω 2 x1 . dt dt

(23.1.27')

The characteristic equation −λ

1

− ω2

−λ

= λ2 + ω 2 = 0

has only purely imaginary roots λ1,2 = ± ωi , the singular point θ = 0 being a centre of stable equilibrium. To study the position of equilibrium θ = π , we make the change of variable θ = π + ϕ , being led to the differential equations ϕ − ω 2 sin ϕ = 0 ;

(23.1.26')

the equivalent system of differential equations is dx 1 dx = x 2 , 2 = ω 2 sin x1 , dt dt

(23.1.28)

leading to the linear form dx 1 dx = x 2 , 2 = ω 2 x1 . dt dt

(23.1.28')

The characteristic equation −λ

1

ω2

−λ

= λ2 − ω 2 = 0

has real roots of opposite signs λ1,2 = ± ω , the singular point being a saddle point; hence, the equilibrium is instable, the known results being thus verified.


525

23.1.1.5 Stability of the Equilibrium of an Autonomous Discrete Mechanical System with n Degrees of Freedom, in Linear Approximation Let be, in general, an autonomous system of differential equation dx i = Xi (x1 , x 2 ,..., x n ), i = 1, 2,..., n , dt

(23.1.29)

for which the point ( α1 , α2 ,..., αn ) corresponds to a position of equilibrium; the system of equations Xi ( α1 , α2 ,..., αn ) = 0, i = 1, 2,..., n ,

(23.1.29')

specifies the set of these points. Starting from one of these points, the origin (0, 0,..., 0) becomes a position of equilibrium by the translation x i → x i + αi , i = 1, 2,..., n . We can assume, in general, that the system of differential equations (23.1.29) has a position of equilibrium at the origin, this one being a singular point. An expansion into a Maclaurin series leads to Xi (x1 , x 2 ,..., x n ) =

n

∑ aij x j j =1

+ Ri (x 1 , x 2 ,..., x n ), i = 1, 2,..., n ,

(23.1.30)

where R1 , R2 ,..., Rn are remainders (powers of higher order of the variables x1 , x 2 ,..., x n ) and where aij , i , j = 1, 2,..., n , are constants. In a linear approximation, we consider the system (called the system of the first approximation too) dx i = dt

n

∑ aij x j , i j =1

= 1, 2,..., n .

(23.1.31)

Let us search solutions of the form n

x i = ui eλt , i = 1, 2,...., n , ∑ ui

2

i =1

> 0;

(23.1.32)

introducing in (23.1.31), we get n

∑ aij u j j =1

= λui , i = 1, 2,..., n ,

(23.1.33)

after simplifying with eλt ≠ 0 , or n

∑ ( aij j =1

− λδij ) u j = 0, i = 1, 2,..., n ,

(23.1.33')


526

where δij is the symbol of Kronecker. Because at least one of the solutions u j is non-zero, the determinant of the coefficients of the system of linear algebraic equations (23.1.33') must vanish, so that D (λ ) = det [ aij − λδij

] = 0,

(23.1.34)

obtaining thus for λ an algebraic equation of n th degree a 0 λn + a1λn −1 + ... + an −1λ + an = 0 ,

(23.1.35)

which has the roots λi , i = 1, 2,..., n . The equation (23.1.34) is the characteristic (secular) equation of the matrix of coefficients A = [ aij ] ;

(23.1.36)

introducing the roots λi , called eigenvalues, in the system (23.1.33), we find n sets of solutions ui , which form the eigenvectors. We introduce also the column matrices (column vectors) ⎡ x1 ⎢ x2 x = ⎢ ⎢ # ⎢ ⎣⎢ x n

⎤ ⎡ u1 ⎤ ⎥ ⎢ u2 ⎥ ⎥,u = ⎢ ⎥, ⎥ ⎢ # ⎥ ⎥ ⎢ ⎥ ⎦⎥ ⎣⎢ un ⎥⎦

(23.1.36')

and the system of differential equations can be written in the form dx = Ax , dt

(23.1.31')

x = ueλt .

(23.1.32')

its solutions being given by

The eigenvectors are given by the equation Au = λ u , u ≠ 0 ,

(23.1.33'')

and the eigenvalues by the equation D (λ ) = det [ A − λδ ] = 0 ,

(23.1.34')

where δ is the unit matrix. One must make a study of the eigenvalues, to can decide on the stability or instability character of the position of equilibrium.


527

Let us suppose firstly that all the roots are simple; to each root λk corresponds an eigenvector uk and a solution uk eλk t , k = 1,2,..., n ; a linear combination of these solutions

x=

n

∑ C k uk eλ t , k

k =1

(23.1.37)

where C k , k = 1, 2,..., n , are arbitrary constants, is – as well – a solution of the system of linear differential equations (23.1.31). To show that (23.1.37) is the general solution of the mentioned system, one must show firstly that the eigenvectors uk , k = 1, 2,..., n , form a system of linear independent vectors. Let be n

∑ ck uk

k =1

= 0;

multiplying at the left by the matrix A and taking into account the equation (23.1.33''), written for the index k , we obtain n

∑ λk ck uk

k =1

= 0.

Eliminating c1 between the last two relations, we get n

∑ ( λk

− λ1 ) ck uk = 0 .

k =2

Multiplying further, at the left, this last relation by the matrix A and taking into account the equation (23.1.33''), we can eliminate c2 between the equation just obtained and the previous one; proceeding – further – analogously, we can eliminate all the constants ck , obtaining, finally,

( λn − λ1 ) ( λn − λ2 ) ... ( λn − λn −1 ) cn un = 0 , wherefrom cn = 0 , all the brackets being non-zero. Because all constants ck are equivalent, it follows that c1 = c2 = c1 = ... = cn = 0 , the eigenvectors u1 , u2 ,..., un being thus linearly independent. Putting t = 0 in (23.1.37), we can write x0 =

n

∑ ck uk ;

k =1

(23.1.37')

hence, assuming the initial conditions x(0) = x0 , in a Cauchy problem, we can determine, univocally, the constants ck , k = 1, 2,..., n , because the vectors uk are linearly independent and form a basis in an n -dimensional space. Thus, the generality of the solution (23.1.37) is proved.


528

If one of these roots, let be λk , is double, then one can show that, besides the solution uk eλk t , t uk eλk t is a solution too; further, one can show that to a root λk with a multiplicity of order m , corresponds a solution of the form

( C k0

+ C k1t + C k2t 2 + ... + C km −1t m −1 ) uk eλk t .

(23.1.37'')

The formulae (23.1.37) and (23.1.37') lead to important conclusions concerning the problem enounced above. Thus: (i) If all the roots of the equation (23.1.35) are real and negative or are complex conjugate with a negative real part ( Re λk < 0, k = 1, 2,..., n ), then appear factors of the form e− αt and e− αt cos( βt − α ), α, β > 0 , respectively, so that lim x(t ) = 0 ; the t →∞

trajectory of the representative point remains in the neighbourhood of the origin to which it tends, so that this one is position of asymptotically stable equilibrium. If some of the roots are simple and purely imaginary ( Re λk ≤ 0, k = 1, 2,..., n ), then appear also terms of the form cos( βt − α ) , which do not vanish for t → ∞ , but remain bounded for any t ; the position of equilibrium is simply stable. (ii) If at least one root is real and positive or if at least two complex conjugate roots have their real part positive or if at least two complex conjugate roots purely imaginary are multiple (at least double), then appear terms of the form eαt , eαt cos( βt − α ) , t cos( βt − α ), α , β > 0 , which are not bounded for t → ∞ ; the position of equilibrium is instable. Figuratively, let be the complex plane of the roots λk ; if all the roots are at the left side of the imaginary axis, then the equilibrium is asymptotically stable, if on the imaginary axis too are one or several distinct roots, then the equilibrium is simply stable, while if all the roots are at the right side of the imaginary axis or if on the imaginary axis there are multiply roots, then the equilibrium is instable. Let us consider, e.g., the case of a damped linear oscillator, of equation (see Sect. 8.2.2.6) x + 2 μx + ω 2 x = 0, μ > 0 ,

(23.1.38)

with the notations 2 μ = k ′ / m , ω 2 = k / m , where m is the mass, k is an elastic coefficient and k ′ is a coefficient of viscous dumping. We can write the equation in the form of the system dx 1 dx = x 2 , 2 = − ω 2 x1 − 2 μx 2 ; dt dt

the characteristic equation is −λ

1

−ω

−2 μ − λ

2

= λ2 + 2 μλ + ω 2 = 0 ,

(23.1.38')


wherefrom λ1,2 = − μ ±

529 μ2 − ω 2 . The roots are real and negative (if μ2 > ω 2 ) or

complex conjugate with the real part negative (if μ2 < ω 2 ) or we have a negative double root ( μ = ω ); obviously, the position of equilibrium is asymptotically stable, the results obtained in Sect. 8.2.2.6 being thus verified. If we have μ < 0 (self-sustained motions) in the equation (23.1.38), then one can make an analogous study, finding the same roots, which have a positive real part; the position of equilibrium is instable, corresponding to the results in Sect. 8.2.2.7.

23.1.1.6 Asymptotic Stability Criteria for the Systems of Linear Differential Equations We have seen above that the solution of the system of linear differential equations (23.1.31) is asymptotically stable at the origin if and only if all the roots of the characteristic equation have the real part negative; hence, the problem is put to find the conditions which must be fulfilled by the coefficients of the polynomial function Pn (λ ) = a 0 λn + a1λn −1 + ... + an −1λ + an ,

(23.1.35')

so that its zeros do verify the mentioned property. We assume that a 0 > 0 and multiply the equation by −1 if a 0 < 0 . Let be λk < 0, k = 1, 2,..., m ,

the

real

zeros

and

let

be

αk ± βk i, αk < 0 ,

k = 1, 2,...,(n − m )/ 2 , the complex conjugate zeros, which are all at the left of the

imaginary axis; we can write m

Pn (λ ) = a 0 ∏ ( λ − λk ) k =1

( n − m )/ 2

∏

m

k =1 ( n − m )/ 2

k =1

k =1

= a 0 ∏ ( λ − λk )

( λ − αk − iβk ) ( λ − αk + iβk )

∏ ( λ2

− 2 αk λ + αk2 + β k2 ).

(23.1.35'')

Because all the terms at the right side of the expression (23.2.35'') are positive, it results that all the coefficients of the polynomial function (23.1.35') must be positive (and non-zero, a j > 0, j = 0,1, 2,..., n , the polynomial being non-lacunar); this is a necessary condition that its zeros have a negative real part. In 1875, Routh gave an algorithm by means of which, using only the coefficient of the polynomial Pn (x ) , one can show if all the considered zeros have their real part negative; in 1895, Hurwitz stated, independently, an analogous algorithm, expressed in a more convenient form, by means of some determinants, called Hurwitz’s determinants, obtained as principal determinants from Hurwitz’s matrix (Routh, E.J., 1892)


530 ⎡ a1 ⎢a ⎢ 0 ⎢0 H = ⎢ ⎢0 ⎢ ⎢ ... ⎢ ... ⎣

a3

a5

a7

... a2 j −1

...

a2

a4

a6

... a2 j − 2

...

a1

a3

a 5 ... a2 j − 3

...

a0

a2

a4

...

... ...

... ...

... ... ... ...

... a2 j − 4 ... ...

... ...

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥, 0 ⎥ ⎥ ... ⎥ an ⎥⎦

(23.1.39)

where a p = 0 everywhere one has p > n or p < 0 , in the form a1 Δ1 = a1 , Δ2 = a 0

a1 a3 a2 , Δ3 = a 0 0

a3 a2

a5 a 4 ,..., Δn = det H = an Δn −1 .

a1

a3

(23.1.39')

We can thus state Theorem 23.1.3 (Routh–Hurwitz criterion). If all Hurwitz’s determinants corresponding to the polynomial Pn (λ ) , non-lacunary and with all coefficients positive, are positive Δ1 > 0, Δ2 > 0,..., Δn > 0 ,

(23.1.39'')

then and only then all the zeros of this polynomial have their real part negative and the system of differential equations (23.1.31) has an asymptotically stable position of equilibrium at the origin. In particular, if n = 2 , then it results Δ1 = a1 > 0, Δ2 = a2 Δ1 = a1a2 > 0 ,

so that it is necessary and sufficient that the three coefficients be positive. If n = 3 , then we obtain a1 Δ1 = a1 , Δ2 = a 0

a3 a2 = a1a2 − a 0a 3 > 0, Δ3 = a 3 Δ2 > 0 ;

besides the positivity of the coefficients, the necessary and sufficient condition a1a2 − a 0a 3 > 0

must be fulfilled too. If n = 4 , then we have

(23.1.40)


531 Δ1 = a1 > 0, Δ2 = a1a2 − a 0a 3 > 0,

a1 Δ3 = a 0

a3 a2

a5 a 4 = a 3 Δ2 − a12a 4 > 0, Δ4 = a 4 Δ3 > 0;

0

a1

a3

as one can easy see, the condition Δ3 > 0 can be fulfilled only if the condition Δ2 > 0 is also fulfilled. Hence, besides the positivity of the coefficients, only the

necessary and sufficient condition

a 3 ( a1a2 − a 0a 3 ) − a12a 4 > 0

(23.1.40')

must be added. For instance, corresponding to what has been said before, the position of equilibrium is asymptotically stable for the equation (23.1.38), because all the coefficients of the characteristic equation (of second degree) are positive. The Routh–Hurwitz criterion is easy to apply if the characteristic equation has numerical coefficients; but if these coefficients contain also literal parameters, then the problem becomes complicated from a practical point of view. It is useful to consider also the conditions established by Liénard and Chipart in 1914, where the number of inequalities is reduced to a half with respect to the conditions (23.1.39''). As a matter of fact, these conditions come from the observations made on Hurwitz’s determinants, assuming – further – the positivity of the coefficients of the characteristic equation. One sees easily from (23.1.39') that Δn > 0 leads to Δn −1 > 0 . One can show also that the condition Δn −1 > 0 includes also the condition Δn − 2 > 0 , as it has been seen in the preceding particular case; then, Δn − 3 > 0 includes the condition Δn − 4 > 0 a.s.o. We can state Theorem 23.1.4 (Liénard–Chipart criterion). If Hurwitz’s determinants, corresponding to the polynomial Pn (λ ) , non-lacunary and with all coefficients positive, verify the conditions Δn −1 > 0, Δn − 3 > 0, Δn − 5 > 0, ... ,

(23.1.39''')

then and only then all zeros of this polynomial have the real part negative and the system of differential equations (23.1.31) admit an asymptotically stable position of equilibrium at the origin. Let us write the polynomial Pn (λ ) in the form n

Pn (λ ) = a 0 ∏ ( λ − λk ) , k =1

(23.1.35''')


532

where λk , k = 1, 2,..., n , are its n zeros. Then, let us replace λ by ω i and let us make ω vary from −∞ to ∞ ; the increase of the angle θ = arg Pn ( ωi) is given by ∞ Δ−∞ θ(ω ) =

n

∞ arg(iω − λk ) . ∑ Δ−∞

k =1

We notice that (we assume that no one of the roots is on the imaginary axis) ⎧⎪ π for λk < 0, ∞ arg(iω − λk ) = ⎨ Δ−∞ ⎪⎩ − π for λk > 0.

Denoting by nl and nr the number of zeros at the left and at the right of the imaginary axis, respectively ( nl + nr = n ), it results ∞ Δ−∞ θ ( ω ) = ( nl − nr ) π .

Let us consider the hodograph (curve) described by the affix of the complex number Pn ( ωi) , when ω varies from −∞ to ∞ ; this hodograph has two branches: one for ω > 0 and one for ω < 0 , the two branches being symmetric with respect to the real axis, because Pn ( ωi) and Pn ( − ωi) are complex conjugate numbers. Denoting by Δ0∞

the increase corresponding to the variation of ω from 0 to ∞ , we obtain Δ0∞ θ ( ω ) =

1 ∞ 1 Δ−∞ θ ( ω ) = ( nl − nr ) π . 2 2

If nl = 0 and nr = 0 , when all the zeros are situated at the left of the imaginary axis. We can thus state Theorem 23.1.5 (geometrical criterion). All the zeros of the polynomial Pn (λ ) (non-lacunary and with all coefficients positive) have their real part negative if and only if the hodograph of the polynomial Pn ( ωi) , for ω varying from 0 to ∞ , does not pass through the origin (so that Pn (λ ) does not have purely imaginary zeros) and if the condition Δ0∞ θ ( ω ) =

n π 2

(23.1.41)

is fulfilled. We notice that for a stable polynomial (called Hurwitz polynomial too) the argument has a monotone variation when ω varies from 0 to ∞ . This criterion has been applied by Mikhailov to the study of the systems with control function; this criterion is called Mikhailov’s criterion too.


533

23.1.1.7 Instability of Equilibrium. Lyapunov’s and Chetaev’s Theorems In 1892, in his famous doctor thesis “General problem of the stability of motion”, A.M. Lyapunov has put the problem of the reciprocal of the Lagrange-Dirichlet theorem, problem which was not yet completely solved; other studies have been made by N.G. Chetaev (Lyapunov, A.M., 1949; Chetaev, N.G., 1963). Let us return to the notations in Sect. 23.1.1.3 and let us suppose, further, that q1 = q2 = ... = qs = 0 and U (0, 0,..., 0) = 0 for the position of equilibrium. Let be an expansion of the potential function into a power series after the generalized co-ordinates U = U m (q1 , q2 ,..., qs ) + U m + 1 (q1 , q2 ,..., qs ) + ... ,

(23.1.42)

where U k = U k (q1 , q2 ,..., qs ) is a homogeneous function of degree k , k = m , m + 1,... , and U m is not identical to zero and where the smallest term of the expansion corresponds to m ≥ 2 , because at the origin ( ∂U / ∂q j )0 = 0 , this one being a position of equilibrium. Let us expand into a power series, with respect to the generalized co-ordinates, the coefficients g jk = g jk (q1 , q2 ,..., qs ) of the expression (23.1.14) of the kinetic energy 0 T ; we denote T0 = (1/ 2)g jk q j qk , where g 0jk = g jk (0, 0,..., 0) . It results

T = T0 + O , U = U 2 + O ,

(23.1.43)

where O represents the terms of higher order with respect to the generalized co-ordinates and the generalized velocities, respectively. Because T0 is a positive definite quadratic form with constant coefficients, there exists a transformation of co-ordinates η j = αij qi , j = 1, 2,..., s , so that – simultaneously – the two quadratic forms be reduced to two sums of squares, i. e. T =

1 s 2 1 s ηk + O , U = ∑ λk ηk2 + O , ∑ 2 k =1 2 k =1

(23.1.43')

where the coefficients λk , k = 1, 2,..., s , are given by the secular equation 0 det ⎡⎣a jk − λk g jk ⎤⎦ = 0 ;

(23.1.43'')

there exists at least one λk < 0 , because the quadratic form U 2 can take some negative values too. The co-ordinates ηk are called normal co-ordinates or principal co-ordinates. Lagrange’s equations in normal form (written by means of the normal co-ordinates) are of the form ηk = − λk ηk + O (!), k = 1, 2,..., s .

(23.1.43''')


534

Let be the auxiliary quadratic form (V does not represent the potential energy)

V = −

μ2 ⎞ 2 μ2 ⎞ 2 ⎤ 1 s ⎡⎛ 2 ⎛ + + + − + + + 1 1 ε λ λ η μ λ η η λ η , ( ) k ⎢⎜ k k k k k k ⎜ 2 k∑ 2 ⎠⎟ k 2 ⎠⎟ k ⎦⎥ ⎣⎝ ⎝ =1 (23.1.44)

where ⎧⎪ −1 for λk < 0, εk = ⎨ ⎪⎩ 1 for λk ≥ 0,

(23.1.44')

with k = 1, 2,..., s and μ > 0 . It is easy to see that d − μt (e V dt

) = e− μt

⎡ s ⎢ μ ∑ εk ⎣ k =1

μ2 ⎞ 2 ⎛ 2 ⎜ λk + 4 ⎟ ( ηk + ηk ) + O ⎝ ⎠

⎤ ⎥, ⎦

(23.1.44'')

for a mechanical system in motion. Without any loss of generality, let us assume that λ1 < 0 , λ1 being the negative number with the greatest modulus; we choose μ > 0 , so that the conditions λ1 +

μ2 μ2 < 0, λ12 + λ1 + >0 4 2

be fulfilled. In this case, the sum in the expression (23.1.44'') is a positive definite quadratic form. If we choose, in a Δ -neighbourhood, | ηk |< Δ, | ηk |< Δ , k = 1,2,..., s , hence quantities sufficiently small in absolute value, then it results d − μt (e V dt

) > 0,

wherefrom V>V0 eμ ( t −t0 ) ,

(23.1.44''')

with V (t0 ) = V0 . Let us take all the initial values equal to zero, excepting η10 , which will be taken so that | η10 |< Δ . From (23.1.44) and from the second condition put above, it results V0 > 0 , the motion being in the limits of the considered Δ -neighbourhood, as small could be | η10 | ; otherwise, from (23.1.44''') would result lim V = ∞ , even if the quadratic form V is bounded in this Δ -neighbourhood.

t →∞

We can thus state Theorem 23.1.6 (Lyapunov; first theorem). If the potential energy −U (q1 , q2 ,..., qs ) of a conservative discrete mechanical system has not a minimum at a position of


535

equilibrium, which is put into evidence from the quadratic form U 2 (q1 , q2 ,..., qs ) of the expansion (23.1.42) (hence, m = 2, U 2 having negative terms – may be ones of them are positive), then the respective position of equilibrium is instable. For m > 2 in the expansion (23.1.42), without proof, we state Theorem 23.1.7 (Lyapunov; second theorem). If the potential energy −U (q1 , q2 ,..., qs ) of a conservative discrete mechanical system has a local maximum for q1 = q2 = ... = qs = 0 , which can be shown starting from the term U m (q1 , q2 ,..., qs ) , m ≥ 2 , with

the smallest power (in a neighbourhood of the origin, excluding the origin, −U m < 0 , which is possible only if m is even) in the expansion (23.1.42), then the position of equilibrium is instable. As well, we state Theorem 23.1.8 (Chetaev). If the potential energy −U (q1 , q2 ,..., qs ) of a conservative discrete mechanical system is a homogeneous function in the generalized co-ordinates and if at the position of equilibrium q1 = q2 = ... = qs = 0 does not take place a minimum, then this position is instable. Let be, e. g., U = Cq1q2 ...qs , C = const ; Chetaev’s theorem shows that the origin is an instable position of equilibrium.

23.1.1.8 Asymptotic Stability of Equilibrium. Dissipative Discrete Mechanical Systems The notion of asymptotic stability has been introduced in Sect. 23.1.1.1. Thus, a position of equilibrium is called asymptotically stable if it is stable and if, for values (in modulus) sufficiently small of the initial generalized co-ordinates and of the generalized velocities, all the generalized co-ordinates and the generalized velocities tend to zero when the time t increases indefinitely, hence if it exists a number δ0 > 0 so that

lim q j (t ) = 0, lim q j (t ) = 0, j = 1,2,..., s ,

t →∞

t →∞

(23.1.45)

when the inequalities q j0 < δ0 , q j0 < δ0 , j = 1, 2,..., s ,

(23.1.45')

hold. The phenomenon is graphically illustrated in Fig. 23.12 for s = 1 in the space (q , q ) . Let be a holonomic and scleronomic discrete mechanical system subjected to the action of some potential generalized forces ∂U / ∂q j ,U = U (q1 , q2 ,..., qs ) and of some non-potential generalized forces Q j = Q j (q1 , q2 ,..., qs , q1 , q2 ,..., qs ), j = 1, 2,.., s , of the form (23.1.13'). Lagrange’s equations can be written in the normal form (18.2.47'), hence in the form qj = G j (q1 , q2 ,..., qs , q1 , q2 ,..., qs ), j = 1, 2,..., s .

(23.1.46)


536

The total derivative with respect to E = E (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) is given by

time

to

the

dE ∂E ∂E q j + G = E ′(q1 , q2 ,..., qs , q1 , q2 ,..., qs ) ; = dt ∂q j ∂q j j

mechanical

energy

(23.1.47)

hence, at each point of the space (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) , not only the mechanical energy has a finite value, but also its total derivative with respect to time.

Fig. 23.12 Graphical representation of asymptotic stability in the phase plane

If the non-conservative generalized forces Q j are dissipative, then their power is negative (see Sect. 18.2.1.3). In this case, dE / dt ≤ 0 if the discrete mechanical system is in motion, so that the function E is non-positive in the domain of the space in which the motion takes place; one has equality only in the case in which the generalized velocities vanish (q1 = q2 = ... = qs = 0) . Let us suppose that the position of equilibrium of the discrete mechanical system is isolated, hence that in the neighbourhood there are not other positions of equilibrium. One can state Theorem 23.1.9. If the potential energy −U of a dissipative holonomic and scleronomic discrete mechanical system has a strict minimum for an isolated position of equilibrium, then this position is asymptotically stable. Let us consider again the case of a damped linear oscillator of equation (23.1.38), considered in Sect. 23.1.1.5, written in the form mx + k ′x + kx = 0, m , k , k ′ > 0 ;

the mechanical energy is given by E =

1 1 mx 2 + kx 2 , 2 2

wherefrom dE = mxx + kxx = ( mx + kx ) x = − k ′x 2 < 0 . dt


537

Hence, the oscillator is dissipative, the position of equilibrium being isolated, as it results from the equation of motion; using the Theorem 23.1.9, we can state that the position of equilibrium is asymptotically stable, as it has been shown in Sect. 23.1.1.5 too. To generalize the Lagrange-Dirichlet theorem, Lyapunov replaces the mechanical energy E by any function V = V (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) of class C 1 (with respect to the generalized co-ordinates and to the generalized velocities), which has a minimum at the position of equilibrium and which does not increase for any motion of the mechanical system. Let us calculate the total derivative with respect to time dV ∂V ∂V q + G = V ′(q1 , q2 ,..., qs , q1 , q2 ,..., qs ) . = dt ∂q j j ∂q j j

(23.1.48)

Observing that q j = qj = 0, j = 1, 2,..., s , at the origin, the function V ′ vanishes at this point. Because V does not increase for any motion, we have dV / dt = V ′ ≤ 0 , so that V ′ has a maximum at the position of equilibrium; if this maximum is a local one, then in the neighbourhood of the origin (excepting the point O ) we have V ′ < 0 , so that at the origin the function V is strictly decreasing. One can now follow the proof given to the Lagrange-Dirichlet theorem, replacing the function E by the function V . Thus, one can state Theorem 23.1.10 (Lyapunov). Knowing the position of equilibrium of a holonomic and scleronomic discrete mechanical system, subjected to the action of forces which do not depend explicitly on time, if there exists a function V (q1 , q2 ,..., qs , q1 , q2 ,..., qs ) of class C 1 which has a strict extremum at the position of equilibrium and if the derivative V ′ with respect to time (calculated by means of the equations of motion) has an extremum of opposite type, at the same position of equilibrium, then this position is stable. If the extremum of the derivative is strict too, then the position of equilibrium is asymptotically stable. We notice that, in the formulation of the stability criterion, the words “minimum” and “maximum” can be interchanged, because one can replace the function V by the function −V , which leads to the previous formulation. The function V is called Lyapunov’s function.

23.1.2 Stability of Motion In what follows, one makes a general study of the problem by the stability of the motion, with the aid of Lyapunov’ function. A special attention is given to the systems with cyclic co-ordinates and to the limit cycles. We mention also the case of rheonomous systems (Lyapunov, A.M., 1949).

23.1.2.1 General Problem of the Stability of Motion. Lyapunov’s Theorems The stability of the position of equilibrium of a discrete mechanical system is put in evidence, in general, by the inequalities (23.1.1), (23.1.1'). But if these inequalities take place only for some generalized co-ordinates q1 , q2 ,..., qs and generalized velocities


538

q1 , q2 ,..., qs or, more general, for some functions of these variables x i = x i (q1 , q2 ,..., qs , q1 , q2 ,..., qs ), i = 1, 2,..., n ,

(23.1.49)

chosen so that x i (0, 0,..., 0, 0, 0,..., 0) = 0 , then we have to do with a problem of conditioned stability; we assume that these new functions verify the non-autonomous system of differential equations of first order dx i = Xi ( x1 , x 2 ,..., x n ; t ), i = 1,2,..., n , dt

(23.1.49')

where Xi are functions of class C 0 in the domain x i ≤ Δ, t ≥ t0 , i = 1,2,..., n ,

(23.1.49'')

t0 being the initial moment.

To the position of equilibrium correspond x i , i = 1, 2,..., n , which implies the conditions Xi (0, 0,..., 0; t ) ≡ 0, i = 1, 2,..., n .

(23.1.49''')

The stability problem of the null solution imposes the existence of δ = δ ( ε ) > 0 for any ε > 0 , so that if x i (t0 ) < δ , i = 1,2,..., n ,

(23.1.50)

x i (t ) < ε, i = 1,2,..., n ,

(23.1.50')

then the inequalities

must take place for any t ≥ t0 . In the case of asymptotic stability, there must exist δ0 > 0 so that lim x i (t ) = 0, i = 1, 2,..., n ,

(23.1.51)

x i (t0 ) < δ0 , i = 1, 2,..., n .

(23.1.51')

t →∞

if

In the case of a problem of non-conditioned stability of the position of equilibrium, one can take as variables x i , i = 1, 2,..., n , the generalized co-ordinates and the generalized velocities; thus, the system (23.1.49') becomes Lagrange’s system of


539

equations (18.2.29), written in the form of a system of differential equations of first order. If we take these variables as generalized co-ordinates and generalized moments, then one obtains Hamilton’s equations (19.1.14). Let be two important cases of functions Xi , i = 1, 2,..., n : (i) The stationary case in which t does not appear explicitly in these functions ∂Xi X i = = 0, i = 1,2,..., n , ∂t

(23.1.52)

the equations being autonomous. (ii) The periodic case, of period τ with respect to the temporal variable t , for which Xi (x1 , x 2 ,..., x n ; t + τ ) = Xi (x1 , x 2 ,..., x n ; t ), i = 1,2,..., n .

(23.1.52')

By a demonstration analogue to that used for the Lagrange-Dirichlet theorem, one obtains an extension of the Theorem 23.1.10. Thus, we state Theorem 23.1.11 (Lyapunov). If, in case of the stationary or periodic system of differential equations of first order (23.1.49'), there exists a function V (x1 , x 2 ,..., x n ; t ) of class C 1 in the domain (23.1.49''), which – for t considered as a parameter – has a strict extremum at the point x i = 0, i = 1,2,..., n , and if, at the same point and at any moment t , its total derivative with respect to time V ′(x1 , x 2 ,..., x n ; t ) =

n

∂V

∑ ∂xi Xi

i =1

+

∂V ∂t

(23.1.53)

has an extremum of opposite type, then the null solution of the system is stable. If the extremum of the derivative V ′ is strict too, then the solution is asymptotically stable. In a stationary case, the function V cannot depend explicitly on time, while in a periodic case, it must be periodic too, of the same period τ as the functions Xi , i = 1,2,..., n . In the general case (non-stationary and non-periodical) Lyapunov’s function V must fulfill more rigorous conditions. Let us consider a particular case in which one can use the Theorem 23.1.11 for a simple (non-asymptotic) stability. Let us suppose that the function V (x1 , x 2 ,..., x n ; t ) is a first integral of the system of equations (23.1.49'), hence that it becomes a constant (independent of t ), after replacing the variables x i , i = 1, 2,..., n , by a solution of this system; in this case dV / dt = V ′ = 0 has an extremum at the origin for any t . Starting from the Theorem 23.1.11, we can state Corollary 23.1.1. If the system of differential equations (23.1.49') admits a first integral V (x1 , x 2 ,..., x n ; t ) which has a strict extremum at x 1 = x 2 = ... = x n = 0 for any fixed t , then this solution is stable.


540

In a stationary case, Lyapunov’s function does not depend explicitly on time, while in a periodic case this function is periodic with respect to time, of period τ . As we have seen in Sect. 23.1.1.1, the study made on the system of differential equations (23.1.49') is equivalent to a study on the system of equations of perturbed motions, of the form (23.1.7), hence it is equivalent to a study on an arbitrary system of differential equations of first order (e. g., of the form (23.1.6)). The Theorem 23.1.11 can be thus used not only to the study of the stability of a position of equilibrium, but also to the study of the stability of motion.

23.1.2.2 Stability of Systems of Non-linear Differential Equations We consider the system of non-linear differential equations (23.1.29), being limited to the autonomous case; as in Sect. 23.1.1.1, we use the notation ρ (t ) =

x12 (t ) + x 22 (t ) + ... + x n2 (t ) .

(23.1.54)

The mechanical systems which lead to such systems of differential equations are scleronomic ones.

Fig. 23.13 Graphical representation of the stability problem of autonomous systems of non-linear differential equations

Let be a function V (x1 , x 2 ,..., x n ) positive definite (otherwise, one considers – for the proof – the function −V ); we assume that in the spherical domain ρ (t ) ≤ h we have V > 0 and V ′ ≤ 0 . As well, let be also the sphere S ε , specified by ρ (t ) = ε, 0 < ε < h , contained in the sphere Sh , of equation ρ (t ) = h (Fig. 23.13). We

denote by l the inferior bound of the function V on S ε , so that, on this sphere, V ≥ l > 0 . The surface S of equation V (t ) = C , 0 < C < l , will, obviously, be interior to the sphere S ε . Let be also the number δ sufficiently small, so that S δ of equation ρ (t ) = δ be interior to the surface S , non having any common point with this


541

one (which is possible, because S does not pass through the origin, having V > 0 ). Observing that V ≤ 0 (see Sect. 23.1.1.2), the trajectory of the representative point P , which starts from the initial position P0 , interior to the sphere S δ , does not pierce the surface S 0 , specified by the equation V (t0 ) = V0 ; having V0 < C , the surface S 0 is interior to the surface S , the trajectory of the point P being thus interior to this surface. We can state Theorem 23.1.12 (Lyapunov; first theorem of stability). If for a system of non-linear differential equations (23.1.29) (of the perturbed motion), we can build up a function V (x1 , x 2 ,..., x n ) of definite sign, for which – due to the system – the derivative V ′ is a function of constant sign, opposite to the sign of V , or is identically zero, then the non-perturbed motion (the trivial solution) is stable. We use previous notations and assume that V ′ < 0 (strict inequality); in this case, the function V (t ) is strictly monotonically decreasing and inferior bounded (by zero). Hence lim V (t ) = α ≥ 0 . In the closed domain contained between the surfaces t →∞

V = V0 and V = α (we suppose that α ≠ 0 ) the negative function V ′ has a superior

bound −l (l > 0) ; we have l ≠ 0 because V ′ is negative definite (it vanishes only at the origin). It results V ′ ≤ −l in the mentioned domain, for any t ; in this case, the identity V − V0 =

t

∫t V ′dt

leads to V ≤ V0 − l (t − t0 ) . For t sufficiently great, V

0

becomes negative, which contradicts the hypothesis according to which V > 0 . Hence, one can have only α = 0 . Thus, we state Theorem 23.1.13 (Lyapunov; second theorem of stability). If, for the system of non-linear differential equations (23.1.29) (of the perturbed motion) we can build up a function V (x1 , x 2 ,..., x n ) of definite sign, for which – due to the system – the derivative V ′ is a function of definite sign (not only of constant sign), opposite to that of V , then the unperturbed motion (the trivial solution) is asymptotically stable. This theorem gives sufficient conditions of asymptotic stability for small initial perturbations. Barbashin and Krasovski gave sufficient conditions of asymptotic stability for any initial perturbations. As we have seen in Sect. 23.1.1.7, a study of the instability of equilibrium, hence of the motion too, is – as well – important; a proof, analogue to those previously given, allows to state. Theorem 23.1.14 (Lyapunov; first theorem of instability). If, for the system of non-linear differential equations (23.1.29) (of the perturbed motion), we can build up a function V (x1 , x 2 ,..., x n ) for which – according to the system – the derivative V ′ is a function of definite sign, the function V being not of constant sign (and opposite to that of V ′ ), then the unperturbed motion is instable. We can state also Theorem 23.1.15 (Lyapunov; second theorem of instability). If, for the system of non-linear differential equations (23.1.29) (of the perturbed motion) we can set up a function V (x1 , x 2 ,..., x n ) , so that – according to the system – to obtain a derivative


542

dV = λV + W , λ > 0, λ = const , dt

(23.1.55)

W (x1 , x 2 ,..., x n ) being a function identically zero or of constant sign, and if V is not

of constant and opposite to W sign, then the unperturbed motion is instable. We can put together the two theorems of Lyapunov in Theorem 23.1.16 (Chetaev). If, for the system of non-linear differential equations (23.1.29) (of the perturbed motion), we can build up a function V ( x1 , x 2 ,..., x n ) so that, in an arbitrary small neighbourhood of the origin, there exists a domain V > 0 (on the frontier of which we have V = 0 ), for which – according to the system – the derivative V ′ be positive definite on this domain, then the unperturbed motion is instable.

23.1.2.3 Rheonomous Mechanical Systems In the case of rheonomous discrete mechanical systems the temporal variable appears explicitly, the non-linear differential equations being of the form dx i = Xi ( x1 , x 2 ,..., x n ; t ), i = 1,2,..., n ; dt

(23.1.56)

they define a non-autonomous system of differential equations. In the following we say that a function V (x1 , x 2 ,..., x n ; t ) is positive definite if, in the spherical domain Sh definite by n

∑x2

i =1

≤ h 2 , t > t0 ,

i

(23.1.57)

we have V (x1 , x 2 ,..., x n ; t ) ≥ W (x1 , x 2 ,..., x n ), V (0, 0,..., 0; t0 ) = 0 ,

(23.1.57')

the function W (x1 , x 2 ,..., x n ) being positive definite in the domain Sh . Analogously, a function V (x1 , x 2 ,..., x n ; t ) is negative definite if, in the domain Sh , we have V (x1 , x 2 ,..., x n ; t ) ≤ −W (x1 , x 2 ,..., x n ), V (0, 0,..., 0; t0 ) = 0 .

(23.1.57'')

The total derivative with respect to time of the function V (x1 , x 2 ,..., x n ; t ) is calculated in the form dV = dt

n

∂V

∑ ∂xi Xi

i =1

+

∂V . ∂t

(23.1.57''')

In connection with the system of differential equations (23.1.56), we assume that the


543

conditions Xi (0, 0,..., 0; t ) ≡ 0, t > t0 , i = 1, 2,..., n ,

(23.1.56')

hold; taking into account (23.1.56'), we can state that the system (23.1.56) is verified by the solution x i (t ) = 0, i = 1,2,..., n , which is the unperturbed solution, called sometimes the trivial solution. Let V (x1 , x 2 ,..., x n ; t ) be a positive definite function and l be the inferior limit of the function W (x1 , x 2 ,..., x n ) in the spherical domain S ε , 0 < ε < h ; it results V (x1 , x 2 ,..., x n ; t ) ≥ l ,

(23.1.58)

for t ≥ t0 . Taking into account the continuity, we can determine a quantity δ so that V (x 10 , x 20 ,..., x n0 ) < l for x i0 < δ , i = 1, 2,..., n ,

(23.1.58')

according to the second relation (23.1.57''). Let us suppose that the functions x i = fi (t ; x 10 , x 20 ,..., x n0 ) represent the solution corresponding to the initial conditions

x i (t0 ) = x i0 ,| x i0 |< δ , i = 1,2,..., n . Let us suppose that, for t > t0 , takes place a relation of the form n

∑ fi2

i =1

≥ ε2

and let be t ′ > t0 the moment for which takes place the equality (the moment t ′ exists, because the solutions fi can be as close as possible to the trivial solution for which n

∑ fi2

i =1

= 0 ). In this case, according to the relation (23.1.58), we can have

V ( f1 (t ′), f2 (t ′),..., fn (t ′); t ′) ≥ l ; in the hypothesis in which the derivative dV / dt is non-positive, the preceding inequality is contradictory to (32.1.58'), because V could not increase in the interval of time from t to t ′ . Hence, n

∑ fi2

i =1

< ε2

the trivial solution being thus stable. We can state Theorem 23.1.17 (Lyapunov; first theorem). If, for the system of non-linear differential equations (23.1.56), which verifies the conditions (23.1.56'), as well as the conditions of existence and uniqueness of the solution, relative to the initial conditions (at the moment t0 ) x i0 , i = 1, 2,..., n , in the spherical domain Sh , defined by (23.1.57), one can set up a function V (x1 , x 2 ,..., x n ; t ) of definite sign, so that – according to the system – one obtains a derivative dV / dt , which does not have the sign of V in the domain Sh , then the unperturbed motion is stable.


544

Analogously, one can state Theorem 23.2.18 (Lyapunov; second theorem). If, for the system of non-linear differential equations (23.1.56), which verify the conditions (23.1.56'), as well as the conditions of existence and uniqueness of the solution, relative to the initial conditions (at the moment t0 ) x i0 , i = 1, 2,..., n , in the spherical domain Sh , defined by (23.1.57), we can set up a function V (x1 , x 2 ,..., x n ; t ) positive definite, with the superior limit arbitrarily small, so that – according to the system – to obtain a derivative dV / dt negative definite, then the unperturbed motion (the trivial solution) is asymptotically stable.

23.1.2.4 Stability in First Approximation Considering the autonomous system of equations (23.1.29), we get the system of the first approximation (23.1.31), by an expansion into a Maclaurin series (23.1.30). The problem is put to see in what conditions the stability in first approximation, considered in Sect. 23.1.1.2, is sufficient for the system (23.1.29) too; a study in this direction has been made be Lyapunov too. Let V (x1 , x 2 ,..., x n ; t ) be a form of degree m , for which V ′ = λV ,

(23.1.59)

where V ′ is the derivative calculated according to the system (23.1.31). If m = 1 , then the linear form can be written

V =

n

∑ αi xi , αi

i =1

= const .

(23.1.60)

Replacing in (23.1.59) and taking into account the system (23.1.31), we get n

∑ aij αi

i =1

= λα j , j = 1, 2,..., n ;

(23.1.60')

to can obtain the non-zero coefficients αi , we determine the parameter λ by means of the characteristic equation D (λ ) = 0 , given by (23.1.34). If m > 1 , then we denote by α1 , α2 ,..., αN the coefficients of the form V ; proceeding as above, it results a system of linear algebraic equations of the form N

∑ Aij αi

i =1

= λα j , j = 1, 2,..., N ,

(23.1.61)

where Aij are linear combinations of coefficients aki ; this system has non-zero solutions if and only if λ is a root of the characteristic equation


545 A11 − λ A21

Dm (λ ) =

A12

...

A1 N

A22 − λ ...

A2 N

...

...

AN 1

AN 2

...

...

= 0.

(23.1.61')

... ANN − λ

Lyapunov showed that all the roots of this equation are given by λ = m1λ1 + m2 λ2 + ... + mn λn ,

(23.1.62)

where λ1 , λ2 ,..., λn are the roots of the characteristic equation D (λ ) = 0 , while mi are non-negative integers linked by the relation

m1 + m2 + ... + mn = m .

(23.1.62')

We will determine a form V of degree m , which does satisfy the equation V′ =W ,

(23.1.63)

the derivative being calculated according to the linear system (23.1.31), while W is a form of degree m too. If α1 , α2 ,..., αN are the coefficients of the form V and β1 , β2 ,..., βN are the coefficients of the form W , then the relation (23.1.63) leads to

the system of linear algebraic equations N

∑ Aij αi

i =1

= β j , j = 1, 2,..., N ,

(23.1.63')

where Aij are the coefficients which intervene in the characteristic equation (23.1.61'). The determinant of the system is det[ Aij ] = Dm (0) . Hence, if no one of the roots λ given by (23.1.62) is zero, then – for any form W – there exists a form V and only one which satisfies the equation (23.1.63); the coefficients αi , i = 1,2,..., N , are obtained from (23.1.63'), because det[ Aij ] ≠ 0 . We can thus state Lemma 23.1.1 If all the roots of the characteristic equation associated to the linear system (23.1.31) have the real part negative, then – for any form W (x1 , x 2 ,..., x n ) of definite sign – there exists a unique form V ( x1 , x 2 ,..., x n ) of the same degree as W , which satisfies the condition (23.1.63) and which is of definite sign, opposite to that of W . Let be now the quadratic form V , defined by the equation n

n

∂V

∑ ∑ aij ∂xi x j

i = 1 j =1

n

= − ∑ x i2 ; i =1


546

on the basis of Lemma 23.1.1, this quadratic form exists and is positive definite. Taking into account this equation, the equation (23.1.29) and the Maclaurin series (23.1.30), we can express the derivative V ′ in the form V′ =

∑ ⎛⎜⎝ − xi2 n

i =1

+

∂V R ⎞; ∂x i i ⎟⎠

because in the second term of the above bracket intervene powers at least of the third degree in the variables x i , the derivative is negative definite and we may state

Theorem 23.1.19 (Lyapunov; first theorem). If all the roots of the characteristic equation which correspond to the system of first approximation (23.1.31) have the real part negative, then the unperturbed motion of the autonomous system of differential equations (23.1.29) is asymptotically stable (for any terms Ri of higher order). We denote A = [ Aij ] and δ = [ δij ] and consider the equation

)

(

α det ⎡ A − + μ δ ⎤ = 0, α > 0 , ⎣ ⎦ m

(23.1.63'')

the roots of which are linked to the roots of the characteristic equation D (λ ) = 0 by the relations μi = λi −

α , i = 1, 2,..., n . m

(23.1.63''')

We can choose the number α so that the equation (23.1.63'') have, as the characteristic equation D (λ ) = 0 , at least a root with positive real part, the number

m1 μ1 + m2 μ1 + ... + mn μn being non-zero; we deduce from here that there exists a function V of the same degree as W and not of a constant opposite sign, so that n

∑ ⎡⎣ ai 1x1 + ai 2 x 2

i =1

(

+ ... + aii −

)

α x + ... + ain x n ⎤ = 0 , ⎦ m i

for any form W of definite sign. But – according to Euler’s theorem – we have n

∑ ( ∂V / ∂xi ) = mV , so that we obtain, finally

i =1

V ′ = αV + W , α > 0 .

(23.1.64)

We can thus state Lemma 23.1.2 If, between the roots of the characteristic equation associated to the linear system (23.1.31) there exists at least one root with a positive real part, then – for any given form W , of definite sign and of degree m – there exists always a function V of the same degree and a positive number α , so that the relation (23.1.64) does take place, the function V being not of constant sign, opposite to that of W .


547

Let be now the quadratic form V , defined by the equation n

n

∂V

∑ ∑ aij ∂xi x j

i = 1 j =1

= αV +

n

∑ xi2 , α > 0 ;

i =1

on the basis of the Lemma 23.1.2, this quadratic form exists and there are points at which it is positive. Taking into account this equation, the equation (23.1.29) and the expansion into a Maclaurin series (23.1.30), we can express the derivative V ′ in the form V ′ = αV +

∑ ⎜⎝⎛ xi2 n

i =1

+

∂V R ⎞; ∂x i i ⎟⎠

in a sufficiently small neighbourhood of the origin, the sign of V ′ is specified by the sum αV + x 12 + x 22 + ... + x 2n , which is positive. In these conditions, the theorems of instability of Lyapunov (see Sect. 23.1.1.7) allow to state Theorem 23.1.20 (Lyapunov; second theorem). If there exists at least a root of the characteristic equation corresponding to the system of the first approximation (23.1.31), with a positive real part, then the unperturbed action of the autonomous system of differential equations (23.1.29) is instable (for any terms Ri of higher order). To be more specific, Lyapunov showed that the terms Ri can be neglected if Ri (x1 , x 2 ,..., x n ) < M ( x 12 + x 22 + ... + x 2n

)1/ 2 + μ , μ > 0, M

= const .

(23.1.65)

If the characteristic equation considered above has not roots with a positive real part, but has purely imaginary roots (with zero real parts), then one cannot decide on the stability without taking into consideration the terms Ri , i = 1, 2,..., n , of higher order; the respective cases are critical ones. Studies in this direction have begun in the last part of nineteenth century by H. Poincaré; they are contained in a paper on the system of differential equations dx 1 dx = − λx 2 + X1 (x 1 , x 2 ), 2 = − λx1 + X 2 ( x1 , x 2 ) dt dt

(23.1.66)

and have been continued by Lyapunov and other researchers. 23.1.2.5 Lyapunov’s Function Practically, to apply the theorems stated above, one must effectively build up functions V of Lyapunov; there does not exists a general method in this direction, but one can give some indications. We consider only autonomous systems of differential equations. If the equations (23.1.29) of the perturbed motion admit a first integral F (x1 , x 2 ,..., x n ; t ) , then we can take V (x1 , x 2 ,..., x n ) = F (x1 , x 2 ,..., x n ) − F (0, 0,..., 0) ,

(23.1.67)


548

if the difference in the second member is a function of definite sign; the derivative V ′ is – obviously – identically zero. If one can determine several first integrals Fj (x1 , x 2 ,..., x n ), j = 1, 2,.., m , for the equations (23.1.29), so that for no one of them the difference Fj (x1 , x 2 ,..., x n ) − Fj (0, 0,..., 0) is a function of definite sign, then – after Chetaev – one can search the function V in the form m

∑ λj [ Fj (x1 , x 2 ,..., xn ) − Fj (0, 0,..., 0) ]

V (x1 , x 2 ,..., x n ) =

j =1

(23.1.68)

or in the more general form V (x1 , x 2 ,..., x n ) =

m

∑ {λj [ Fj (x1 , x 2 ,..., xn ) − Fj (0, 0,..., 0) ] j =1

+ μj ⎡⎣ Fj (x1 , x 2 ,..., x n ) − Fj2 (0, 0,..., 0) ⎤⎦} , 2

(23.1.68')

the constants λj , μj , j = 1, 2,..., m , being determined so that the function V be of definite sign. Also in these cases, the derivative V ′ is identically zero. In the case of mechanical systems, the first integrals can be obtained with the aid of the universal theorems. We have seen in Sects. 23.1.1.5 and 23.1.1.6 that, in case of linear systems, the stability problem can be directly studied; in this case, the functions V and V ′ will be quadratic forms in the co-ordinates x1 , x 2 ,..., x n . In a matric formulation, let be the autonomous system of differential equations dx = ax , dt

(23.1.69)

where we have introduced the matrices ⎡ x1 ⎢ x2 x= ⎢ ⎢ # ⎢ ⎣⎢ x n

⎤ ⎡ a11 a12 ⎥ ⎢a a22 ⎥ , a = ⎢ 21 ... ⎥ ⎢ ... ⎥ ⎢ a n 1 an 2 ⎣ ⎦⎥

... a1n ⎤ ... a2 n ⎥ ⎥ ... ... ⎥ . ... ann ⎥⎦

(23.1.69')

We choose Lyapunov’s function in the form V = xT αx ,

where the matrix

(23.1.70)


549 ⎡ α11 ⎢ α21 α = ⎢ ... ⎢ ⎢ αn 1 ⎣

α12 α22 ... αn 2

... α1 n ⎤ ... α2 n ⎥ ⎥ ... ... ⎥ ... αnn ⎥⎦

(23.1.70')

is symmetric. The total derivative with respect to time is V ′ = x T αx + xT αx ;

taking into account the matric equation (23.1.69), we can write V ′ = x T a T αx + xT αax = xT ( aT α + αa ) x

too. We put the condition that −(x 12 + x 22 + ... + x 2n ) , hence that

this

quadratic

form

be

of

the

form

V ′ = −xT x ;

in this case, the matrix α will be determined by the matric equation aT α + αa = − δ .

(23.1.71)

If the equation (23.1.71) admits a positive definite solution for the symmetric matrix α , then the solution x = 0 is asymptotically stable. Let us consider now Hamilton’s canonical equations (19.1.14) in the space Γ 2s , the unknown functions being the generalized co-ordinates q j and the generalized momenta p j , j = 1, 2,..., s , for a conservative, holonomic, scleronomic, discrete mechanical

system;

Hamilton’s

function

H = H (q1 , q2 ,..., qs , p1 , p2 ,..., ps ) = T − U = h ,

h = const , is a first integral of this system of differential equations. The kinetic energy T is a positive definite form; if the potential energy −U is a positive definite form too; then H = E ( E is the mechanical energy) is – as well –a positive definite form in a domain situated in the neighbourhood of the origin (0, 0,..., 0, 0, 0,..., 0) .

The condition that the potential energy −U be a positive definite form in the neighbourhood of the origin (we have U = 0 at the origin) is equivalent to the condition that −U have a minimum at this point, obtaining thus again the Dirichlet-Lagrange theorem (see Sect. 23.1.1.3); in exchange, if the potential energy −U has a maximum, then we find again Lyapunov’s Theorem 23.1.7 of instability. Let us choose as Lyapunov’s function the expression (we use the summation convention of dummy indices) V = − Hq j p j ,

(23.1.72)


550 Hamilton’s function being given by H = T −U =

1 β p p − U (q1 , q2 ,..., qs ) ; 2 ij i j

(23.1.73)

we express then H in the form H =

1 0 1 β p p + α 0 q q + θ (q1 , q2 ,..., qs , p1 , p2 ,..., ps ) , 2 ij i j 2 ij i j

(23.1.73')

where αij0 , βij0 are constants, while the function θ contains terms of higher order than 2 . Assuming that −U has not a minimum at the origin, it results that, for small values of the generalized momenta, there exists a neighbourhood of the origin for which H < 0 , i.e. the neighbourhood for which −U < 0 . Let us denote by D the domain for which H < 0 and q j p j > 0 . We have U > 0 in D , while on its frontier we have U = 0 , because H = 0 or q j p j = 0 ; we notice that the origin is a point of this

frontier. We can write ∂H ⎞ ⎛ ∂H −H ⎜ pj − qj ∂ ∂ p q j ⎟⎠ j ⎝ + θ ( q1 , q2 ,..., qs , p1 , p2 ,..., ps ) ,

V ′ = H ′q j p j − H ( q j p j + q j p j = − H ( β ij0 pi p j − αij0 qi q j

)

)=

where we took into account Hamilton’s equations (19.1.14), the fact that H is a first integral of these equations and the expression (23.1.73'); we denoted by θ the set of terms of degree greater than 2 . Taking into account that in the domain D we have U < 0 , the principal part of the kinetic energy being positive definite, it results that the principal part of the potential energy must be negative to have V ′ > 0 ; the function θ does not have any influence on this result. The potential energy cannot have a minimum (V ′ is a function positive definite), obtaining thus again Chetaev’s Theorem 23.1.8 of instability. 23.1.2.6 Systems of Cyclic Co-ordinates. Routh’s Theorem Let us consider a natural, holonomic and scleronomic discrete mechanical system, for which the first r generalized co-ordinates which intervene in Routh’s autonomous system of equations (19.1.31), (19.1.31') are cyclic co-ordinates, so that ∂H / ∂q α = 0, α = 1, 2,..., r ; in this case, p α = 0 , resulting the first integrals pα = cα , α = 1,2,..., r . Routh’s function is of the form (19.1.30); we can state that we have – as well – ∂L / ∂q α = 0 , hence ∂R / ∂q α = 0, α = 1,2,..., r , too. The

equations (19.1.31) lead – in this case – to the same result as the Routh–Helmholtz theorem (see Sect. 18.2.3.7). There remain to be integrated the equations (19.1.31'). which are of Lagrange type.


551

We call stationary motion that motion in which all the non-cyclic co-ordinates q j , j = r + 1, r + 2,..., s , have constant values (equal to that at the initial moment); without any restriction on the generality of this motion, we can suppose that, for some values of the constants cα , the system (19.1.31') admits the particular solution qr + 1 = qr + 2 = ... = qs = 0 ,

(23.1.74)

to which corresponds the unperturbed motion (for all the co-ordinates q1 , q2 ,..., qs ). The initial generalized velocities which correspond to the non-cyclic co-ordinates are – as well – zero, while the cyclic generalized velocities are constant. Let us consider now some initial perturbations of this system, without changing the constants cα . As in the general case of Lagrange’s equations, we multiply each equation by the generalized velocity q j , j = r + 1, r + 2,..., s , sum and obtain a first integral of Jacobi type ∂R q j − R = h , j = r + 1 ∂q j s

∑

(23.1.75)

where h = const is the energy constant; let be the function V =

∂R q j − R − h0 , ∂ j = r + 1 q j s

∑

(23.1.76)

where h0 corresponds to the solution (23.1.74). If h0 is an extreme value (minimal or maximal), then the function V will be of definite sign (positive or negative). Calculating the total derivative with respect to time and taking into account the first integral (23.1.75), we obtain dV / dt = 0 ; according to the Theorem 23.1.17, we can state Theorem 23.1.21 (Routh). The stationary (unperturbed) motion of a natural holonomic and scleronomic discrete mechanical system, relative to the cyclic co-ordinates is stable if the mechanical energy of this system, after exclusion of the cyclic velocities, admits an extremum relative to the perturbation of the constants cα . This result has been obtained by Routh in 1877 (Routh, E.J., 1898). We must notice that the stability with respect to Routh’s variables q j , q j , j = r + 1, r + 2,..., s , and pα , α = 1,2,..., r , does not also mean stability with respect to the cyclic co-ordinates qα , α = 1, 2,..., r . 23.1.2.7 Limit Cycles Let be the autonomous system of differential equations with one degree of freedom (23.1.18), considered in Sect. 23.1.1.4; in certain conditions, this system admits periodic solutions to which correspond closed curves, called limit cycles. The denomination results from the behaviour of some solutions of the system.


552

To fix the ideas, we consider the particular system dx 1 = − ωx 2 + ( R2 − x12 − x 22 ) x1 , dt dx 2 = ωx1 + ( R2 − x12 − x 22 ) x 2 , dt

(23.1.77)

to which corresponds as a limit cycle the circle x12 + x 22 = R2 . Passing to polar co-ordinates ( x 1 = r cos θ , x 2 = r sin θ ), we obtain r cos θ − r θ sin θ = − ωr sin θ + ( R2 − r 2 ) r cos θ , r sin θ − r θ cos θ = ωr cos θ + ( R2 − r 2 ) r sin θ ;

(23.1.77')

hence, it results

r = ( R2 − r 2 ) r , θ = ω .

(23.1.77'')

Fig. 23.14 Stable limit circles: (a) r0 < R ; (b) r0 > R

Integrating, we can write r =

R 1 + A e −2 R t 2

, θ = ωt + B ;

(23.1.78)

the initial conditions r (0) = r0 , r0 ≠ 0, θ (0) = 0 lead to r =

We also notice that

R 1 + ( R2 / r02 − 1 ) e −2 R t 2

, θ = ωt ;

(23.1.78')


553 lim r = R ;

t →∞

(23.1.78'')

both for r0 < R (Fig. 23.14a) and for r0 > R (Fig. 23.14b); the circle r = R represents, in this case, a stable limit cycle. In particular, if r0 = R , then the motion takes place only on the circle. The origin corresponds to a case of instability. There exist also systems of differential equations for which the limit cycle is instable, the trajectory moving away from it; in this case, the origin can be a stable position “in small” (Fig. 23.15a) or an instable position “in great” (Fig. 23.15b). We mention also the possibility that the limit cycle be semi-stable, so that to have the situation in Figs. 23.14b and 23.15a or the situation in Figs. 23.14a and 23.15b.

Fig. 23.15 Instable limit cycle around: (a) a stable position “in small”; (b) an instable position “in great”

In general, the limit cycles which are encountered in practice have the property that to each of them comes close not only one trajectory in spiral, but an infinity of such trajectories, which pass through the ordinary points of the phase plane and which fill in at least a domain which contains the cycle C . If one takes into account that any ordinary point P0 of the phase plane can be considered to be a given initial position and that the motion on the limit cycle does not depend on the initial conditions, in the sense that all the trajectories which pass through various points of the phase plane tend to a same limit cycle. The important distinction between the closed trajectories of conservative mechanical systems and the closed trajectories of limit cycle type consists in the fact that the first ones appear always in the form of a continuous family of curves, while the other ones appear – in general – in the form of isolated curves. In conclusion, if C is a closed trajectory of limit cycle type, then do not exist other closed curves which be trajectories and differ arbitrarily little of C . This can be easily seen by comparing a conservative mechanical system (e.g. a mathematical pendulum for which the motion is completely definite by means of the


554

prescribed initial conditions) to a mechanical system with a limit cycle (e.g., a watch for which the motion of regime does not depend on the initial conditions). In general, the problem of putting in evidence the limit cycles is very difficult. We mention, in this direction, the theory of indices, initiated by H. Poincaré in 1892 in his treatise about new methods in celestial mechanics (Poincaré, H., 1892–1899) and developed by I. Bendixon in 1901. We state thus Theorem 23.1.22 (Bendixon). If x1 = x1 (t ), x 2 = x 2 (t ), t > t0 , is the parametric representation of the trajectory C ′ and if, for t → ∞ , the curve C remains in the interior of a bounded domain D , without coming close to a singular point, then C ′ constitutes either a closed trajectory C or tends to such a trajectory in D . This theorem gives a necessary and sufficient condition of existence of a closed trajectory.

Fig. 23.16 Graphical representation of the Bendixon theorem in a two-dimensional space

Let us return to the preceding example. Obviously, for r sufficiently great the trajectories are directed towards the interior. Using the substitution x1 = x10 + ξ1 ,

x 2 = x 20 + ξ2 , the equations are brought to the form

ξ1 = ξ1 − ωξ2 + ..., ξ2 = − ωξ1 + ξ2 + ... ,

(23.1.77''')

where we have neglected powers of higher order. The origin of the co-ordinates represents an instable focus, so that the trajectories start from the origin and enter in the domain D through the internal frontier C 1 (Fig. 23.16). As well, the trajectories enter in the domain D from the exterior through the external frontier C 2 . We notice also that the domain D does not contain singular points; hence, in the interior of this domain there exists a stable limit cycle. But it is not always possible to define such a domain D , so that this theorem can no more be used in this case.

23.1.3 Applications In what follows, we present some applications to the motion of the rigid solid, especially to the motion of rotation about the principal axes of inertia; as well, we consider the motion of a projectile in rotation. We mention also the motion in a central field.


555

23.1.3.1 Motion of Rotation of the Rigid Solid We consider firstly the motion of a rigid solid with a fixed point O , subjected to the action of its own weight G = M g , applied at the centre of mass C , in the hypothesis C ≡ O (see Sect. 15.1.1.6). The equations of motion (15.1.40) can be written in the form I 1 ω 1 = ( I 2 − I 3 ) ω2 ω3 , I 2 ω 2 = ( I 3 − I 1 ) ω3 ω1 ,

(23.1.79)

I 3 ω 3 = ( I 1 − I 2 ) ω1 ω2 ,

with respect to the axes of the non-inertial frame of reference, rigidly connected to the rigid solid, which are principal axes of inertia; Euler’s equations (15.1.11'') have the same form if we equate to zero the moment MO of the external forces ( MO 1 = MO 2 = MO 3 = 0 ). One makes a direct study of the permanent rotations in Sect. 15.1.2.7. We assume that the principal moments of inertia are ordered in the form I 1 > I 2 > I 3 . The unknown functions are the components of the rotation vector ω . We are in the Euler–Poinsot case of integrability. Let us consider a rotation about the axis Ox 3 ; a unperturbed motion is characterized by

ω1 = 0, ω2 = 0, ω3 = ω 03 , ω 03 = const ,

(23.1.80)

which – obviously – verifies the system of equations (23.1.79). Let be the perturbed motion ω1 = ε1 , ω2 = ε2 , ω3 = ω 03 + ε3 ,

(23.1.80')

where ε1 , ε2 , ε3 are arbitrarily small angular velocities, functions of time. Replacing in (23.1.79), we obtain I 1 ε1 = ( I 2 − I 3 ) ( ω 03 + ε3 ) ε2 , I 2 ε2 = ( I 3 − I 1 ) ( ω 03 + ε3 ) ε1 ,

(23.1.79')

I 3 ε3 = ( I 1 − I 2 ) ε1 ε2 .

Linearizing, we can write I 1 ε1 = ( I 2 − I 3 ) ω 03 ε2 , I 2 ε2 = ( I 3 − I 1 ) ω 03 ε1 , . I 3 ε3 = 0.

(23.1.79'')


556

The characteristic equation (of the form (23.1.34)) −λ I 3 − I1 0 ω3 I2 0

I2 − I 3 0 ω3 I1

0

−λ

0

0

−λ

=0

leads to λ ⎡⎣ I 1I 2 λ2 − ( I 2 − I 3 ) ( I 3 − I 1 ) ( ω30

)2 ⎤⎦

= 0;

(23.1.81)

one of the roots vanishes, while the other ones are given by λ1,2 = ±

( I 2 − I 3 ) ( I 3 − I1 ) I 1I 2

ω30 , ω30 > 0 .

(23.1.81')

Assuming for the principal moments of inertia the order mentioned above, there result two imaginary roots; hence, the motion of rotation about the axis of minimal principal moment of inertia is simply stable. In case of a rotation about the axis Ox1 , we proceed analogously and find the purely imaginary roots of the characteristic equation in the form (excepting the zero root) λ1,2 = ±

( I 3 − I1 ) ( I1 − I 2 ) I2I 3

ω10 , ω10 > 0 ;

(23.1.81'')

hence, the motion of rotation about the axis of maximal principal moment of inertia is, as well, simply stable. Let be now a rotation about the axis Ox 2 ; after an analogous study, excepting the zero root, one obtains the real roots λ1,2 = ±

( I1 − I 2 ) ( I 2 − I 3 ) I 3 I1

ω20 , ω20 > 0 ,

(23.1.81''')

one of them being positive, so that we can state that the motion of rotation about the axis of mean principal moment of inertia is instable. We find thus again the results in Sect. 15.1.2.7 for permanent rotations. In the case of the system (23.1.79'), we multiply the first equation by (I 3 − I 1 )ε1 and the second equation by (I 2 − I 3 )ε2 and subtract the equations thus obtained one of the other; it results I 1 ( I 3 − I 1 ) ε1 ε1 − I 2 ( I 2 − I 3 ) ε2 ε2 = 0 ,


557

wherefrom we obtain the first integral I 1 ( I 1 − I 3 ) ε12 + I 2 ( I 2 − I 3 ) ε22 = const .

In this case, we can choose the function V = I 1 ( I 1 − I 3 ) ε12 + I 2 ( I 2 − I 3 ) ε22

(23.1.82)

as Lyapunov function. Using the same order of the principal moments of inertia, we can state that V ( ε1 , ε2 ) is a positive definite function in the plane of the variables ε1 , ε2 ; on the other hand, V ′ = 0 , and we can state that the motion of rotation about the principal axis of minimal principal moment of inertia is stable for the non-linear system too. Proceeding analogously with the equations which are obtained taking the Ox 1 -axis as axis of rotation, we can show that the motion of rotation about the principal axis of maximal moment of inertia is stable for the non-linear system too. Choosing as axis of rotation the principal axis Ox 2 and maintaining the order I 1 > I 2 > I 3 , we write the system of non-linear equations of the perturbed motion in

the form I 1 ε1 = ( I 2 − I 3 ) ( ω2 + ε2 ) ε3 , I 2 ε2 = ( I 3 − I 1 ) ε3 ε1 ,

(23.1.83)

I 3 ε3 = ( I 1 − I 2 ) ( ω 2 + ε3 ) ε1 . 0

Let us choose the Lyapunov function V = I 1I 3 ε1 ε3 ;

(23.1.83')

using the equations (23.1.83), we can calculate V ′ = ( ε2 + ω 20 ) ⎡⎣ I 1 ( I 1 − I 2 ) ε12 + I 3 ( I 2 − I 3 ) ε22 ⎤⎦ .

(23.1.83'')

If ε2 + ω20 = 0 , the instability results – obviously – even from the system of equations (23.1.83). If ε2 + ω20 > 0 , then V ′ is positive definite and we are in the conditions of Chetaev’s theorem 23.1.16 of instability. Let us consider now the motion of the heavy rigid solid in the case in which the centre of mass C does not coincide with the fixed point O , being in the Lagrange–Poisson case of integrability (for which I 1 = I 2 = J > I 3 , hence in the case of an oblate spheroid, the mass centre C ( ρ1 , ρ2 , ρ3 ) being on the Ox 3 -axis, ρ1 = ρ2 = 0, ρ3 > 0 ). The corresponding equation of motion (15.2.1) admits the

solution


558

ω1 (t ) ≡ 0, ω2 (t ) ≡ 0, ω3 (t ) ≡ ω30 , α1 (t ) ≡ 0, α2 (t ) ≡ 0, α3 (t ) ≡ 1.

(23.1.84)

for a rotation about the Ox 3 -axis. Let us introduce the perturbed motion ω1 = ε1 , ω2 = ε2 , ω3 = ω30 + ε3 ,

(23.1.84')

α1 = η1 , α2 = η2 , α3 = 1 + η3 ,

Using the first integral (15.1.44), (15.2.1') and (15.2.1''), we can introduce the functions (we neglect the constant terms) V1 = J ( ε12 + ε22 ) + I 3 ( ε32 + 2 ω30 ε3 ) + 2mg ρ3 η3 , V2 = J ( ε1 η1 + ε2 η3 ) + I 3 ( ε3 + ω30 η3 + ε3 η3 ) , V3 = η12 + η22 + η32 + 2 η3 ,

(23.1.84'')

V4 = ε3 .

We search now the Lyapunov function in the form V = V1 + 2λV2 − ( mg ρ3 + I 3 ω30 λ )V3 − 2 I 3 ( ω30 + λ )V4 + μV42 = J ε12 + 2J λε1 η1 − ( mg ρ3 + I 3 ω30 λ ) η12 +J ε22 + 2J λε2 η2 − ( mg ρ3 + I 3 ω30 λ ) η22

(23.1.84''')

+ ( I 3 + μ ) ε32 + 2 I 3 λε3 η3 − ( mg ρ3 + I 3 ω30λ ) η32 .

According to Silvester’s criterion, the first two quadratic forms (with the same coefficients) are positive definite if and only if their discriminant is positive Jλ

J

J λ − ( mg ρ3 + I 3 ω30 λ )

> 0,

hence if and only if P (λ ) ≡ J λ2 + I 3 ω30 λ + mg ρ3 < 0 ;

(23.1.85)

this is possible only if the zeros of this polynomial are real, hence only if I 32 ( ω30

)2

> 4Jmg ρ3 .

The last quadratic form in (23.1.84''') is positive definite if and only if

(23.1.86)


559

I 3 + μ > 0,

I3 + μ

I 3λ

I 3λ

− ( mg ρ3 + I 3 ω30 λ )

> 0,

hence if and only if I 3 + μ > 0,

I 32 λ 2 + I 3 ω30 λ + mg ρ3 < 0 . I3 + μ

(23.1.85')

The last inequality is reduced to the inequality (23.1.86) if one chooses μ in the form μ =

I3 ( I3 − J ) ; J

(23.1.86')

I 32 > 0. J

(23.1.86'')

in this case, I3 + μ =

Hence, if the inequality (23.1.86) holds and μ is given by (23.1.86') then Lyapunov’s function V is positive definite for any λ contained between the two zeros of the polynomial P (λ ) . We can thus state that the rotation about the Ox 3 -axis is stable with respect to all the variables ω1 , ω2 , ω3 , α1 , α2 , α3 . 23.1.3.2 Motion of Rotation of a Projectile We will consider the motion of rotation of a projectile in a grazing case, assuming – after A.N. Krylov who studied the problem – that its centre of gravity C ≡ O has a rectilinear and uniform motion, of velocity v along the horizontal fixed axis Ox1′ . Thus, we will be in the Lagrange–Poisson case of integrability; the resistance R of the air, supposed to be constant, will have its point of application on the dynamic axis of symmetry Ox 1 . Let Ox1′x 2′ be the vertical plane in which one shoots and let Ox1′x 2′ x 3′ be the inertial frame of reference (Fig. 23.17). To specify the position of the Ox 1 -axis, we project it on the plane Ox1′x 2′ along Oξ and denote by α the angle made by the latter axis with Ox 1′ and by β the angle made by it with Ox 1 ; thus, one passes from the inertial frame of reference Ox1′x 2′ x 3′ to the non-inertial frame Ox 1x 2 x 3 (the axes Ox 2′ and Ox 2 form the angle α and the axes Ox 3′ and Ox 3 form the angle β ). A supplementary rotation of angle ϕ about the axis Ox 1 allows to pass from the frame Ox 1x 2 x 3 to a frame rigidly linked to the rigid solid. Hence, the generalized co-ordinates are α, β and ϕ . The angular velocity ω of the projectile will thus have the components α , β and ϕ ; the projections of this velocity on the principal axes of inertia Ox1 ,Ox 2 ,Ox 3 are expressed in the form


560

ω1 = ϕ + α sin β , ω2 = − β , ω3 = α cos β .

(23.1.87)

The kinetic energy is given by T =

1 1 ⎡ I ω 2 + J ( ω22 + ω32 ) ⎤⎦ = ⎡⎣ I ( ϕ + α sin β )2 + J ( β 2 + α 2 cos2 β ) ⎤⎦ , 2⎣ 1 2 (23.1.88)

Fig. 23.17 Motion of rotation of a projectile

where I 1 = I is the principal moment of inertia with respect to the axis, while J is the principal equatorial moment of inertia. Besides the own weight which acts at the gravity centre C , we apply a force of resistance R of the air, at the centre of pressure A , on the Ox1 -axis, at the distance OA = l ; we mention that R − R (v ), v = const , so that R = const , the force being along the direction of the Ox 1′ -axis, in a sense opposite to that of the velocity v . If we denote by γ the angle made by the axes Ox 1′ and Ox1 , then the moment of the force R with respect to the centre O will be given by Rl sin γ (in modulus), resulting the elementary work dW = Rl sin γ dγ = − d(Rl cos γ ) ; the potential energy can be written in the form

−U = Rl cos γ = Rl cos γ cos β ,

(23.1.88')

where we took into account that the arcs α, β , γ form a rectangular spherical triangle with γ as hypotenuse. We obtain the first integral of energy 1 ⎡ I ( ϕ + α sin β )2 + J ( β 2 + α 2 cos2 β ) ⎤⎦ + Rl cos α cos β = h , h = const. 2⎣ (23.1.89)

We notice that the generalized momentum I ω1 is constant, because the generalized co-ordinate ϕ is cyclic (does not intervene in H = T − U ); we can thus eliminate the


561

term I (ϕ + α sin β ) from the first integral (23.1.89) (we notice that ϕ + α sin β = const is the projection of the angular velocity of the rigid solid on the axis along which advances the mass centre). The components of the moment of momentum with respect to the pole O , along the axes of the non-inertial frame of reference, are KO 1 = I ω1 , KO 2 = J ω2 , KO 3 = J ω3 ,

so that we can write a second first integral (the component of the moment of momentum along the Ox1′ -axis) in the form KO 1′ = KO 1 cos ( Ox1′ ,Ox1 ) + KO 2 cos ( Ox1′ ,Ox 2 ) + KO 3 cos ( Ox1′ ,Ox 3 ) = I ω1 cos α cos β − J ω2 sin α − J ω3 cos α sin β = I ω1 cos α cos β + J ( β sin α − α cos α cos β sin β ) = const,

(23.1.89')

where we have used the fact that the arcs α, β + π / 2 and (x1′ , x 3 ) form a rectangular spherical triangle with sides α and β + π / 2 , the hypotenuse of which is given by cos(x1′ , x 3 ) = cos α cos( β + π / 2) = − cos α sin β . The total derivatives with respect to time of these first integrals are – obviously – zero. Expanding into series after the power of the variables α and β , we can introduce the functions 1 1 J ( α 2 + β 2 ) − Rl ( α2 + β 2 ) , 2 2 1 + αβ ) − I ω1 ( α 2 + β 2 ) . V2 ( α, β , α , β ) = J ( βα 2 V1 ( α, β , α , β ) =

(23.1.90)

We set up a Lyapunov function of the form V = V1 − λV2 , so that V =

1 1 + ( I ω1λ − Rl ) α 2 ⎤. + ( I ω1λ − Rl ) β 2 ⎦⎤ + ⎣⎡J β 2 − 2J λβα ⎡J α 2 + 2J λαβ ⎦ 2⎣ 2 (23.1.90')

We notice that the two expressions in the rectangular brackets have similar forms, i. e. J ( x ± λy ) − y 2 ( J λ 2 − I ω1λ + Rl ) ;

indeed, for x = α , y = β and the sign + we obtain the first bracket, while for x = β , y = α and the sign – we get the second bracket. This form is positive definite if the condition J λ2 − I ω1λ + Rl < 0

(23.1.91)


562

holds; hence, the zeros of the trinomial in λ must be real and distinct. It results the condition I 2 ω12 > 4JRl

(23.1.91')

and the constant λ must be contained between the two zeros. As a matter of fact, the inequality (23.1.91') represents a condition for ω1 . In this case, the motion of rotation of the projectile is stable, having a strict minimum for V at α = α = β = β = 0 . 23.1.3.3 Circular Motion in a Central Field Let us consider a particle of mass m acted upon by a force of intensity F (r ) < 0 , where r is the distance to a fixed point O (a centre of attraction); obviously, this is a central force (see Sects. 8.1.1.1, 18.3.2.2 and 19.2.3.1). The trajectory of the particle being plane, we can write the kinetic energy and the potential energy in polar co-ordinates (r , θ ) in the form T =

1 m ( r 2 + r 2 θ2 ) , − U (r ) = − ∫ F (r )dr ; 2

(23.1.92)

it results Hamilton’s function H = T −U =

(

)

1 1 1 m ( r 2 + r 2 θ2 ) − U (r ) = pr2 + pθ2 , 2 2m 2

(23.1.92')

where we have introduced the generalized momenta pr =

∂L ∂L = mr , pθ = = mr 2 θ, L = T + U , ∂r ∂θ

corresponding to the generalized co-ordinates (r , θ ) . We obtain thus the canonical equations r =

p pr p2 θ , θ = θ 2 , pr = + F (r ), p θ = 0 , m mr mr 3

(23.1.93)

where θ is a cyclic co-ordinate; the solutions

r = R, θ =

−

F (r ) t + θ0 , pr = 0, pθ = R − mRF (R ) , mR

(23.1.93')

corresponding to a uniform circular motion, which is unperturbed, are verified. Let be the perturbed motion


563

F (R ) t + θ0 + ε2 , mR pr = η1 , pθ = r − mRF ( R ) + η2 .

r = R + ε1 , θ =

−

(23.1.93'')

Replacing in the equations (23.1.93), we obtain the perturbed equations ε1 =

R − mRF (R ) + η2 η1 , ε = − m 2 m ( R + ε1 )2

−

F (R ) , mR

(23.1.94)

2

⎡⎣ R − mRF (R ) + η2 ⎤⎦ + F ( R + ε1 ) , η2 = 0, η1 = ( R + ε1 )3

corresponding to Hamilton’s function H =

2 1 ⎧⎪ 2 ⎡⎣ R − mRF (R ) + η2 ⎤⎦ ⎫⎪ + η ⎨ ⎬ − U ( R + ε1 ) − 2 2m ⎪ 1 ( ) R + ε 1 ⎪⎭ ⎩

−

F (R ) η . mR 2

(23.1.94')

We notice that the potential energy −U ( R + ε1 ) does not admit a local minimum for ε1 = 0 , because it does not contain the co-ordinate ε2 , so that an eventual minimum of this function for ε1 = 0 could take place for any ε2 and could no more be a local minimum; hence, the motion is instable in Lyapunov’s sense. We notice, from (23.1.94), that η2 = η20 = const , where η20 can be taken arbitrarily small, hence H = H ( ε1 , η1 ) . Neglecting the corresponding to the kinetic energy, we consider the function 2

⎡⎣ R − mRF (R ) + η20 ⎤⎦ − U ( R + ε1 ) − V ( ε1 ) = 2m ( R + ε1 )2

−

term

F (R ) 0 η ; mR 2

(1/ 2m )η12 ,

(23.1.95)

we have 2

⎡⎣ R − mRF (R ) + η20 ⎤⎦ dV =− − F ( R + ε1 ) = 0 dε1 m ( R + ε1 )2

(23.1.95')

for the position of equilibrium. The condition of minimum is 2

3 ⎡⎣ R − mRF (R ) + η20 ⎤⎦ d2V = − F ′ ( R + ε1 ) > 0 . dε12 m ( R + ε1 )2

(23.1.95'')

Eliminating the rectangular brackets between the last two relations, we can write −3 F ( R + ε1 ) − ( R + ε1 ) F ′ ( R + ε1 ) > 0 ;

(23.1.96)


564 if ε1 → 0 , then this condition becomes

3 F ( R ) + RF ′ ( R ) < 0 .

(23.1.96')

In the important particular case in which U =

k ks , F (r ) = − s + 1 , k > 0, s ∈ ], k , s = const , rs r

(23.1.97)

we have F ′(r ) = ks (s + 1)/ r s + 2 , and the condition (23.1.96') gives s < 2;

(23.1.97')

in this case, the motion of the particle is orbitally stable in Poincaré’s sense (e.g., in case of the Keplerian motion for which s = 1 ). We will consider now the more general case of motion on a sphere, having a circular motion on the equator (the radius R and the angle ϕ in the equatorial plane – the longitude) and on the meridian (the radius R and the angle θ , measured on the meridian from the equator towards the pole – the latitude). We denote (unlike the preceding case, we introduce in these expansions also the mass m of the particle – besides the constant k – for subsequent simplifications) U =

km kms s , F (r ) = − s + 1 , k , m > 0, s ∈ ], k , m , s = const , r r

(23.1.98)

for a central force of attraction; the equation of motion admit a solution of the form (see Sect. 8.1.1.1) r ≡ R, ϕ ≡ ϕ 0 , F (R ) + mRϕ 02 = 0 .

(23.1.98')

We will make a study of the stability of the motion with respect to all variables r , r , θ , θ and ϕ . The mechanical system is conservative, while ϕ

is a cyclic co-ordinate

( ∂H / ∂ϕ = ∂T / ∂ϕ = 0) , so that we obtain the first integral E = T −U =

1 km 1 m ( r 2 + r 2 θ2 + r 2 cos2 θϕ 2 ) − s = mh , 2 r 2 ∂T = mr 2 cos2 θϕ = ma , ∂ϕ

(23.1.99)

where h , a = const . The perturbed motion will be characterized by the variables η1 , η2 , η3 in the form r = R + ε1 , θ = ε2 , r = η1 , θ = η2 , ϕ = ϕ 0 + η3 ;

(23.1.100)


565

the corresponding first integral will be F1 = η12 + ( R + ε1 )2 η22 + ( R + ε1 )2 ( ϕ 0 + η3 )2 cos2 ε2 −

2k

( R + ε1 )s

= h,

(23.1.99')

F2 = ( R + ε1 )2 ( ϕ 0 + η3 ) cos2 ε2 = a ,

but no one of them is of definite sign with respect to the variables ε1 , ε2 , η1 , η2 , η3 . Hence, we search a Lyapunov function in the form V = F1 − F1 (0) + λ [ F2 − F2 (0) ] + μ ⎡⎣ F22 − F22 (0) ⎤⎦ ,

(23.1.101)

trying to determine the constants λ and μ so that this function be of definite sign; we also use the relation Rϕ 02 =

ks , Rs + 1

(23.1.98'')

obtained from (23.1.98), (23.1.98'). Expanding the non-polynomial terms into a Maclaurin series and neglecting the terms of a degree higher then 2 , we can write F1 − F1 (0) = 4 Rϕ 02 ε1 + 2 R2 ϕ 0 η3 − sϕ 02 ε12 − R2 ϕ 02 ε22 + η12 + R2 ( η22 + η32 ) + 4 Rϕ 0 ε1 η3 , F2 − F2 (0) = 2 Rϕ 0 ε1 + R2 η3 + ϕ 0 ε12 − R2 ϕ 0 ε22 + 2 Rε1 η3 , F22 − F22 (0) = 4 R 3 ϕ 02 ε1 + 2 R 4 ϕ 0 η3 + 6 R2 ϕ 02 ε12 −2 R 4 ϕ 02 ε22 + R 4 η32 + 8 R 3 ϕ 0 ε1 η3 ;

we replace in (23.1.101) and equate to zero the linear terms (the origin must be a point of extremum), which leads to the condition λ = −2ϕ 0 ( 1 + μR2 ) ,

(23.1.102)

remaining only the determination of the parameter μ . We can express now Lyapunov’s function in the form V = V1 + V2 , where V1 = ( 4 μR2 − s − 2 ) ϕ 02 ε12 + 4 μR 3 ϕ 0 ε1 η3 + R2 ( 1 + μR2 ) η32 , V2 = R2 ϕ 02 ε22 + η12 + R2 η22 .

.

We notice that quadratic form V2 is positive definite with respect to the variables ε2 , η1 , η2 . To have also the quadratic form V1 positive definite with respect to the


566

variables ε1 , ε2 , η1 , η2 , η3 (it is sufficient to be positive definite with respect to ε1 and η3 ), we use Sylvester’s criterion; hence, it is necessary and sufficient to have 4 μR2 > s + 2,

( 4 μR2

− s − 2 ) ϕ 02

3 μR 3 ϕ 0

2 μR 3 ϕ 0 R2 ( 1 + μR2

)

> 0,

which leads to the conditions

μ >

2+s 2+s ,μ > . 2 ( 2 − s ) R2 4R

(23.1.103)

In the particular case in which s = 1 (Keplerian motion), both inequalities are satisfied if μ > 3 / R2 ; we can thus state that the Keplerian motion is stable with respect to any perturbations, on a circular orbit.

23.2. Vibrations of Mechanical Systems In many domains of physics, chemistry, biology appear oscillatory phenomena; the mechanical oscillations can be also vibrations. From a mathematical point of view, the development of the knowledge in this direction has led to a development of the qualitative theory of differential equations; indeed, together with vibrations appear problems of stability of motion. We mention that the respective phenomena are encountered everywhere in techniques and technology. The basic notions of the theory of oscillations can be found in Christian Huygens’s fundamental work “Horologium oscillatorium sive de motu pendulorum ad horologia adaptato demonstrationes geometricae”, dedicated to the French king Louis XIV in 1673, and in his famous “Undulatory theory of light”, presented at the Academy of Sciences in Paris, in 1678. The knowledges in this direction are enriched by Leonhard Euler in “Mechanica sive motus analyticae exposita” in 1736, by J.-L. Lagrange, in 1788, in “Mécanique analytique”, the first systematic presentation of this new method of calculation, by E. J. Routh in his “Dynamics of particles”, in 1898, by A. M. Lyapunov in his famous doctor thesis, in 1892, and by others; the actual possibilities of calculation allow, besides a linear study of vibrations, a non-linear one too (Huygens, Cr., 1920; Lagrange, J.-L., 1788; Lyapunov, A.M., 1949; Routh, E.J., 1898). In what follows, we consider undamped and damped, free and forced small oscillations about a stable position of equilibrium, non-linear vibrations and limit cycles; the results thus obtained are followed by applications with theoretical and practical character.

23.2.1 Small Free Oscillations About a Stable Position of Equilibrium Many motions of discrete mechanical systems with one or several degrees of freedom are periodical; in such a motion, the geometric parameters which determine the position of a mechanical system are periodic function of time. The oscillatory motion is a motion with a single degree of freedom, in which the geometric parameter which determines the position of the discrete mechanical system changes periodically the sense of its variation. An important rôle is played by the small oscillations about a


567

stable position of equilibrium; the case of a heavy particle constrained to stay on a fixed surface has been considered in Sect. 7.2.3.3. The oscillatory motion is periodical if the constitutive oscillations of the motion and the parameters (the geometric parameter which determines the position of the mechanical system, the velocity and the acceleration) are repeated with the same detail, in the same order after a certain interval of time T , called period. The simplest periodic oscillation is the harmonic oscillation – a periodic motion in which the periodicity is expressed by the sinus or by the cosines of a linear function of time, e. g., x = a cos ( ωt − ϕ ) ,

(23.2.1)

where x is the elongation, a is the amplitude, ϕ is the phase shift, ω is the pulsation (circular frequency), T = 2 π / ω is the period and f = 1/T = ω / 2 π is the frequency. Thus, in Sects. 23.2.1 and 23.2.2, we have considered the oscillations of the discrete mechanical systems with two degrees or with one degree of freedom, respectively. After establishing the equations of motion, we put in evidence the proper forms of vibration, we pass to normal co-ordinates and give some methods to determine proper pulsations; we mention, as well, the introduction of the influence of a supplementary constraint. The results thus obtained are applied to the study of small oscillations with one degree or two degrees of freedom, as in the case of two harmonic oscillations. The small oscillations of damped free oscillations will be considered too. The theory of small oscillations with which we deal corresponds to the cases of motion of the discrete mechanical systems which remain close to a stable position of equilibrium or – more general – to a state of motion in permanent regime. 23.2.1.1 Equations of Motion Let us consider a conservative holonomic and scleronomic discrete mechanical system with s degrees of freedom. We assume that the potential energy V = V (q1 , q2 ,..., qs ) has an isolated minimum at the position of equilibrium q j = q j0 , j = 1, 2,..., s ; because the generalized forces are given by Qj =

∂V , j = 1, 2,..., s , ∂q j

(23.2.2)

these ones will vanish at the respective point, the position of equilibrium being thus a stable one, Without any loss of generality, we assume that q j0 = 0, j = 1, 2,..., s ; as well, because the potential energy V is determined neglecting an arbitrary constant, we can take this minimum equal to zero (V = V (0, 0,..., 0) = 0 ), so that in the neighbourhood of the origin we have V > 0 . Expanding V in a power series at the origin, we can write (we notice that V = V (0, 0,..., 0) = 0 and V > 0 , so that the linear terms must disappear) V =

1 ⎛ ∂2V ⎞ 1 q q = bij qi q j , bij = bji = const , 2 ⎜⎝ ∂qi ∂q j ⎟⎠0 i j 2

(23.2.3)


568

the sum being made – obviously – from 1 to s in the hypothesis of small motions, we neglect the terms of higher order. It results that the quadratic form V is positive definite. It can happen that, by neglecting some terms of higher order, the minimum at the origin does no more be strict (e. g., 2 2 2 4 4 4 V = a (q1 + q2 + ... + qs ) + b (q1 + q2 + ... + qs ) , a , b > 0 ); we will not take into account such special cases. The kinetic energy is given by T =

1 g q q , g = gij ( q1 , q2 ,..., qs ) ; 2 ij i j ij

expanding the coefficients gij into a power series and neglecting all the terms which contain generalized co-ordinates, because we have to do with small oscillations, we admit that gij = aij = const, i , j = 1, 2,..., s . In this case T =

1 a q q , a = a ji , 2 ij i j ij

(23.2.3')

the kinetic energy being, as well, a quadratic positive definite form, assuming that the origin is not a singular point. If, at the initial moment, the position of the discrete mechanical system is sufficiently close to the position of stable equilibrium and if the initial velocities are sufficiently small in absolute value, then – during the motion – both the deviations from the position of equilibrium and the generalized velocities will be small in absolute value. One can thus justify the above linearization of the problem. Lagrange’s function L = T − V leads to Lagrange’s equations in the form aij qj + bij q j = 0, i = 1, 2,..., s ,

(23.2.4)

corresponding to the small motions of a discrete mechanical system, subjected to holonomic and scleronomic constraints and to conservative forces, around a stable position of equilibrium; we will try to determine a motion described by particular solutions of the form

q j (t ) = a j cos( ωt − ϕ ) ,

(23.2.5)

which – as a matter of fact – are harmonic oscillations. Introducing in the system of equations (23.2.4) and using the notation λ = ω2 ,

(23.2.6)

we obtain the homogeneous system of linear algebraic equations

( bij

− λaij ) a j = 0, i = 1, 2,..., s ,

which has non-trivial solutions for values given by the secular equation

(23.2.7)


det [ bij − λaij

569

]=

b11 − λa11

b12 − λa12

... b1s − λa1s

b21 − λa21

b22 − λa22

... b2 s − λa2 s

...

...

bs 1 − λas 1 bs 2 − λas 2

...

...

= 0.

(23.2.8)

... bss − λass

Developing this determinant, we obtain a polynomial equation of degree s , which is the frequencies equation, with s roots λk = ωk2 > 0, k = 1,2,..., s (the link between the frequency (circular frequency) fk and the pulsation ωk is given by ωk = 2 π fk ); we find, for any root λk , a solution of the homogeneous system (23.2.7), which corresponds to the amplitudes of the harmonic oscillations. We can thus state that there are at the most s distinct frequencies. If λk < 0 , then the corresponding normal mode is not oscillatory (the generalized co-ordinates q j increase or decrease exponentially in time). The general solution will be a linear superposition of such particular solution as it will be shown in the following subsection. Let us introduce the matrices a11

b12

... b1s

b21 b22 B = , ... ... ... ... ... ass bs 1 bs 2

... b2 s

a12

... a1s

a21 a22 A = ... ... as 1 as 2

... a2 s

b11

...

...

(23.2.9)

... bss

and the column vectors ⎡ q1 ⎤ ⎡ q1 ⎤ ⎢ q ⎥ ⎢ q2 ⎥ ⎢ 2⎥ ⎢ ⎥ , q = ⎢ ⎥ . q = ⎢# ⎥ # ⎢ ⎥ ⎢ ⎥ ⎢⎣ qs ⎥⎦ ⎣⎢ qs ⎦⎥

(23.2.9')

We say that the symmetric matrices A and B are positive definite because the corresponding quadratic forms are positive definite. In this case, the kinetic and the potential energy can be written in the form T =

1 T 1 q Aq , V = q T Bq , 2 2

(23.2.3'')

being led to the kinetic potential

L = T −V and to Lagrange’s equations

(23.2.3''')


570

+ Bq = 0 ; Aq

(23.2.4')

the normal mode of motion is given by q(t ) = a cos( ωt − ϕ ) .

(23.2.5')

Replacing in (23.2.4'), we obtain the matric equation ( B − λA ) a = 0 ,

(23.2.7')

which has non-zero solutions if and only if

det [ B − λ A ] = 0 ;

(23.2.8')

we obtain thus the amplitude vectors a .

23.2.1.2 Secular Equations To can find the nature of the roots of the secular equation, we will put in evidence some properties of the quadratic forms with real coefficients (we assume that aij = a ji , i , j = 1,2,..., s ). We notice first of all that to any quadratic form aij ui u j corresponds the bilinear form A ( u, v ) = aij ui v j ,

(23.2.10)

where u and v are the column vectors ⎡ u1 ⎤ ⎡ v1 ⎤ ⎢ u2 ⎥ ⎢ v2 ⎥ u = ⎢ ⎥, v = ⎢ ⎥ ; ⎢ # ⎥ ⎢ # ⎥ ⎢ ⎥ ⎢ ⎥ ⎣⎢ us ⎦⎥ ⎣⎢ vs ⎥⎦

(23.2.10')

in this case, the quadratic form is written in the form

A ( u, u ) = aij ui u j ;

(23.2.10'')

Following properties can be easily verified: (i) A ( u1 + u2 , v ) = A ( u1 , v ) + A ( u2 , v ) ; (ii) A (λu, v ) = Aλ ( u, v ) , λ scalar; (iii) A ( u, v ) = A ( v, u ) ; (iv) A ( u, u ) = A ( v, v ) + A ( w , w ), with u = v + iw , u = v − iw is a real number; (v) if A ( u, u ) is a positive definite quadratic form and u ≠ 0 is an arbitrary


571

complex vector, then A ( u, u ) > 0 ; (vi) if λ is a root amplitude of the secular equation det[ B − λ A ] = 0 and a is the corresponding amplitude vector ( Ba = λ Aa, a ≠ 0 ), then for any vector v one has B( a, v ) = λ A ( a, v ) . One can now easily show that the relation A ( a, a ′ ) = 0

(23.2.11)

takes place for two amplitude vectors a and a ′ , which correspond to two different roots λ and λ ′, λ ≠ λ ′ , of the secular equation. Let λ be a complex root of the secular equation (23.2.8') with λ ≠ λ and let a be a corresponding complex vector; λ is, in this case, also a root of this equation, to which corresponds the amplitude vector a . Because λ ≠ λ , it results the equality (23.2.11), in contradiction to the property (v). One can thus state that the secular equation (23.2.8') has only real roots. One can choose a real amplitude vector a to a real roots λ . Using the property (vi), putting v = a and observing that A ( a, a ) is a positive definite quadratic form, we may write λ =

B ( a, a ) ; A ( a, a )

but B ( a, a ) is a positive definite quadratic form too, so that λ > 0 . Taking into account (23.2.6), we can state that the secular equation (23.2.8') has s positive roots λj , to which correspond the positive pulsation ω j = λj and the real amplitude vectors a j , j = 1,2,..., s . Assuming firstly that all the roots of the characteristic equation are real roots, to each λj corresponds q j = a j cos ( ω j t − ϕ j ) , j = 1,2,..., s ,

(23.2.12)

where the amplitude vectors must satisfy the linear system ( B − λA ) a = 0 .

(23.2.7'')

Because the system (23.2.4') is linear, any linear combination of the solutions (23.2.12) is a solution of this system; hence q =

s

∑ C j a j cos ( ω j t − ϕj ) , j =1

where C j , j = 1, 2,..., s , are arbitrary constants, is a solution of this system. Let be

(23.2.12')


572

c j a j = 0, c j = const, j = 1, 2,..., s ;

is this case A ( ak , c j a j ) = c j A ( ak , a j ) ,

for any fixed k , 0 < k < s . But ⎧⎪ 0, k ≠ j , A ( ak , a j ) = ⎨ ⎪⎩ > 0, k = j ,

so that c j = 0, j = 1,2,..., s ; we can thus state that the amplitude vectors are linear independent. We impose Cauchy type initial conditions q (0) = q (0) =

s

∑ C j a j cos ϕj

j =1 s

= q0 ,

∑ ω jC j a j sin ϕj = q 0 .

(23.2.13)

j =1

The vectors a j being linear independent, we determine uniquely C j cos ϕ j and C j ω j sin ϕ j ; because ω j ≠ 0 , we obtain then the arbitrary constants C j and ϕ j (for the latter constants, till a multiple of 2 π ). Hence, in the absence of multiple roots of the secular equation, the formula (23.2.12') gives the general solution of the system of equations of motion (23.2.4'). Lagrange considered, erroneously, that in case of multiple roots appear solutions of the form

( a + a ′t

+ a ′′t 2 + ... ) cos ( ωt − ϕ ) ;

but Weierstrass showed further that to each root λ of m th order of multiplicity one can determine m linearly independent amplitude vectors ak , the final solution being of the same form (23.2.12'). The vibrations C j a j cos( ω j − ϕ j ), j = 1, 2,..., s , are also called principal vibrations of the discrete mechanical system.

23.2.1.3 Vibrations of a Discrete Mechanical System In particular, we can assume that the system of equations of motion (23.2.4) or (23.2.4') corresponds to a discrete mechanical system of particles. If this system has only one degree of freedom, specified by the variable x , we can write T = (1/ 2)mx 2 and V = (1/ 2)kx 2 , where m is a mass, while k is a coefficient of elasticity. In the case in which the mechanical system has n degrees of freedom, by generalizing, we can write


T =

573

1 n n 1 n n mij xi x j , V = ∑ ∑ kij x i x j , ∑ ∑ 2 i =1 j =1 2 i =1 j =1

(23.2.14)

where the variables x i , i = 1, 2,..., n , correspond to the degrees of freedom, mij = m ji are quantities of the nature of a mass, while kij = k ji , i , j = 1, 2,..., n , are quantities of the nature of a coefficient of elasticity. Introducing the matrix of inertia (the matrix of masses) and the matrix of rigidity (the matrix of elastic coefficients)

M =

m11

m12

m21

m22

... mn 1

... mn 2

k11

k12

... k1 n

k21 ... ... , K = ... ... mnn kn 1

k22

... k2 n

...

...

... m1n ... m2 n

kn 2

...

,

(23.2.15)

... knn

as well as the column vectors ⎡ x1 ⎢ x2 x = ⎢ ⎢ # ⎢ ⎣⎢ x n

⎡ x1 ⎤ ⎤ ⎢ x ⎥ ⎥ 2 ⎥ , x = ⎢ ⎥ , ⎢ # ⎥ ⎥ ⎢ ⎥ ⎥ ⎢⎣ xn ⎥⎦ ⎦⎥

(23.2.15')

we can write T =

1 T 1 x Mx , V = xT Kx , 2 2

(23.2.14')

Lagrange’s equations taking the form + Kx = 0 . Mx

(23.2.16)

The coefficients mij = m ji , i ≠ j , if they are not null, are called coefficients of dynamical coupling and the coefficients kij = k ji , i ≠ j , if they are not zero, are called coefficients of statically coupling. If mij = 0, i ≠ j , and if there exist coefficients kij ≠ 0, i ≠ j , then the mechanical system is statically coupled. If kij = 0, i ≠ j existing coefficients mij ≠ 0, i ≠ j , then the mechanical system is

dynamically coupled. If mij = kij = 0, i ≠ j

then the mechanical system is

uncoupled, each differential equation (23.2.16) having only one degree of freedom (only one unknown function) mii xi + kii x i = 0, i = 1, 2,..., n ,

(23.2.16')

unlike the system (23.2.4). We assumed that i , j = 1,2,..., n , in all the above considerations.


574 Starting from the solution

x = C cos ( ωt − ϕ )

(23.2.17)

and using the results in the preceding subsection, one can state that the general solution is obtained as a superposition of n proper modes, depending on 2n constants of integration, which are expressed by means of the initial positions x i (0) = x i0 and of the initial velocities x i0 (0) = x i0 , i = 1,2,..., n , in a Cauchy type problem. The column vector ⎡ C1 ⎢ ⎢C2 C= ⎢ # ⎢ ⎢C n ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(23.2.15'')

is given by the linear and homogeneous matric equation 2 ⎣⎡ K − ω M ⎦⎤ C = 0 ,

(23.2.18)

which leads to the characteristic equation det ⎡⎣ K − ω 2 M ⎤⎦ = 0 ;

(23.2.18')

to each eigenvalue ωi , i = 1,2,..., n , will correspond a proper form of vibration Ci , of components C 1i ,C 2i ,...,C ni , i = 1,2,..., n , having – in total – n such forms.

23.2.1.4 Proper Forms of Vibration Let be two proper forms of vibration, denoted by “prime” and “second”. By writing the equation (23.2.18) for the two cases, it results KC ′ − ω ′2 MC ′ = 0, KC ′′ − ω ′′2 MC ′′ = 0;

multiplying the first equation at the left by C ′′T and the second one by C ′T , we obtain C ′′T KC ′ − ω ′2 C ′′T MC ′ = 0, C ′T KC ′′ − ω ′′2 C ′T MC ′′ = 0.

The matrices K and M are symmetric with respect to the principal diagonal; it results that the scalars C ′′T KC ′ and C ′′T MC ′ are symmetric with respect to C i′,C j′ , i , j = 1,2,..., n , so that we can replace C ′ by C ′′ and C ′′ by C ′ . It results C ′′T KC ′ = C ′T KC ′′, C ′′T MC ′ = C ′T MC ′′ .


575

We can thus write

( ω ′2

− ω ′′2 ) C ′′T MC ′ = 0 ;

but ω ′′ ≠ ω ′ , so that, finally C ′′T MC ′ = 0 .

(23.2.19)

C ′′T KC ′ = 0 ,

(23.2.19')

Analogously,

obtaining thus the property of orthogonality of the proper modes. Developing these relations, we get n

n

n

n

∑ ∑ mijC i′C j′′ = 0, ∑ ∑ kijC i′C j′′ = 0 .

i = 1 j =1

(23.2.20)

i =1 j = 1

If the mechanical system is statically coupled, then it results n

n

n

∑ miiC i′C i′′ = 0, ∑ ∑ kijC i′C j′′ = 0 ,

i =1

(23.2.20')

i =1 j =1

while, if the system is dynamically coupled, we can write n

n

n

∑ ∑ mijC i′C j′′ = 0, ∑ kiiC i′C i′′ = 0 ;

i =1 j = 1

(23.2.20'')

i =1

finally, if the discrete mechanical system is uncoupled, we have n

n

i =1

i =1

∑ miiC i′C i′′ = 0, ∑ kiiC i′C i′′ = 0 .

(23.2.20''')

Let be n

n

j =1

j =1

∑ ( −mij xj ) = ∑ mijC j′ ω ′2 cos ( ω ′t − ϕ ′ ), i

= 1,2,..., n ,

the forces of inertia corresponding to the “prime” mode, where we took into account the displacements x j′ = C j′ cos( ω ′t − ϕ ′), j = 1, 2,..., n ; if we consider also the displacements x j′′ = C j′′ cos( ω ′′t − ϕ ′′), j = 1,2,..., n , corresponding to the “second” mode, and if we calculate the work of the mentioned forces of inertia, when they travel through these displacements, with all their intensity, then we obtain


576 W =

n

n

∑ ∑ mijC i′C j′′ cos ( ω ′t − ϕ ′ ) cos ( ω ′′t − ϕ ′′ ) .

i =1 j =1

The orthogonality condition (23.2.20) (or, in particular, the condition (23.2.20'')) leads to W = 0;

(23.2.21)

we can thus state Theorem 23.2.1 In the case of a discrete mechanical system of particles, the work of the forces of inertia corresponding to a proper mode of vibration vanishes if they travel through, with all their intensity, the displacements corresponding to another mode. Observing that this work is equal to the variation of the kinetic energy, we can also state Theorem 23.2.1' In the case of a discrete mechanical system of particles there does not take place a transfer of energy from a proper mode of vibration to another one.

23.2.1.5 Normal Co-ordinates We state, without proof, Theorem 23.2.2 (J. J. Sylvester). If A and B are two real and symmetric matrices and if A is positive definite, then it exists a non-singular real matrix S , so that

ST AS = E, S T BS = Λ ,

(23.2.22)

where E is the unit matrix and Λ is a diagonal matrix

Λ =

λ1

0

...

0

0

λ1 ...

0

...

... ... ...

0

0

,

(23.2.22')

... λs

the non-zero elements (the eigenvalues) of which are the roots of the equation

det [ B − λ A ] = 0 .

(23.2.22'')

We notice that det [ Λ − λE ] = det ⎡⎣ S T ( Β − λ A ) S ⎤⎦ = det S T det [ Β − λ A ] det S = det S T det S det [ Β − λ A ] = ( det S )2 det [ Β − λ A ] ,

where det S ≠ 0 , the matrix S being non-singular; hence det[ B − λ Λ ] vanishes together with det[ Λ − λE ] .


577

The change of co-ordinates q ′ = S −1 q

(23.2.23)

transforms the equation (23.2.4') in the form ′ + Λq ′ = 0 , q

(23.2.24)

to which – taking into account the above observation – corresponds a secular equation with the same roots. We can thus state that Lagrange’s equations are invariant to the point transformation (23.2.23), defined by the matrix S , the form (23.2.24) being the normal form, while the co-ordinates q j′ are normal co-ordinates. We mention that the matrices A and B are simultaneously diagonal if all the co-ordinates are normal, hence if between the equations of motion there is not any coupling. The solution of the system of equations (23.2.24) will be thus independent, of the form q ′j (t ) = a ′j cos( ω j t − ϕ j ), j = 1, 2,..., s ,

(23.2.24')

Any independent normal co-ordinate q j′ of (23.2.24') depends on the eigenvalues λj , hence on the pulsation ω j =

λj , on the proper form a ′j and on the arbitrary

phase shift ϕ j ; by means of the arbitrary constants C j′ , we obtain the general solution q (t ) =

n

∑ C j′a ′j cos ( ω j t − ϕj ) , j =1

(23.2.24'')

corresponding to the solution (23.1.12'). We notice that this solution is general, even if the characteristic equation has multiple roots. We can thus state Theorem 23.2.2 (Daniel Bernoulli). The permanent small motions of a discrete mechanical system, subjected to holonomic and scleronomic constraints and to conservative forces around a stable position of equilibrium can be obtained by the superposition of a finite number of independent harmonic vibrations. The kinetic energy and the potential energy can be expressed in normal co-ordinates in the form T =

1 s 2 1 s q j′ , V = ∑ ω 2j q ′2 . ∑ 2 j =1 2 j =1

(23.2.3''')

There are two principal problems in the theory of oscillations: the determination of the possible frequencies of the normal modes and the determination of the normal co-ordinates as functions of the co-ordinates in which the problem has been initially formulated. The frequencies are given by the secular equation (23.2.8'), while it is more


578

difficult to obtain the normal co-ordinates, because one must determine the matrix S for the transformation (23.2.22). As a matter of fact, by the determination of the matrix S one obtains also the frequencies of the normal modes directly from the relation Λ = S T BS . In both problems, one must know all the masses of the particles, as well as the constants which characterize the forces, that is the matrices A and B . From symmetry considerations, we have the possibility to determine the limits of the number of possible frequencies, the superior limit of the number of constants necessary to specify T and V , a partial factorization of the secular equation and the co-ordinates in which the matrices A and B are expressed in a reduced form (not necessary diagonal). Let us suppose that one can find a matrix D ≠ E , of the same dimension as the vector q , so that V ( Dq ) = V ( q ), ∀q ;

(23.2.25)

in this case, the configurations q and Dq have the same potential energy. Because 2V ( Dq ) = ( Dq )T B ( Dq ) = q T DT BDq = 2V ( q ) ,

it results that DT BD = B .

(23.2.26)

At the same time, the relations V ( D2 q ) = V ( D ( Dq ) ) = V ( Dq ) = V ( q ) ,

take place and, replacing q by D−1 q in (23.2.25), we get V ( D−1 q ) = V ( q ) , so that the group G generated by D lets invariant the potential V . The set of matrices D which satisfy the relation (23.2.25) forms a representation of the group G . Let us suppose that the relation DT AD = A is satisfied too, hence that the Lagrangian (23.2.3''') is invariant with respect to the transformations of the group G . If the representation D is orthogonal and completely reducible, having each irreducible component orthogonal, then the equation (23.2.26) becomes BD = DB ,

(23.2.26')

where ⎡ D(1) ⎢ D= ⎢ ⎢ ⎢⎣

D(2)

⎤ ⎥ −1 T ⎥ , ( D( k ) ) = ( D( k ) ) , DT = D−1 . ⎥ %⎥ ⎦

(23.2.27)


579

Be a permutation of the rows of the matrix D (which is equivalent to a transformation of similitude), we may group all the equivalent irreducible elements of the reduced form (23.2.27) in the form ⎡ D(1) ⎢ ⎢ ⎢ ⎢ D= ⎢ ⎢ ⎢ ⎢ ⎢ ⎣⎢

D(1) % D(2 ) D(2 )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ % ⎦⎥

(23.2.27')

It results, from the relation (23.2.26'), that the matrix B commutes with the matrices of the representation D and, according to Schur’s lemma, must be of the form ⎡ B(1) ⎢ B = ⎢ ⎢ ⎢⎣

B(2 )

⎤ ⎥ ⎥, ⎥ %⎥ ⎦

(23.2.28)

where B( k ) is a submatrix of the same dimension as the block of all the matrices D( k ) form (23.2.27'); as well, the matrix B( k ) must be given by

B( k ) = ( B( k ) )

T

(k )

βij

(k ) (k ) ⎡ β11 E β12 E ... ⎤ ⎢ ⎥ (k ) (k ) = ⎢ β21 E β22 E ... ⎥ = β( k ) × E, ⎢ ⎥ ... ... ⎥ ⎢ ... ⎥⎦ ⎣⎢

= β ji , i , j = 1,2,..., s ,

(23.2.28')

(k )

where we have put in evidence the Cartesian product of the matrix β( k ) by the unit matrix and where dim E = nk , dim β( k ) = c( k ) ,

( β( k ) )

T

= β( k ) , while c( k ) shows

how many times is D( k ) contained in D . As well, ⎡ A(1) ⎢ A = ⎢ ⎢ ⎢⎣

A(2 )

⎤ ⎥ ⎥ , A( k ) = α( k ) × E , ⎥ %⎥ ⎦

α( k ) being a matrix; we obtain thus the secular equation

(23.2.28'')


580 det [ B − λ A ] =

∏ ( det ⎡⎣ β( k ) − λα( k ) ⎤⎦ )

nk

k

= 0.

(23.2.28''')

Hence, the number of the different frequencies is at the most equal to the number of the irreducible components of the representation D and the degeneration of each frequency is at least equal to the dimension of the corresponding irreducible component. Because the representation D can be reduced by a real transformation, we can assume that the transformed matrix B is real. Thus, the number of the real constants necessary to can determine the matrices A and B is given by 1 r (i ) (i ) c (c + 1) , 2 i∑ =1

(23.2.29)

corresponding to the decomposition in a direct sum

D=

∑ ⊕ c( i )D( i ) .

(23.2.29')

If an irreducible component D ( k ) ∈ D appears only once in (23.2.29'), then A( k ) and B( k ) are diagonal matrices, while the corresponding symmetry co-ordinates (in which the representation D is completely reduced) are normal co-ordinates. Let be two systems of normal co-ordinates q ′ and q ′′ . Let us suppose that, in the normal co-ordinates q ′ , the matrices A and B have the form ⎡ A1 ⎢ A = E, B = Λ = ⎢ ⎢ ⎢⎣

A2

⎤ ⎥ ⎥, ⎥ % ⎥⎦

(23.2.30)

where each matrix Λ i is a scalar matrix (which has only one element), with Λ i ≠ Λ j , i ≠ j . If q ′′ = R −1 q ′ , then, in these co-ordinates, the matrices A and B

will be of the form A ′ = R T AR and B ′ = R T ΛR ; because the co-ordinates q ′′ are normal, so as to have A ′ = E and B ′ = Λ , the matrix R must satisfy the relations

R T R = E, R T ΛR = Λ ,

(23.2.31)

that is [ R , Λ ] = RΛ − ΛR = 0 . It results that the most general form of the matrix R , which transforms a system of normal co-ordinates in another system of co-ordinates, normal too, maintaining the order of eigenvalues in the matrix Λ , is ⎡ S1 ⎢ R = ⎢ ⎢ ⎢⎣

S2

⎤ ⎥ ⎥. ⎥ % ⎥⎦

(23.2.31')


581

If the representation D given by (23.2.29') is real, before taking the reduced form, then we can suppose that it took this form by a real transformation, so that the matrices α( k ) and β( k ) of the relations (23.2.28'), (23.2.28'') must be real. Because these matrices are symmetric, while α( k ) is positive definite, there exists a non-singular real matrix Sk , so that SkT α( k ) Sk = E( k ) and SkT β( k ) Sk = b( k ) , b( k ) being a diagonal matrix. Hence,

( Sk × Ek )T A( k ) ( Sk × Ek ) = Ek × Ek , ( Sk × Ek )T B( k ) ( Sk × Ek ) = bk × Ek ,

(23.2.32)

if the rows of the Kronecker products are correspondingly ordered. We notice that the system of co-ordinates Q = S −1 q is a system of normal co-ordinates for S =

∑ ⊕ Sk

×E,

k

(23.2.32')

while S −1 DS = D . In normal co-ordinates Q , the representation D is completely reduced to irreducible components. The matrix (23.2.31') transforms each irreducible component of D is an equivalent irreducible component. Hence, all the systems of normal co-ordinates for which Λ is of the form (23.2.30) are systems of symmetry co-ordinates. The normal co-ordinates which correspond to different frequencies cannot be transformed one into another because, if Q j is a normal co-ordinate, obtained by a transformation, then Qj + ω 2j Q j = 0 , and if Q j′ is also a normal co-ordinate, obtained by a transformation, then Qj′ + ω 2j Q j′ = 0 and it is impossible to appear, in the expression of Q j′ , other frequencies than f j = ω j / 2 π .

23.2.1.6 Case of Two Harmonic Oscillators Let us consider a discrete mechanical system formed of two harmonic oscillators of equal masses m , linked at the ends by identical springs of elastic constants k ′′ , the spring between them being of elastic constant k ′ (Fig. 23.18). The kinetic energy and the potential energy of this system are

Fig. 23.18 Problem of two harmonic oscillators of equal masses

1 m ( x12 + x22 ) , 2 1 1 + k ′ ( x1 − x 2 )2 = k ( x12 + x 22 ) − k ′x1x 2 , 2 2 T =

V =

1 k ′′ ( x12 + x 22 2

)

(23.2.33)


582

where k = k ′ + k ′′ , x1 and x 2 being the displacements of the masses from the position of equilibrium ( x1 = x 2 = 0 ). When the oscillators are in the position of equilibrium, then the mechanical system is symmetric with respect to the centre O . The reflection of the co-ordinates with respect to the centre, followed by interchange of masses, defined by the relation x1′ = − x 2 , x 2′ = − x1 ,

(23.2.34)

lets invariant the expressions of the energies T and V ; hence, the Lagrangian of the discrete mechanical system is invariant to the transformation (23.2.34). The matrix associated to this transformation is ⎡ 0 −1 ⎤ D= ⎢ ⎥. ⎣ −1 0 ⎦

(23.2.34')

From the secular equation det[ D − μE ] it results μ = ±1 , hence the representation D can be reduced to the diagonal form ⎡1 0 ⎤ D0 = ⎢ ⎥, ⎣ 0 −1 ⎦

which corresponds to the decomposition of the representation D into two irreducible ones. The determined symmetry does not reduce, in this case, the number of the possible frequencies. To determine the matrix S , we use the relation ⎡ 0 −1 ⎤ ⎡1 0 ⎤ S −1 = ⎢ S = ⎢ ⎥ ⎥, ⎣ −1 0 ⎦ ⎣ 0 −1 ⎦

which leads to ⎡ S11 S −1 = ⎢ ⎣⎢ S21

− S11 ⎤ ⎥. S 21 ⎦⎥

Hence, the symmetry co-ordinates are given by the relations ⎡ Q1 ⎤ −1 ⎢ ⎥ =S ⎢⎣Q2 ⎦⎥

⎡ x1 ⎤ ⎡ S11 (x1 − x 2 ) ⎤ ⎢ x 2 ⎥ = ⎢ S (x + x ) ⎥ . ⎣ ⎦ ⎣⎢ 21 1 2 ⎦ ⎥

We obtain −k ′ ⎤ ⎡ k ⎡m 0 ⎤ ,B = ⎢ A = ⎢ ⎥, ⎥ ⎢⎣ − k ′ k ⎥⎦ ⎣ 0 m⎦


583

from (23.2.33), hence ⎡ m ⎢ 2S 2 ST AS = ⎢ 11 ⎢ 0 ⎣⎢

Because S11 = S21 =

⎡k + k′ 0 ⎤ 0 ⎤ ⎢ 2S 2 ⎥ ⎥ T 11 ⎢ ⎥. , S BS = ⎥ m k − k′⎥ ⎢ ⎥ 0 2 2 ⎥ ⎢ 2S 21 ⎦⎥ 2S11 ⎣ ⎦

m / 2 , it results

⎡k + k′ 0 ⎤ ⎢ m ⎥ T T S AS = E, S BS = ⎢ ; k − k ′ ⎥⎥ ⎢ 0 ⎣ m ⎦

the symmetry co-ordinates (23.2.35) become Q1 =

m ( x − x 2 ) , Q2 = 2 1

m ( x + x2 ) 2 1

(23.2.35)

and the frequencies of the normal mode are ω12 =

k + k′ 2 k − k′ , ω2 = . m m

(23.2.35')

There result thus two modes of symmetry. One of them is Q1 ≠ 0, Q2 = 0 , where x 2 = − x1 , the particles oscillating in the same sense; the second mode of symmetry ( Q1 = 0, Q2 ≠ 0 , where x 2 = x1 ) corresponds to the case in which the particles oscillate in opposite senses.

23.2.1.7 Case of Three Harmonic Oscillators Let be now a discrete mechanical system formed of three harmonic oscillators of equal masses m , linked between them and at the ends by identical springs of elastic constants k (Fig. 23.19). We will make a direct study of this problem, in which – for the sake of simplicity – we have chosen, as mentioned above, equal elastic constants.

Fig. 23.19 Problem of three harmonic oscillators of equal masses

The displacements of the masses from the position of equilibrium are x1 , x 2 and x 3 , so that we can write T = V =

1 ( x 2 + x22 + x32 ) , 2m 1

1 k ⎡ x 2 + ( x 2 − x1 )2 + ( x 3 − x 2 )2 + x 32 ⎦⎤ . 2m ⎣ 1

(23.2.36)


584 We obtain Lagrange’s equations

x1 + p 2 ( 2 x1 − x 2 ) = 0, x2 − p 2 ( x1 − 2 x 2 + x 3 ) = 0, ,

(23.2.37)

x3 − p ( x 2 − 2 x 3 ) = 0, 2

where we have introduced the notation p 2 = k / m . The characteristic equation is ⎡ 2 p2 − λ ⎤ 0 − p2 ⎢ ⎥ det ⎢ − p 2 2 p2 − λ − p2 ⎥ = 0 , ⎢ ⎥ 2 2 ⎢ ⎥ 0 p 2 p − − λ ⎣ ⎦

(23.2.37')

which leads to the roots ω1 =

λ1 =

ω2 = ω3 =

2−

λ2 =

λ3 =

2 p = 0.766 p ,

2p = 1.414 p ,

2+

(23.2.38)

2 p = 1.848 p ;

there correspond the proper modes x1 = a1 cos( ω1t − ϕ1 ), x 2 = a1 2 cos( ω1t − ϕ1 ), x 3 = a1 cos( ω1t − ϕ1 ), x1 = −a2 cos( ω2t − ϕ2 ), x 2 = 0, x 3 = a2 cos( ω2t − ϕ2 ),

(23.2.39)

x1 = a 3 cos( ω3t − ϕ3 ), x 2 = −a 3 2 cos( ω3t − ϕ3 ), x 3 = a 3 cos( ω3t − ϕ3 ).

Using the results in Sect. 23.2.1.4, one can easily verify the orthogonality of these proper modes. By superposition, we obtain the general solution x1 = a1 cos( ω1t − ϕ1 ) − a2 cos( ω2t − ϕ2 ) + a 3 cos( ω3t − ϕ3 ), x 2 = a1 2 os( ω1t − ϕ1 ) − a 3 2 cos( ω3t − ϕ3 ),

(23.2.39')

x 3 = a1 cos( ω1t − ϕ1 ) + a2 cos( ω2t − ϕ2 ) + a 3 cos( ω3t − ϕ3 ).

Denoting x1′ = a1 cos( ω1t − ϕ1 ), x 2′ = a2 cos( ω2t − ϕ2 ), x 3′ = a 3 cos( ω3t − ϕ3 ),

(23.2.39'')


585

we get the linear transformation x1 = x1′ − x 2′ + x 3′ , x2 =

2 ( x1′ − x 2′ ) ,

(23.2.39''')

x 3 = x1′ + x 2′ + x 3′ .

The expressions (23.2.36) take the form T = m ( x1′2 + x2′2 + x 3′2 ) , V = 2k ⎡⎣ ( 2 −

2 ) x1′2 + x 2′2 + ( 2 +

2 ) x 3′2 ⎦⎤.

(23.2.36')

Lagrange’s system of equations becomes x1′ + ( 2 −

2 ) p 2 x1′ = 0,

x2′ + 2 p 2 x 2′ = 0, x3′ + ( 2 +

(23.2.37'')

2 ) p x 3′ = 0, 2

being an uncoupled system; obviously, we find again the proper pulsations (23.2.38).

23.2.1.8 Small Oscillations with One Degree of Freedom In the case of a discrete mechanical system with only one degree of freedom ( s = 1 ), corresponding to the generalized co-ordinate q , we have 1 2 1 aq ,V = bq 2 , a , b > 0 ; 2 2

T =

(23.2.40)

we obtain the equation of motion q + ω 2q = 0, ω 2 =

b , a

(23.2.41)

with the general solution q (t ) = A cos( ωt − ϕ ), q (t ) = − ωA sin( ωt − ϕ ), A > 0 ,

(23.2.41')

where the integration constants can be determined by the initial conditions q (0) = q 0 , q (0) = q0 in the form

A=

q 02 +

q02 q , tan ϕ = 0 . 2 ω q0 ω

(23.2.41'')

Obviously, the motion is given by a harmonic oscillation of pulsation ω , which does not depend on the initial conditions.


586

Eliminating the time t between the relations (23.2.41'), we obtain a family of ellipses q2 q 2 + = 1, 2 A ( ωA )2

(23.2.41''')

in the phase plane (Fig. 23.20); these ellipses are concentric, of semi-axes A and ωA , and are obtained from one of them by a dilatation of modulus A . The representative point P (q , q ) describes counterclockwise the corresponding ellipse.

Fig. 23.20 Small oscillations with one degree of freedom

We supposed in (23.2.40) that d2V / dq 2 ≠ 0 ; to make vanish this derivative, we choose V =

1 2n bq , n ∈ `, n > 1, b > 0 , 2

(23.2.42)

so that the position q = 0 does correspond to a minimum of the potential energy. The theorem of conservation of the mechanical energy leads to 1 2 1 2n 1 aq + bq = h , 2 2 2

(23.2.43)

where h is the energy constant, which may be determined by initial conditions. Hence, we can write q = ±

h b 1 − q2n , a h

(23.2.43')

wherefrom it results −q1 < q < q1 , q1 =

( ) h b

1/ 2 n

, q1 > 0 ;

(23.2.43'')


587

the discrete mechanical system will oscillate between the positions which are symmetric with respect to the stable position of equilibrium q = 0 , with the period T =4

a q1 h ∫0

dq 2n

q 1 − ⎛⎜ ⎞⎟ q ⎝ 1⎠

.

(23.2.43''')

We mention that these oscillations are harmonic only in the particular case n = 1 , previously considered.

23.2.1.9 Small Oscillations with Two Degrees of Freedom In the case of a discrete mechanical system with two degrees of freedom ( s = 2 ), we have 1 ( a q2 + 2a12q1q2 + a22q22 ) , 2 11 1 1 V = ( b11q12 + 2b12q1q2 + b22q22 ) . 2

T =

(23.2.44)

We assume that b11 ≠ 0 , otherwise the form V is no more positive definite; in this case, we can make the substitution q1 = q1 −

a12 q , q = q3 , b11 2 3

following to which the term q1q2 disappears from the expression of the potential energy. We can thus consider b12 = 0 in the expression of this energy. One obtains thus Lagrange’s equations

a11q1 + a12q2 + b11q1 = 0, a12q1 + a22q2 + b22q2 = 0. We pass to normal co-ordinates by relations of the form q1 = α11q1′ + α12q2′ , q2 = α21q1′ + α22q2′ ,

(23.2.45)

where αij , i , j = 1, 2 , are real constants, which can be determined so as to obtain T =

1 2 1 ( q ′ + q2′2 ) , V = 2 ( ω12q1′2 + q2′2 ) ; 2 1

replacing (23.2.45) in (23.2.44) and identifying, we get the relations

(23.2.44')


588

2 2 + 2a12 α11 α21 + a22 α21 = 1, a11 α11 2 2 + 2a12 α12 α22 + a22 α22 = 1, a11 α12

(23.2.45')

a11 α11 α12 + a12 ( α11 α22 + α12 α21 ) + a22 α21α22 = 0, b11 α11 α12 + b22 α21 α22 = 0, 2 2 b11 α11 + b22 α21 = ω12 , 2 b11 α12

+

2 b22 α22

=

(23.2.45'')

ω22 .

The last two relations (23.2.45') lead to a11b22 − a22b11 α11 α12 , a12b22 b = − 11 ( α11 α12 )2 . b22

α11 α22 + α12 α21 = − α11 α22 α12 α21

We can thus express the unknowns of the transformation (23.2.45) in the form α21 = k1 α11 , α22 = k2 α12 , where k1 and k2 are the roots of the equation of second degree

a12b22 k 2 + ( a11b22 − a22b11 ) k − a12b11 = 0 ,

(23.2.46)

that is

k1,2 = −

1 ⎡ a b − a22b11 ± 2a12b22 ⎣ 11 22

2 b11b12 ⎤⎦ . ( a11b22 − a22b11 )2 + 4a12

(23.2.46')

We notice that b11 , b22 > 0 , the form V being positive definite; hence, the discriminant of the equation is positive, the roots k1 and k2 being thus real. The two roots are of different sign, let be – for instance – k1 < 0, k2 > 0 . Replacing in the first two relations (23.2.45'), we obtain 2 α11 =

a22 k12

1 1 2 . = , α12 2 + 2a12 k1 + a11 a22 k2 + 2a12 k2 + a11

The kinetic energy being positive definite, we can state that α11 , α12 are real quantities; in this case, we can calculate also α21 , α22 , which will be real quantities too. The relation (23.2.45'') will give, finally, the pulsations, in the form ω12 =

b22 k12 + b11 b22 k22 + b11 2 = , ω ; 2 a22 k12 + 2a12 k1 + a11 a22 k22 + 2a12 k2 + a11

(23.2.46'')

these quantities are real too, because the potential energy is positive definite. We are thus led to the normal co-ordinates


589 qi′ = ai cos( ωi t − ϕi ), i = 1, 2 ,

the amplitudes a1 , a2 and the phase shifts ϕ1 , ϕ2 being determined by the initial conditions. Taking into account (23.2.46), (23.2.46'), we can calculate ω12 − ω22 =

1

( a12b22 ) P 2

2 b11b12 ⎤⎦ ⎡⎣ ( a11b22 − a22b11 )2 + 4a12

3/2

,

where we have denoted P = ( a22 k12 + 2a12 k1 + a11 )( a22 k22 + 2a12 k2 + a11 ) .

Obviously, this difference is positive, so that we can state ω1 > ω2 ; this result corresponds to the signs chosen for k1 and k2 . The generalized co-ordinates (23.2.45) will thus be given by q1 (t ) = α11a1 cos( ω1t − ϕ1 ) + α12a2 cos( ω2t − ϕ2 ), q2 (t ) = α21a1 cos( ω1t − ϕ1 ) + α22a2 cos( ω2t − ϕ2 ),

being a linear combination of the normal co-ordinates. We can assume that α11 , α12 > 0 ; in this case α21 < 0 and α22 > 0 . We can also write q1 (t ) = A11 cos ω1t + A12 cos( ω2t − ϕ ), q2 (t ) = − A21a1 cos ω1t + A22 cos( ω2t − ϕ ),

(23.2.47)

where Aij , i , j = 1, 2 , are amplitudes (positive quantities) and where we have conveniently changed the origin of the time, so that to appear a phase shift only in the second normal co-ordinate. If the two vibrations have the same direction, then one can make a study analogue to that in Sect. 8.2.2.4 for the interference phenomena, the beats etc. 23.2.1.10 Methods to Determinate the Proper Pulsations If the number of degrees of freedom of a discrete mechanical system is great, then can appear difficulties in solving the secular equation; in this case, it is convenient to use some approximate methods of calculation. In the case of a discrete mechanical system, we start from the matric equation (23.2.18), written in the form KC − ω 2 MC = 0 .

(23.2.48)

Multiplying at the left by the transpose matrix CT to obtain a scalar, it results ω =

CT KC ; CT MC

(23.2.49)


590 we can also write n

ω =

n

∑ ∑ kijC iC j

i =1 j =1 n n

∑ ∑ mijC iC j

.

(23.2.49')

i =1 j =1

In the case of statically coupled discrete mechanical systems ( mij = 0, i ≠ j ), we obtain n

ω =

n

∑ ∑ kijC iC j

i =1 j =1 n

∑

i =1

.

(23.2.49'')

miiC i2

These formulae are exact; but we must know the proper forms of vibrations, hence the coefficients C i , i = 1, 2,..., n . In Rayleigh’s method one introduces the generalized forces

Fi =

n

∑ kij x j , i

i =1

= 1, 2,..., n ;

replacing these forces by mii and the displacements x j by the coefficients C j , it results mii =

n

∑ kijC j , i

= 1, 2,..., n .

j =1

Multiplying by C i and summing, we can write n

∑ miiC i =

i =1

n

n

∑ ∑ kijC iC j .

i =1 j =1

Finally, we get the approximate formula n

ω =

∑ miiC i

i =1 n

∑

i =1

,

(23.2.49''')

miiC i2

where C i are the displacements produced by the generalized forces Fi , numerically equal to mii .


591

Let us return to the case of three harmonic oscillators, considered in Sect. 23.2.1.7; we have m11 = m22 = m33 = m , mij = 0, i ≠ j , k11 = k22 = k33 = 2k , k12 = k21 = k23 = k32 = − k , k13 = k31 = 0 . The coefficients C i , i = 1, 2, 3 , will be given by

the system 2kC 1 − kC 2 = − kC 1 + 2 kC 2 − kC 3 = − kC 2 + 2kC 3 = m ;

it results C 1 = C 3 = 3m / 2k , C 2 = 2m / k . The formula (23.2.49''') leads to ω =

10k = 0, 767 p ; 17 m

comparing with the minimal pulsation corresponding to the fundamental proper form given by (23.2.38), we have an error of 1.31‰, hence a very good approximation. If det K ≠ 0 , then we can calculate K −1 , and (23.2.48) leads to C = ω 2 αC, α = K −1 M ;

(23.2.48')

we have always det M ≠ 0 , so that M −1 does exist and we can obtain analogously ω 2 C = β C, β = M −1 K .

(23.2.48'')

Starting from (23.2.48) or from (23.2.48'), we can use a method of matric iteration. We consider a column matrix C0 for the proper form; one of the mentioned formulae allows us to calculate a matrix C1 a.s.o., till an approximation of order n , when the forms Cn and Cn −1 are in direct proportion, with a good approximation. The formula (23.2.48') leads to the fundamental proper form, corresponding to the smallest proper pulsation, while the formula (23.2.48'') allows to calculate the fundamental proper form, corresponding to the greatest proper pulsation. One can thus appreciate the spectrum of the proper pulsations. In the previously considered problem, we have ⎡ 2k ⎡m 0 0 ⎤ ⎢ ⎢ ⎥ M = ⎢ 0 m 0 ⎥ , K = ⎢ −k ⎢ ⎢ ⎥ ⎢⎣ 0 0 m ⎥⎦ ⎢⎣ 0

−k 2k −k

0 ⎤ ⎥ −k ⎥ , ⎥ 2k ⎦⎥

so that ⎡ 3 / 4 1/ 2 1/ 4 ⎤ ⎡ 2 −1 0 ⎤ ⎥ ⎢ ⎥ 1 ⎢ 1 1/ 2 ⎥ , β = p 2 ⎢ −1 2 −1 ⎥ . α = 2 ⎢ 1/ 2 p ⎢ ⎥ ⎢ ⎥ ⎢⎣ 1/ 4 1/ 2 3 / 4 ⎥⎦ ⎣⎢ 0 −1 2 ⎦⎥


592

To determine the smallest proper pulsation, we start from C 10 = 1, C 20 = 2,C 30 = 1 ; it results C 1 = 2 ω 2 / p 2 ,C 2 = 3 ω 2 / p 2 ,C 3 = 2 ω 2 / p 2 . Because 2 /1 = 2, 3 / 2 = 1.5 , (1)

we make another iteration, starting from C 1

= 2,C 2

= 3,C 3

(1)

(1)

C 1 = (7 / 2)ω / p ,C 2 = 5 ω / p ,C 3 = (7 / 2)ω / p . 2

2

2

previous results, we have (2 ) C1

=

(2) 7,C 2 = 2 2

(2) 10,C 3

2

2

with

the

7 / 2 = 3.50, 10 / 3 = 3.33 , so that, starting from

= 7,

we

make

a

C 1 = 12 ω / p ,C 2 = 17 ω / p ,C 3 = 12 ω / p ; 2

= 2 ; we obtain thus

Comparing

2

2

2

new in

2

this

iteration

and

get

12 / 7 = 1.714 ,

case

17 /10 = 1.700 , and we can consider that the result is sufficiently exact. Applying the

formula (23.2.48'), we can write, e. g., C2 =

(

)

ω2 1 1 C1 + C 2 + C 3 , 2 p2 2

wherefrom 17 =

we obtain ω =

(

)

ω2 1 1 12 + 17 + 12 ; 2 p2 2

17 / 29p = 0.766 p , hence the same result as that given by (23.2.38).

For the greatest pulsation, we start from C 10 = 1,C 20 = −1,C 30 = 1 and obtain (1)

successively C 1

= 3 p 2 / ω 2 ,C 2

(1)

= −4 p 2 / ω 2 ,C 3

(1)

(2)

= −14 p 2 / ω 2 ,C 3

( 3)

= 17 p 2 / ω 2 ; observing that 17 / 5 = 3.40, 20 / 7 = 3.43 , we can stop to this

C3

(2)

operation. Finally, it results that ω =

( 3)

C1

= 17 p 2 / ω 2 ,C 2

= 10 p 2 / ω 2 ,

(2 )

C2

= 10 p 2 / ω 2 ,

= 3 p2 / ω2 , C1

( 3)

= −24 p 2 / ω 2 ,

41/12p = 1.848 p , hence the name result as

that given by (23.2.38). Various other methods of calculation have been considered, useful for particular problems; we mention thus Holzer’s method, which allows writing the characteristic equation in a general form in the case of a discrete mechanical system with several masses and springs disposed in series.

23.2.1.11 Vibration of Discrete Mechanical Systems Subjected to Supplementary Holonomic Constraints Let be a holonomic and scleronomic mechanical system with s degrees of freedom, for which the kinetic and the potential energy, expressed by means of the normal co-ordinates q1′ , q2′ ,..., qs′ , are given by (23.2.3'''). We suppose that a supplementary holonomic and scleronomic constraint of the form f (q1′ , q2′ ,..., qs′ ) = 0

(23.2.50)


593

intervenes. Because we have to do with small oscillations around a stable position of equilibrium, we can remain with the terms of first degree, in a power series expansion, so that α1q1′ + α2q2′ + ... + αs qs′ = 0,

s

∑ α2j j =1

> 0,

(23.2.50')

where α j , j = 1, 2,..., s , are known constants; the condition imposed to these coefficients ensures us that they are not all zero, hence that the linear constraint relation is effective. The corresponding equations of Lagrange are of the form q j′ + ω 2j q j′ + λα j = 0, j = 1,2,..., s ,

(23.2.51)

where ω j are the proper frequencies of the initial discrete mechanical system, while λ is a non-determined multiplier of Lagrange. The s normal co-ordinates and the parameter λ are determined by means of the s equations (23.2.51) and of the constraint relation (23.2.50'). We search a particular solution of the form q j′ = a j cos ωt , λ = μ cos ωt ;

(23.2.51')

replacing in (23.2.50') and (23.2.51), we obtain the relations s

∑ αj a j j =1

= 0,

a j ( ω 2j − ω 2 ) + μα j = 0, j = 1, 2,..., s ,

wherefrom s

∑ ω2 j =1

j

α2j − ω2

= 0,

assuming that μ ≠ 0 . This algebraic equation has s − 1 roots for ω 2 , corresponding to the s − 1 degrees of freedom of the new mechanical system (after the intervention of the supplementary holonomic constraint); these roots are situated in the intervals between the squares of the pulsations ω12 , ω22 ,..., ωs2 , ordered after their magnitude. Assuming the existence of m holonomic and scleronomic linear constraints of the form (23.2.50'), we can state Theorem 23.2.3 (Rayleigh). The intervention of m supplementary holonomic and scleronomic constraints in a holonomic and scleronomic discrete mechanical system, subjected to small motions around a stable position of equilibrium cannot bring down the fundamental note, nor can carry it up over the value of the harmonic frequency of order m + 1 .


594

23.2.1.12 Symmetry Properties of a Conservative Discrete Mechanical System We assume, further, that ω1 ≤ ω2 ≤ ... ≤ ωs . If A and B are the matrices (23.2.9), then the kinetic energy and the potential energy, respectively, can be expressed in the form T =

1 1 A( q, q ), V = B ( q, q ) , 2 2

(23.2.53)

where we have used the notation (23.2.10''). We will use now normal co-ordinates, while in the expression of the kinetic energy we will replace the generalized velocities by normal co-ordinates; we consider thus the quadratic forms A( q ′, q ′) = q1′2 + q2′2 + ... + qs′2 , B ( q ′, q ′) = ω12q1′2 + ω22q2′2 + ... + ωs2qs′2 .

(23.2.53')

Let be the ratio ρ =

B ( q ′, q ′) . A( q ′, q ′)

(23.2.54)

It can be calculated for all the values of the normal co-ordinates, which are not simultaneously zero; obviously, ρ > 0 . Replacing all the pulsations by ω1 (which is the smallest one) or by ωs (which is the greatest one), we notice that ω12 ≤ ρ ≤ ωs2 ;

(23.2.54')

as well, making q2′ = q 3′ = ... = qs′ = 0 or q1′ = q2′ = ... = qs′ −1 = 0 , we obtain min ρ = ω12 , max ρ = ωs2 .

(23.2.54'')

If we fix the points ω12 , ω22 ,..., ωs2 on the real axis and if, at these points, we concentrate the masses m1 = q1′2 , m2 = q2′2 ,..., ms = qs′2 , then we notice that ρ is the co-ordinate of the barycentre corresponding to these masses, the relations (23.2.54') and (23.2.54'') being thus justified. Let us suppose now that one introduces the supplementary holonomic and scleronomic constraint (23.2.50'), which will be denoted shortly α = 0 . One can always determine values q1′ , q2′ , which – together with q 3′ = q 4′ = ... = qs′ = 0 – do satisfy the relation (23.2.50'). In this case, ρ =

ω12q1′2 + ω22q2′2 ≤ ω22 ; q1′2 + q2′2


595

as well

min ρ = ω22 for α = 0 . If we make vary the constraint relation (23.2.50'), then the above inequalities remain, further, valid; in particular, for q1′ = 0 ( α1 = 1, α2 = α3 = ... = αs = 0 ) we have

ρ =

ω22q2′2 + ω32q 3′2 + ... + ωs2qs′2 , q2′2 + q 3′2 + ... + qs′2

wherefrom min ρ = ω22 for q1′ = 0 .

Taking into account the previous inequalities, we can state that max min ρ = ω22 for q1′ = 0 .

If one imposes m holonomic and scleronomic linear constraints α1 = 0 , α2 = 0,..., αm = 0 , of the form (23.2.50'), then one can show that 2 max min ρ = ωm + 1 for α j = 0, j = 1,2,..., m , m = 1,2,..., s − 1 ;

(23.2.55)

this is the property of maximinimum of the frequencies of the discrete mechanical system. Analogously, together with the second formula (23.2.54''), one can show that min max ρ = ωs2−m for α j = 0, j = 1, 2,..., m , m = 1, 2,..., s − 1 ;

(23.2.55')

this is the property of minimaximum of the frequencies of the same discrete mechanical system. We have thus put in evidence the properties of extremum of the proper frequencies of a conservative discrete mechanical system. Let be now also a second conservative discrete mechanical system, specified by the quadratic forms A( q ′, q ′) and B ( q ′, q ′) with the principal frequencies ω1 < ω2 < ... < ωs . In this case 2 max min ρ = ωm + 1 for α j = 0, j = 1,2,..., m , m = 1,2,..., s − 1 ,

where we have denoted ρ =

B ( q ′, q ′) . A( q ′, q ′)

Let us suppose that the new system has a greater rigidity, with the same inertia, hence that


596

A( q ′, q ′) = A( q ′, q ′), B ( q ′, q ′) ≥ B ( q ′, q ′) ,

(23.2.56)

or that the new system has a smaller inertia, with the same rigidity, so that A( q ′, q ′) ≤ A( q ′, q ′), B ( q ′, q ′) = B ( q ′, q ′) .

(23.2.56')

In both cases, we have ρ ≤ ρ for non-zero normal co-ordinates. In this case, both the minima and maxima of these ratio are linked by the same inequality; in other words – taking into account the properties of extremum found above – it results that ω j ≤ ω j , j = 1, 2,..., m ,

(23.2.56'')

where we have a strict inequality, at least for one of the indices. We can thus state Theorem 23.2.3' (Rayleigh). The frequencies of a holonomic and scleronomic discrete mechanical system increase if its rigidity increases or if its inertia decreases. This theorem, enounced in 1873, is another form of the Theorem 23.2.3 (the supplementary constraints increase the rigidity of the mechanical system) (Rayleigh, lord, 1899, 1900).

23.2.1.13 Free Damped Small Oscillations of a Discrete Mechanical System We consider the case of oscillations with a damping in direct proportion to the velocity (viscous damping); the matric equation of Lagrange is completed in the form Mx + K ′x + Kx = 0 ,

(23.2.57)

where ′ ⎡ k11 ⎢ ′ ⎢ k21 K′ = ⎢ ... ⎢ ⎢⎣ kn′ 1

′ k12 ′ k22 ... kn′ 2

... k1′n ⎤ ⎥ ... k2′n ⎥ ... ... ⎥ ⎥ ′ ⎥⎦ ... knn

(23.2.57')

is the matrix of the damping coefficients. In general, kij′ ≠ k ji′ , i ≠ j , i , j = 1,2,..., n , the matrix K being asymmetric; if kij′ = k ji′ , i ≠ j , then the matrix K is symmetric and one can introduce the quadratic

form R =

1 T x K ′x , 2

so that the damping forces be given by

(23.2.58)


597 Φk = −

∂R , k = 1, 2,..., n . ∂x k

(23.2.58')

The quadratic form R is called dissipative energy (Rayleigh’s dissipative function). We search solutions of the form x = Aeλt , where A is a column matrix of components Ai , i = 1, 2,..., n ; introducing in the equation (23.2.57), we obtain

( Mλ 2

+ K ′λ + K ) A = 0 .

(23.2.59)

We can write, in a developed form, n

∑ ( mij λ2 j =1

+ kij′ λ + kij )Aj = 0, i = 1,2,..., n ;

(23.2.59')

imposing the condition to have non-trivial solutions Aj , j = 1,2,..., m , we get the characteristic equation det ⎣⎡ mij λ 2 + kij′ λ + kij ⎦⎤ = 0 .

(23.2.59'')

For the damped vibrations, one can make a study analogue to that in Sect. 8.2.1.3, after the position of the roots of this equation in the complex plane. As well, one can make also a study of the self-sustained vibrations (called self-vibrations too), as in Sect. 8.2.1.4.

23.2.2 Small Forced Oscillations If the small oscillations are influenced by the motion of perturbing forces, then they become forced oscillations. We consider, in what follows, various types of such forces: periodic forces, forces which do not depend explicitly on time, dissipative forces etc. As well, we make distinction between undamped and damped small oscillations.

23.2.2.1 Action of Periodic Perturbing Forces on the Small Oscillations of a Conservative Discrete Mechanical System Let be a holonomic and scleronomic discrete mechanical system, acted upon – in general – by viscous damped forces and by periodic perturbing forces. Lagrange’s matric equation (23.2.57) will be completed in the form Mx + K ′x + Kx = F cos ωt ,

(23.2.60)

where F is a column matrix, the elements of which are F1 , F2 ,..., Fs ; the pulsation of this perturbing force is ω . The complete solution of the matric equation (23.2.60) is formed from the solution of the homogeneous equation (a damped proper vibration) over which is superposed a forced harmonic vibration (a particular solution of the complete equation). This latter vibration is obtained as a real part of the solution of the complex equation


598

Mz + K ′z + Kz = Feωt ;

(23.2.61)

we search the particular solution of this equation in the form z = Aeiωt .

(23.2.61')

Calculating z = iω z , z = − ω 2 z , we obtain

( − Mω 2

+ iω K ′ + K ) z = Feiωt ,

wherefrom z = ( − Mω 2 + iω K ′ + K )

−1

Feiωt .

(23.2.61'')

In the absence of the damping force (we take K ′ = 0 ), we notice that the bracket is real, while Re eiωt = cos ωt ; the forced vibration is given by x = ( − Mω 2 + K )

−1

F cos ωt .

(23.2.61''')

If the mechanical system has s degrees of freedom, then we are in the space of configurations Λs ; the passing from the column vector q of the generalized co-ordinates to the vector column q ′ or the normal co-ordinates is made by the formula (23.2.23), written in the form q = Sq ′, S = [ sij ] , det S ≠ 0 .

(23.2.23')

Analogously, we will try to pass from the column vector Q of the generalized forces, expressed by means of the generalized co-ordinates, to the corresponding column vector Q ′ , expressed by means of the normal co-ordinates; to do this, let be the virtual work Qi δqi = Qi′δqi′ .

But δqi = sij δq j′ , i = 1,2,..., s , so that Qi sij δq j′ = Q j′ δq j′ ; the virtual displacements being independent (the constraints of the mechanical system are holonomic), it results Q j′ = Qi sij or Q ′ = ST Q ;

(23.2.62)

one can thus write

Q = ( ST

)−1 Q ′ ;

(23.2.62')

where the generalized forces have a contravariant transformation with respect to the generalized co-ordinates.


599

If S is an orthogonal matrix, then ( S T )−1 = S and the generalized forces are transformed as the generalized co-ordinates. Finally, we can write the system of Lagrange’s equations in normal co-ordinates in the form q j′ + ω 2j q j′ = Q j′ (t ), j = 1, 2,..., s ,

(23.2.63)

the general solution being given by

q j′ (t ) = a j cos ( ω j t − ϕ j

) + q ∗j (t ), j

= 1, 2,..., s ,

(23.2.63')

where q ∗j (t ) is a particular solution of the complete equation (23.2.63). Assuming that Q j′ (t ) is a periodic function of time, which can be expanded into a Fourier series, we choose a term of this series, so that Q j′ (t ) = Aj cos ωt , j = 1,2,..., s ,

(23.2.64)

of period T = 2 π / ω ; we obtain easily q ∗j (t ) =

Aj

− ω2

ω 2j

cos ωt , j = 1, 2,..., s .

(23.2.65)

Analogously, the expansion into a Fourier series

Q j′ (t ) =

∞

∑ Ajn cos ( n ωt − ϕjn ) , j

n =0

= 1,2,..., s ,

(23.2.64')

leads to q ∗j (t ) =

∞

∑ ω2

n =0

j

Ajn

− n 2 ω2

cos ( n ωt − ϕ jn ), j = 1, 2,..., s .

(23.2.65')

If n ω = ω j , while Ajn ≠ 0 , then takes place the phenomenon of resonance for the respective normal co-ordinate. If a j , j = 1, 2,..., s , are the amplitude vectors of components aij , i , j = 1, 2,..., s , then we can write q = q 0 + q∗ ,

(23.2.66)

where the free oscillations are given by q0 =

s

∑ C j a j cos ( ω j t − ϕj ) j =1

(23.2.66')


600 and the forced oscillations by q∗ =

s

∑ a j q ∗j . j =1

(23.2.66'')

23.2.2.2 Small Oscillations of a Holonomic and Scleronomic Discrete Mechanical System Subjected to the Action of Forces Which Do Not Depend Explicitly on Time If the holonomic and scleronomic discrete mechanical system is acted upon by forces which depend only on the generalized co-ordinates and on the generalized velocities, then Lagrange’s equations will be of the form d ⎛ ∂T ⎞ ∂T − = Q j ( q1 , q2 ,..., qs , q1 , q2 ,..., qs ) , j = 1, 2,..., s , dt ⎜⎝ ∂q j ⎟⎠ ∂q j

(23.2.67)

where the kinetic energy is a positive definite quadratic form (23.2.3'). If we expand the generalized forces into a Maclaurin series around the origin, considered to be a stable position of equilibrium, then we get ⎛ ∂Q j ⎞ ⎛ ∂Q j ⎞ Q j = Q j0 + ⎜ ⎟ qk + ⎜ ∂q ⎟ qk , j = 1, 2,..., s , q ∂ ⎝ k ⎠0 ⎝ k ⎠0

where we neglect the terms of higher order, because we assume that the oscillations are small; we take Q j (0, 0,..., 0) = Q j0 = 0, j = 1, 2,..., s , the origin being a position of equilibrium, and denote ( ∂Q j / ∂qi )0 = −bij , ( ∂Q j / ∂qi )0 = −cij , i , j = 1, 2,..., s , so that Q j = −bij qi − cij qi , j = 1, 2,..., s .

(23.2.68)

The system of equations (23.2.67) takes the form aij q j + cij q j + bij q j = 0, i = 1, 2,..., s ,

(23.2.67')

analogue to that in case of small damped free oscillations (see Sect. 23.2.1.13). Introducing the matrices (23.2.9), the matrix ⎡ c11 c12 ⎢ c21 c22 C = ⎢ ... ... ⎢ ⎢ cn 1 cn 2 ⎣

... c1s ⎤ ... c2 s ⎥ ⎥ ... ... ⎥ ... css ⎥⎦

(23.2.9'')

and the column vectors (23.2.9'), we can write Lagrange’s system of equations in the matric form Aq + Cq + Bq = 0 .

(23.2.67'')


601

We search a solution of the form q = aeλt ,

(23.2.68)

where a is a column amplitude vector, while λ is a scalar; introducing in the equation (23.2.67''), we obtain the equation which determines the amplitude vector

( Aλ2

+ Cλ + B ) a = 0 ,

(23.2.69)

+ cij λ + bij ) aij = 0, i = 1, 2,..., s .

(23.2.69')

or, in a developed form,

( aij λ2

These equations have non-trivial solutions if and only if the determinant of the coefficients vanishes det ⎡⎣ Aλ2 + Cλ + B ⎤⎦ = 0 ;

(23.2.70)

developing, we can write a11λ 2 + c11λ + b11

a12 λ 2 + c12 λ + b12

... a1s λ 2 + c1s λ + b1s

a21λ2 + c21λ + b21 ...

a22 λ2 + c22 λ + b22 ...

... a2 s λ 2 + c2 s λ + b2 s = 0, ... ...

as 1λ2 + cs 1λ + bs 1 as 2 λ2 + cs 2 λ + bs 2

...

(23.2.70')

ass λ2 + css λ + bss

obtaining an algebraic equation of degree 2s in λ (the secular equation). Let us suppose that all the roots λ1 , λ2 ,..., λ2 s of this equation are distinct. To each root λk will correspond non-zero amplitude ak , hence a particular solution ak eλk t of the equation (23.2.67'); the general solution is obtained in the form q =

2s

∑ C k ak eλ t . k

(23.2.68')

k =1

If Re λk < 0, k = 1, 2,..., 2s , then the position of equilibrium is asymptotically stable, not only for the linearized system but also for the non-linear initial system. In the case of a conservative discrete mechanical system we have C = 0 , the matrices A and B being positive definite. The secular equation det[ Aλ2 + B ] = 0 may be written also in the from det[ B − μ A ] , where μ is a positive real root; we notice that, in this case λ = ± iμ , so that the equation (23.2.70') has purely imaginary roots in case of a conservative holonomic and scleronomic discrete mechanical system.

23.2.2.3 Small Oscillations of a Holonomic and Scleronomic Discrete Mechanical System Subjected to the Action of Forces Which Depend Explicitly on Time Unlike the case considered at the preceding subsection, we will suppose now that the generalized forces depend explicitly on time. The system of equations (23.2.67') is completed in the form


602

aij q j + cij q j + bij q j = Qi (t ), i = 1, 2,..., s ;

(23.2.71)

in a matric form, we can write Aq + Cq + Bq = Q .

(23.2.71')

The general solution is given by q = q 0 + q∗ =

2s

∑ C k ak eλ t k

k =1

+ q∗ ,

(23.2.71'')

where q 0 is the general solution of the homogeneous equation, while q∗ is a particular solution of the complete equation. We assume that the origin ( q1 = q2 = ... = qs = 0 ) represents an asymptotically stable position of equilibrium, so that Re λk < 0, k = 1, 2,..., 2s ; hence, q 0 → 0 for t → ∞ , so that, for a sufficient long time t , one can neglect q 0 with respect to q∗ . In what follows we make a study of the particular solution q∗ . We assume that only one component of the generalized force Q is non-zero, let be Q1 (t ) ≠ 0,Q j (t ) = 0, j = 2, 3,..., s ; because the system is linear, we can obtain a general result by superposition of effects. Choosing the non-zero component of the generalized force in the form Q1 (t ) = Aeiωt , the system (23.2.71) will be given by a1 j q j + c1 j q j + b1 j q j = Aeiωt , aij q j + cij q j + bij q j = 0, i = 2, 3,..., s .

We search a solution of the form q j = a j eiωt , j = 1, 2,..., s ,

the amplitude a j being given by the system of algebraic equations 2 ⎣⎡a1 j (iω ) + c1 j (iω ) + b1 j ⎦⎤a j = A,

⎡⎣aij (iω )2 + cij (iω ) + bij ⎤⎦a j = 0, i = 2, 3,..., s ;

(23.2.72)

the solution of this system is a j = W1 j (iω )A, j = 1, 2,..., s ,

(23.2.72')

where W1 j (iω ) =

Δ1 j (iω ) , j = 1,2,..., s , Δ (iω )

(23.2.72'')


603

is a function in the form of a rational fraction with real coefficients, depending on iω ; the hodograph of this function in the complex plane (or even the function itself) is called the characteristic frequency or the characteristic amplitude-phase. Let us write the characteristic frequency in the form W1 j (iω ) = R1 j ( ω )eiψ1 j ( ω ) , j = 1, 2,..., s ,

(23.2.72''')

where R1 j ( ω ) > 0 is the characteristic amplitude, while ψ1 j ( ω ) is the characteristic phase; finally, it results q j (t ) = R1 j ( ω )Aei[ωt + ψ1 j ( ω ) ] , j = 1, 2,..., s .

(23.2.73)

If we choose now the non-zero component of the generalized force in the form Q1 (t ) = A cos ωt , we get, analogously, q j (t ) = R1 j ( ω )A cos ( ωt + ψ1 j ( ω ) ) , j = 1, 2,..., s .

(23.2.73')

We remark that, passing from the generalized force Q1 (t ) to the generalized co-ordinate q j (t ) , we amplify the amplitude by the characteristic amplitude and shift the phase by the characteristic phase ψ1 j ( ω ) for j = 1,2,..., s . We can thus state that R1 j ( ω ) increases or decreases the amplitude of the perturbing force; the filter principle is based on this observation.

23.2.2.4 Action of Small Dissipative Forces Upon a Conservative, Holonomic and Scleronomic Discrete Mechanical System Let us suppose that the matrices B = [bij ] and C = [cij ] which intervene in the expression (23.2.68) of the generalized forces are symmetric and positive definite. Introducing the potential energy and Rayleigh’s dissipative function (see Sect. 23.2.1.13 too), we can write Qj = −

∂V ∂R 1 1 − , j = 1, 2,..., s , V = bij qi q j , R = cij qi q j . ∂q j ∂q j 2 2

(23.2.74)

The dissipative forces do not change the stability of the position of equilibrium; moreover, in some cases, this position can be asymptotically stable. To integrate the equation of motion (23.2.67''), we search a solution of the form (23.2.68); the amplitudes a j , j = 1, 2,..., s , will be given by the algebraic system (23.2.69'). Multiplying by the amplitudes ai , complex conjugate to ai , we obtain λ 2aij ai a j + λcij ai a j + bij ai a j = 0 ,

(23.2.75)

or, in compact notation, A( a, a )λ2 + C ( a, a ) + B ( a, a ) = 0 ,

(23.2.75')


604

where we have used the bilinear form A( a, a ) > 0, B ( a, a ) > 0, C ( a, a ) > 0 , so that

(23.2.10); we notice that Re λ < 0 . Let λ = γ + δ i,

λ = γ − δ i be two complex conjugate roots of the secular equation (23.2.70); we notice that these roots verify the equation (23.2.75') too. Denoting a = u + vi, a = u − vi , where u and v are real column vectors, we can write 2 Re λ = λ + λ = −

λ

2

= λλ =

C ( a, a ) C ( u, u ) + C ( v , v ) = − < 0, A( a, a ) A( u, u ) + A( v, v )

B ( a, a ) B ( u, u ) + B ( v , v ) = > 0, A( a, a ) A( u, u ) + A( v, v )

(23.2.75'')

where we have used the property (iv) in Sect. 23.2.1.2. Hence, to two complex conjugate roots λ and λ correspond two complex conjugate oscillations aeλt and aeλt . Taking two complex conjugate constants C = (1/ 2)( α + β i) , C = (1/ 2)( α − β i) , we may write C aeλt + C aeλt = e γt [ ( αu − β v ) cos δt − ( α v + β u ) sin δt ] .

(23.2.75''')

If to the real roots λ and λ ′ correspond the column vectors a and a ′ , respectively, we obtain, analogously, the equation

A( a, a ′)λ2 + C ( a, a ′) + B ( a, a ′) = 0 ,

(23.2.76)

wherefrom λ+λ = −

C ( a, a ′ ) B ( a, a ′ ) . , λλ = A( a, a ′ ) A( a, a ′ )

(23.2.76')

Passing to normal co-ordinates, we can express the kinetic and the potential energy, respectively, in the form (23.2.3'''); Rayleigh’s dissipative function will be given by R =

1 s 1 s s βi qi′2 + ∑ ∑ βij qi′q j′ , ∑ 2 i =1 2 i =1 j = 1

(23.2.77)

where we assume that the coefficients βi > 0, βij , i ≠ j , i , j = 1, 2,..., s , are small (the product and the squares of these coefficients are negligible). We obtain Lagrange’s equations qi′ + βi qi′ +

s

∑ βij q j j =1

= 0, i = 1, 2,..., s ;

(23.2.78)

searching solutions of the form qi′ = αi eλt , i = 1, 2,..., s , we get the system of linear algebraic equations


( λ2

605 s

+ βi λ + ωi2 ) αi + λ ∑ βij α j = 0, i ≠ j , i = 1, 2,..., s . j =1

(23.2.78')

To have non-trivial solutions, one must equate to zero the determinant of the coefficients; it results the characteristic equation λ2 + β1λ + ω12

β12 λ λ + β2 λ +

β21λ

2

...

...

βs 1λ

βs 2 λ

ω22

...

β1s λ

...

β2 s λ

...

...

= 0.

(23.2.79)

... λ 2 + βs λ + ωs2

Developing the determinant and neglecting the product of coefficients βi , βij , as it has been mentioned above, we obtain s

∏ ( λ2 i =1

+ βi λ + ωi2

) = 0,

(23.2.79')

so that we can write, with a good approximation, λj ≅ −

βj ± iω j , j = 1, 2,..., s . 2

(23.2.79'')

Let us consider λ1 = −(1/ 2)β1 + ω1 i and let us calculate the amplitudes αi , i = 1, 2,..., s . Replacing in the last s − 1 equations (23.2.78'), we can write ω 2 − ω22 ⎞ α2 α3 αs ⎛ β21 + ⎜ β2 − β1 + i 1 ⎟ α + β23 α + ... + β2 s α = 0, ω 1 1 1 ⎝ ⎠ 1 2 2 α ω − ω3 ⎞ α3 αs ⎛ β31 + β32 2 + ⎜ β3 − β1 + i 1 ⎟ α + ... + β3 s α = 0, α1 ⎝ ω1 1 ⎠ 1 .............................................................................................. βs 1 + βs 2

α2 α ω 2 − ωs2 ⎛ + βs 3 3 + ... + ⎜ βs − β1 + i 1 α1 α1 ω1 ⎝

⎞ αs ⎟ α = 0, ⎠ 1

obtaining a system of s − 1 equations for the ratios α j / α1 , j = 1, 2,..., s ; neglecting the terms of higher order in βi and βij , we obtain

β j 1 ω1 αj = iε j , ε j = 2 , j = 2, 3,..., s , α1 ω1 − ω 2j

(23.2.80)

εj , j = 2, 3,..., s , being real small quantities, of the same order of magnitude. There

results the normal co-ordinates (we assume that α1 =Ae − αi )


606

q1′ (t ) = Ae − ( β1 / 2 )t ei( ω1t − α ) , q1′ (t ) = εj Ae − ( β1 / 2 )t ei( ω1t − α + π / 2 ) , j = 2, 3,..., s .

(23.2.80')

Taking into account λ1 = −(1/ 2)β − iω1 too, we obtain, by a linear combination, q1′ (t ) = Ae − ( β1 / 2 )t cos ( ω1t − α ) ,

(

q j′ (t ) = εj Ae− ( β1 / 2 )t cos ω1t − α +

)

π , j = 2, 3,..., s . 2

(23.2.80'')

We get analogous expressions for the other principal oscillations. We notice thus that the dissipative forces do not modify the frequencies of the conservative discrete mechanical system; the oscillations induced by these forces tend to zero for t → ∞ . In the principal oscillation of order j , all the co-ordinates are small with respect to the co-ordinate of order j and differ in phase from it with a quarter of a period ( π / 2 = 2 π / 4 ).

23.2.3 Non-linear Vibrations The non-linear vibrations appear in the case of many phenomena of mechanical nature, their mathematical modelling leading to non-linear differential equations or systems of differential equations. In general, a differential equation of these vibrations is of the form x + f (x , x , t ) = 0

(23.2.81)

and is obtained from Newton’s equation, which links the elastic forces to the displacements. In particular, one can consider autonomous vibrations, which lead to equations of the form x + f (x , x ) = 0

(23.2.81')

or to non-autonomous vibrations of the form x + f (x , t ) = 0

(23.2.81'')

x + f (x , t ) = 0 .

(23.2.81''')

or of the form

It has been shown in Sect. 8.2.2.13, in some particular cases, how one can obtain such equations. As well, it has been put in evidence an exact method to determine the period of non-damped free non-linear vibrations, establishing the formula (8.2.86). An important class of non-linear equations is that with small non-linearities (which are put in evidence by a small parameter) of the form (8.2.88), (8.2.88''), and these equations will be called quasilinear.


607

Besides the topological methods of calculation, as S.P. Timoshenko’s method, the Ostrogradskiĭ-Lyapunov method, the least squares method and the method of the harmonic balance. One has considered also the method of the equivalent linearization for a quasilinear equation of the form

x + ω02 x + ε f (x , x ) = 0 and for Van der Pol’s equation x + x − λ ( 1 − x 2 ) x = 0, λ > 0 ;

we mention also the method of perturbations, initiated by Poincaré and used by Krylov and Bogoliubov (Krylov, A.N., 1958; Bogoliubov, N. and Mitropolsky, Y., 1961). We mention also the graphic methods of computation, between them the isoclinal method and the delta method. In what follows, we present the method of the variation of constants too; one makes, as well, some considerations concerning the parametric vibrations and the non-linear self-vibrations.

23.2.3.1 Parametric Vibrations The parametric vibrations are vibrations with variable characteristics, hence vibrations of a discrete mechanical system, the parameters (mass, frequency, dimensions, elastic coefficient, damping coefficient etc.) of which are functions of time. Such an equation is, e. g., the equation (8.2.82') of the mathematical pendulum of variable length l = l (t ) . In the absence of a perturbing force, a mathematical model of such a mechanical phenomenon leads to a differential equation of the form x + α (t )x + β (t )x = 0 ,

which – by a certain substitution – leads to the equation u + γ (t )u = 0, γ (t ) = β (t ) −

1 1 α (t ) − α2 (t ) , 2 4

studied in Sect. 8.2.2.12. In the following, we consider the equation u + [ a + bf (t ) ] u = 0, a , b = const ,

(23.2.82)

where f (t ) is a periodic function of time ( f (t ) = f (t + T ) ). If f (t ) = cos 2 ωt , then one obtains Mathieu’s equation (see Sect. 8.2.2.13 too) for a variation of the rigidity in time. The general solution of the equation (23.2.82) is of the form


608

u (t ) = C 1u1 (t ) + C 2 u2 (t ) ,

(23.2.82')

where u1 (t ) and u2 (t ) are two linear independent particular solutions of this equation, while C 1 and C 2 are two arbitrary integration constants. Obviously, we can write u (t + T ) = C 1u1 (t + T ) + C 2 u2 (t + T )

too, where u1 (t + T ) and u2 (t + T ) are also particular solutions, because f (t ) is a periodic function. Hence, u1 (t + T ) = a11u1 (t ) + a12 u2 (t ), u2 (t + T ) = a21u1 (t ) + a22 u2 (t ),

where u1 (t ) and u2 (t ) , as well as u (t ) , are functions which – in general – are not periodic ones. Let us suppose that u (t + T ) = su (t ) ,

(23.2.82'')

hence, that, after a period, the amplitude of the action is amplified by s ; let be s > 0 . After n periods, the amplitude is amplified by s n . If s > 1 , then the amplitude increases indefinitely, the motion being instable, while if s < 1 , then the amplitude tends to zero, the motion being stable. Replacing in the formula (23.2.82''), we may write s [C 1u1 (t ) + C 2 u2 (t ) ] = C 1 [ a11u1 (t ) + a12 u2 (t ) ] + C 2 [ a21u1 (t ) + a22 u2 (t ) ] ;

this relation is independent on time if

( a11 − s )C 1 + a21C 2 = 0, a12C 1 + ( a22 − s )C 2 = 0 . The constants C 1 and C 2 are non-trivial if the determinant of the coefficients vanishes a11 − s a12

a21

a22 − s = 0 ,

(23.2.83)

hence for s given by s 2 − ( a11 + a22 ) s + a11a22 − a12a21 = 0 ;

(23.2.83')

if | s1 |> 1 or | s2 |> 1 , then the motion is instable, while if | s1 |≤ 1 and | s2 |≤ 1 , then the motion is stable. The difficulty of calculation consists in the searching of the particular solutions u1 (t ) and u2 (t ) .


609

23.2.3.2 Method of the Variation of Constants Let be a discrete mechanical system subjected to non-damped free non-linear oscillations, which lead to the non-linear equation x + x + ε f (x , x ) = 0 ,

(23.2.84)

where ε is a small parameter, the function f (x , x ) containing both linear and non-linear terms in x and x . Denoting x = y , one obtains the equivalent system x = y , y = − x − ε f (x , y ) ,

(23.2.84')

the associated linear system of which is (we make ε = 0 ) x = y, y = −x ,

(23.2.85)

x = C 1 cos t + C 2 sin t , y = −C 1 sin t + C 2 cos t .

(23.2.85')

with the solutions

In the method of variation of constants, we write x = C 1 (t )cos t + C 2 (t ) sin t , y = −C 1 (t )sin t + C 2 (t ) cos t ;

(23.2.85'')

introducing in the non-linear system (23.2.84'), one obtains C 1 (t ) cos t + C 2 (t ) sin t = 0, C 1 (t ) sin t + C 2 (t ) cos t = − ε f ( C 1 (t ) cos t + C 2 (t ) sin t , −C 1 (t ) sin t + C 2 (t ) cos t ) .

Hence, it results C 1 (t ) = ε f ( C 1 (t ) cos t + C 2 (t ) sin t , −C 1 (t )sin t + C 2 (t ) cos t ) sin t , C 2 (t ) = − ε f ( C 1 (t ) cos t + C 2 (t ) sin t , −C 1 (t ) sin t + C 2 (t ) cos t ) cos t ;

(23.2.86)

by integration, we get the exact solution. In general, it is very difficult to integrate this system of equations for an arbitrary function f . In an approximate calculation, we assume that the functions C 1 (t ) and C 2 (t ) are constant during a period 2π , so that they can be approximate by their mean values on that period; we can write ε 2π f ( C 1 (t ) cos t + C 2 (t ) sin t − C 1 (t ) sin t + C 2 (t ) cos t ) sin t d t , 2 π ∫0 ε 2π C 2 (t ) = − f ( C 1 (t ) cos t + C 2 (t ) sin t − C 1 (t ) sin t + C 2 (t ) cos t ) cos t d t . 2 π ∫0 (23.2.86') C 1 (t ) =


610

where C 1 and C 2 are considered to be constant in the integrand. Calculating the integrals, it results a system of the form C 1 (t ) = εF1 ( C 1 (t ),C 2 (t ) ) , C 2 (t ) = εF2 ( C 1 (t ),C 2 (t ) ) ,

(23.2.86'')

where C 1 and C 2 are of new considered as functions of time. Integrating the latter system and replacing in (23.2.85), we get the solution of the non-linear equation (23.2.84). Let us consider, in particular, the equation ( f (x , x ) = x 3 ) x + x + εx 3 = 0 ;

(23.2.87)

the formulae (23.2.86') lead to ε 2π ( C 1 cos t + C 2 sin t 2 π ∫0 ε 2π C 2 (t ) = − ( C 1 cos t + C 2 sin t 2 π ∫0 C 1 (t ) =

3ε C ( C 2 + C 22 ) , 8 2 1 3ε cos t d t = − C 1 ( C 12 + C 22 ) , 8

)3 sin t d t = )3

We notice that C 1 /C 2 = −C 2 /C 1 , wherefrom C 12 + C 12 = a 2 = const , a being an amplitude. Eliminating C 1 and C 2 , respectively, we can write C 1 + ω 2C 1 = 0, C 2 + ω 2C 2 = 0, ω 2 =

9 ε2 4 a , 64

wherefrom C 1 (t ) = A cos ωt + B sin ωt , C 2 (t ) = − A sin ωt + B cos ωt ,

the integration constants A and B being determined by the initial conditions. In this case x (t ) = A cos(1 + ω )t + B sin(1 + ω )t , y (t ) = x (t ) = −(1 + ω )A sin(1 + ω )t + (1 + ω )B cos(1 + ω )t ,

where ω =

3ε 2 a . 8

(23.2.87')

Putting the initial conditions x (0) = a , x (0) = 0 , it results the solution of the equation and the period, respectively, x (t ) = a cos(1 + ω )t , x (t ) = −(1 + ω )a sin(1 + ω )t , T =

2π . 1+ω

(23.2.87'')


611

23.2.3.3 Non-linear Self-vibrations In the vibratory motions considered till now one has introduced resistance forces with a sense opposite to that of the velocity; as a matter of fact, the rôle of these forces is that of damping the free motion or of limiting the increasing of the amplitudes of the forced vibrations. In some cases, one can introduce forces with the same sense as that of the velocity, which amplifies the amplitudes of the vibrations; these forces correspond to a negative friction and lead to self-vibrations. Unlike the case in which we have to do with perturbing forces, external to the discrete mechanical system, which depend explicitly on time and which lead, e.g., to the phenomenon of resonance, the negative friction depends – in general – only on the velocity, being internal to the mechanical system. The self-vibrations can be produced, e.g., by a dry friction. Other self-vibrations can appear in case of the rotation of an axletree in a bearing, in case of the combined action of the resistance by driving and the lifting power on an airplane profile or in case of the fuel injection valves of Diesel motors. In the above mentioned cases, we are led to differential equations of the form mx − k ′x + kx = 0, m , k , k ′ > 0 ,

which have been studied in Sect. 8.2.2.7. The corresponding motions are called self-sustained motions too. But it can happen that the coefficient k ′ be not constant, depending on x or on x ; in this case the vibrations are non-linear. Let thus be the equation of Van der Pol type mx − ( c1 − c2 x 2 ) x + kx = 0

(23.2.88)

and the equation of Rayleigh type mx − ( k1 − k2 x 2 ) x + kx = 0 .

(23.2.88')

In case of the first equation, the coefficient −(c1 − c2 x 2 ) is negative for small values of the elongation ( x
c1 / c2 ), then this coefficient is positive, appearing the phenomenon of damping of the vibrations. One can make analogous considerations for the equation of Van der Pol type, after the magnitude of the velocity ( x < k1 / k2 or x > k1 / k2 ). As a matter of fact, the equations (23.2.88) and (23.2.88') are equivalent; indeed, if we differentiate the equation (23.2.88') with respect to time and if we denote k1 = c1 , 3k2 = c2 and x = y , then we find again the equation (23.2.88). Starting from the equation (23.2.88) and making the change of variable τ = k / mt , we can write dx dx dτ k dx = = , dt d τ dt m dτ dx dx dτ k d2 x = = ; x = dt dτ dt m dτ 2

x =


612 the Van der Pol equation becomes

d2 x (c1 − c2 )x 2 dx − + x = 0. dτ km dτ 2

By the change of function x = form

(23.2.88'')

c1 / c2 z , the differential equation (23.2.88') takes the

d2 z dz − ε (1 − z2 ) +z = 0, dτ dτ 2

(23.2.88''')

with the notation ε = c1 / km .

Fig. 23.21 Non-linear self - vibrations: (a) ε = 0 ; (b) ε = 0.1 , small; (c) ε = 10 , great

The motion is very much influenced by the value of the parameter ε . Thus, if ε = 0 , then the vibrations are harmonic, their graph being given in Fig. 23.21a. If the values of the parameter are small (e. g., ε = 0.1 ), then the motion has, at the beginning, the character of self-vibrations, the amplitudes increasing very much; when the amplitudes attain a sufficiently great value, then appears a phenomenon of limitation of them, hence a phenomenon of saturation (Fig. 23.21b). If the parameter ε is great (e. g., ε = 10 ), then appear the so called relaxation vibrations, characterized by slow increases followed by sudden decreases of the elongation (Fig. 23.21c).

23.2.4 Applications In what follows we present some applications to the results obtained in the case of the linear vibrations. We consider thus two centrifugal regulators (one of them is the Watt regulator), a mechanical system rotor-axletree, the vibrations of the axles of


613

negligible mass, the dynamical absorber without and with damping and the oscillations of the vehicles.

23.2.4.1 Centrifugal Regulator of James Watt A Watt centrifugal regulator is compound of two rods OA and OB of the same length l , hinged at the point O to a vertical axletree; at their ends one has two balls of equal masses m . Other two rods CD and CE are hinged at the points D and E to the first ones and by a clutch C , which slides along the axletree; one assumes that the quadrangle is a rhomb of side a . For the balls A and B is considered a particle modelling (Fig. 23.22). If the angular velocity of the axletree increases, then the rods and the masses raise; as well, the clutch raises, acting by a force P the manoeuvre of a system of levels which decrease the admission of the steam in a motor. The masses of the rods and of the clutch are neglected.

Fig. 23.22 Centrifugal regulator of James Watt

The position of the regulator is determined, at a certain moment, by the angle of rotation θ of the plane of the regulator about the axle OC and by the angle ϕ made by the rods OA and OB with the axis of the axletree, in the plane of the regulator; the discrete mechanical system has thus two degrees of freedom. The moment of inertia of the parts in rotation (excepting the balls A and B ) with respect to the axis of the axletree is IO ; the bringing back moment due to the variation Δϕ = ϕ − ϕ0 of the angle ϕ (made by OA with OC ), with respect to an angle ϕ0 , in case of a constant angular velocity ω0 of the axletree, is − k Δϕ = − k (ϕ − ϕ0 ) , where k is a constant coefficient. The motion of the regulator is composed of a rotation in its plane about an axis normal at O to the plane, by the angular velocity ϕ and by a rotation of the plane about the axis OC , with the angular velocity θ . The two axes are principal axes of inertia, so that we obtain the kinetic energy T =

1 ( I θ2 + I 2 ϕ2 ) , 2 1

(23.2.89)


614 with

I = IO + 2ml 2 sin2 ϕ , I 2 = 2ml 2 ;

finally, it results T =

1 ⎡ ( I + 2ml 2 sin2 ϕ ) θ 2 + 2ml 2 ϕ2 ⎦⎤ . 2⎣ O

Upon the regulator act the weights mg of the balls, the force P in the clutch. The moment − k (ϕ − ϕ0 ) and the constraint forces at O and C , which give a vanishing virtual work. Assuming that only a virtual displacement δθ takes place, we obtain δW = − k (ϕ − ϕ0 )δθ ; it results the generalized force Qθ = − k (ϕ − ϕ0 ) . As well, the virtual displacement δϕ leads to δW = P δzC + mg δz A + mg δz B = P δzC + 2mg δz A ;

but zC = 2a cos ϕ , z A = l cos ϕ , so that δW = −2(aP + mgl ) sin ϕδϕ . It results the generalized force Qϕ = −2 ( aP + mgl ) sin ϕ .

We obtain thus Lagrange’s equations d 1 ∂I ( I θ ) = −k ( ϕ − ϕ0 ) , I 2ϕ − 2 θ2 ∂p1 = −2 ( aP + mgl ) sin ϕ . dt 1

(23.2.90)

We search first of all the position of relative equilibrium of the regulator in its plane, corresponding to a rotation with a constant angular velocity θ = ω0 about the axis of the axletree; let be ϕ0 the angle corresponding to this position. Observing that ϕ = 0 , the second equation (23.2.90) leads to sin ϕ0 (m ω02l 2 cos ϕ0 − aP − mgl ) = 0 ; one obtains thus two positions of relative equilibrium: for cos ϕ0 (aP +

mgl )/ m ω02l 2 .

ϕ0 = 0

and for

The motion with a constant angular velocity ω0 , given by

the second relation for which we assume that aP + mgl < m ω02l 2 , is called motion of regime of the regulator. We use now the equations (23.2.90) to study the small oscillations about this motion of regime. We denote ϕ = ϕ0 + ψ, θ = ω0 + γ . The first equation (23.2.90) is written in the form

4 ml 2 sin ϕ cos ϕϕθ + I 1 θ = − k ( ϕ − ϕ0 ) , wherefrom


615

2ml 2 sin 2 ( ϕ0 + ψ ) ψ ( ω0 + γ ) + ⎡⎣ IO + 2ml 2 sin2 ( ϕ0 + ψ ) ⎤⎦ γ = k ψ ;

neglecting the powers of higher order ( sin ψ ≅ ψ , cos ψ ≅ 1 ), we get

( IO

+ 2 ml 2 sin2 ϕ0 ) γ + 2ml 2 ω0 sin 2ϕ0 ψ + k ψ = 0 .

The second equation (23.2.90) becomes

I 2 ϕ − 2ml 2 θ 2 sin ϕ cos ϕ = −2 ( aP + mgl ) sin ϕ or I 2 ψ2 − ml 2 ( ω0 + γ )2 sin2 ( ϕ0 + ψ ) = −2 ( aP + mgl ) sin ( ϕ0 + ψ ) .

(23.2.90')

In the frame of the same approximation, we obtain ψ − ω0 sin 2ϕ0 γ + ω02 sin2 ϕ0 ψ = 0 .

(23.2.90'')

The solutions of the system of equations (23.2.90'), (23.2.90'') are of the form ψ = A1 eλt , γ = A2 eλt and lead to the characteristic equation (the necessary and sufficient condition that A1 and A2 be non-zero)

a 0 λ 3 + a2 λ + a 3 = 0 ,

(23.2.91)

IO + sin2 ϕ0 , a1 = 0, 2ml 2 I k ω0 sin2 ϕ0 . a2 = ω02 sin2 ϕ0 ⎛⎜ 1 + 3 cos2 ϕ0 + O 2 ⎞⎟ , a 3 = ⎝ 2ml ⎠ 2ml 2

(23.2.91')

where a0 =

To have a stable motion, the real parts of the roots λ must be negative. According to Hurwitz’s criterion, this takes place if a1 a1 > 0, a 0

a3

a2 = a1a2 − a 0a 3 > 0,

a1

a3

0

a0

a2

0 = a 3 ( a1a2 − a 0a 3 ) > 0;

0

a1

(23.2.91'')

a3

but, as it can be seen in the considered case, these conditions are not verified, the motion of regime being instable. This fact – and it can be experimentally seen – imposes the introduction of new elements in the system of regulations.


616

23.2.4.2 Centrifugal Regulator We will study the motion of the centrifugal regulator in Fig. 23.23. Each ball has the mass m1 and the clutch has the mass m2 , the spring is of elastic constant k and the four bars have each one the length l ; the weights of the bars and of the spring are negligible. The moment of inertia of the clutch with respect to the axis of rotation is I . Upon the axis of the regulator acts a moment M , so that the regulator will rotate with an angular velocity ω , the variation of which leads to a change of the distance of the balls from the axis of rotation, to a displacement of the clutch and to a deformation of the spring; a device acts on a valve which regulates the supply with fuel of the engine, so as to have a certain angular velocity. We assume also that the clutch is linked to a hydraulic damper which produces a viscous force of resistance, the damping coefficient being c .

Fig. 23.23 Centrifugal regulator

We choose as generalized co-ordinates the angle ϕ of the rotation about the vertical axis and the angle α , indicated on the figure; hence, the discrete mechanical system has two degrees of freedom. We assume that the regulator is built up so that for α = 0 the spring be non-deformed; the distances s1 and s2 , indicated on the figure, are measured from this position of the clutch and are given by s1 = (2 − cos α )l , s2 = 2(1 − cos α )l . The kinetic energy of the balls and of the clutch, respectively, will be T1 = 2

m1 2 1 1 ν , T = m2 ν22 + I ϕ2 . 2 1 2 2 2

Observing that, for the two balls, the relative velocities and the transportation velocities are orthogonal and that vr = l α, vt = (a + l sin α )ϕ , we get v12 = (a + l sin α )2 + l 2 α2 . The clutch has a motion of rotation about the vertical axis


617

with the angular velocity ϕ and a motion of rotation about the vertical axis with the angular velocity v2 = ds / dt = 2l α sin α . Finally, the kinetic energy of the discrete mechanical system is given by T = T1 + T2 = +

1 ⎡ 2m ( a + l sin α )2 + I ⎦⎤ ϕ2 2⎣ 1

1 ( 2m2l 2 + 4m2l 2 sin2 α ) α2 . 2

(23.2.92)

In a displacement compatible with the constraints, the virtual work is given by δW = Qϕ δϕ + Qα δα = M δϕ − 2 m1g δs1 − m2 g δs2 − ks2 δs2 − cs2 δs2 ,

where ks2 is the elastic force in the spring and cs2 , in the viscous resistance. Calculating δs1 , δs2 and s2 and replacing in the above relation, we may write δW = M δϕ + [ −2m1gl sin α − 2m2 gl sin α − 4l 2 k ( 1 − cos α ) sin α − 4l 2cα sin2 α ⎤⎦ δα ,

so that the generalized forces will be Qϕ = M , Qα = −2l sin α [ ( m1 + m2 ) g + 2lk ( 1 − cos α ) sin α + 2lcα sin α ] .

(23.2.92') Lagrange’s equations are given by ⎡⎣ 2m1 ( a + l sin α )2 + I ⎤⎦ ϕ + 4 m1 ( a + l sin α ) l ϕα cos α = M ,

( m1

+ 2m2 sin2 α ) l 2 α + 2m2l 2 α2 sin α cos α − m1 ( a + l sin α ) l ϕ2 cos α

(23.2.92'')

= −l sin α [ ( m1 + m2 ) g + 2lk ( 1 − cos α ) + 2lcα sin α ]

and form a system of non-linear differential equations. Obviously, the rôle of the regulator is to maintain a constant angular velocity ω0 of the axis. First of all, we will determine the position of relative equilibrium of the regulator, corresponding to this angular velocity; the corresponding motion of the regulator is the motion of regime. In this case, let λ0 be the value of α and ϕ = ω0 = const . Because ϕ = 0 and α = 0 , it results that M = 0 and m1 ( a + l sin α0 ) ω02 cos α0 − 2kl ( 1 − cos α0 ) sin α0 − ( m1 + m2 ) g sin α0 = 0,

(23.2.92''')


618

obtaining thus the connection between the angular velocity ω0 of regime, the position α0 of the regulator and the position s2 = 2(1 − cos α0 )l of the clutch an important relation for design. To put in evidence the stability of the motion of regime, we assume that this one is characterized by ω0 and can be perturbed by the variation of the moment M . We can write ϕ = ω = ω0 + ω1 and α = α0 + α1 , where ω1 and α1 are small, so that we can consider sin ( α0 + α1 ) = sin α0 + α1 cos α0 , cos ( α0 + α1 ) = cos α0 − α1 sin α0 , M ( α0 + α1 ) = M ( α0 ) + α1M ′( α0 ) + ... ≅ α1M ′( α0 ),

because M ( α0 ) = 0 . Replacing in the differential equations (23.2.92'') and taking into account the equation (23.2.92'''), we obtain the system f ω1 + pα1 − M ′α1 = 0, h α1 + bα1 + d α1 − pω1 = 0,

(23.2.93)

where f = 2m1 ( a + l sin α0 )2 + I , p = 4 m1 ( a + l sin α0 ) ω0 cos α0 , b = 4cl 2 sin2 α0 , h = 2 ( m1 + 2m2 sin2 α0 ) l 2 , d = 2m1I ω02 ⎡⎣a sin α0 + l ( 2 sin2 α0 − 1 ) ⎤⎦

(23.2.93')

+4 kl 2 ( cos α0 + 2 sin2 α0 − 1 ) + 2 ( m1 + m2 ) gl cos α0 ;

this system of differential equations determines the oscillations of the regulator around the motion of regime. Searching the solutions of this system in the form α1 = A1 eλt , ω1 = A2 eλt we obtain the characteristic equation p2 ⎞ p ⎛ hλ 3 + bλ2 + ⎜ d + ⎟λ − M ′ f = 0 , f ⎝ ⎠

(23.2.94)

which gives the pulsations; for the damping of the oscillations, the exponential must decreases in time, hence the real part of the roots of this equation must be negative. According to Hurvitz’s theorem, this takes place if the conditions p2 ⎞ p⎞ ⎛ ⎛ h > 0, b > 0, b ⎜ d + ⎟ − h ⎜⎝ − M ′ f ⎟⎠ > 0 , f ⎝ ⎠ p⎡ ⎛ p2 ⎞ p⎤ − M ′ ⎢b ⎜ d + + hM ′ ⎥ > 0 ⎟ f ⎣ ⎝ f ⎠ f⎦

hold; because h > 0 , these conditions can be written in the form

(23.2.94')


619

p2 ⎞ p ⎛ b > 0, M ′ < 0, b ⎜ d + ⎟ > − hM ′ f . f ⎝ ⎠

(23.2.94'')

The condition b > 0 asks the presence of a damping, which must satisfy the last condition (23.2.94''). The condition M ′ < 0 is satisfied if, by an increase of the angle α , the regulator provokes a decrease of the motive moment.

23.2.4.3 Rotor-Axletree System Let us consider the axletree of a rotor supposed on a spherical hinge with the centre at the point O (Fig. 23.24). The weigh of the system rotor-axletree is P , the centre of gravity C being situated over the point O at the distance OC = l . The rotor is rotating with the constant angular velocity ω about the vertical axis of symmetry of the discrete mechanical system. The inferior extremity of the axletree is at the distance OM from the fixed point O . We will study the stability of the motion of rotation of the discrete mechanical system, knowing that the moment of inertia with respect to the axis of symmetry is I 1 and with respect to any other axis normal to the first one at O is I2 .

Fig. 23.24 Rotor - axletree system

We consider the frame of reference Ox1x 2 x 3 with the Ox 3 -axis in vertical position. The position of the axis of rotation at a given moment is specified by the position M ′ of the point M of the axis of the axletree; at a certain moment, it will coincide with the OC -axis, while the functions of time by means of which we study the vibrations are the co-ordinates x1 and x 2 of the point M . Because the elastic forces vanish at the point M ′ , the differential equations of the vibrations are of the form I 2 x1 + I 1 ωx 2 − lPx1 = 0, I 2 x 2 + I 1 ωx1 − lPx 2 = 0;

(23.2.95)


620

multiplying the second equation by i and introducing the complex variable z = x1 + x 2 i , we obtain the differential equation in z I 2 z − iI 1 ωz − lPz = 0 .

(23.2.95')

The characteristic equation I 2 λ 2 − iI 1 ωλ − lPz = 0

(23.2.96)

has the roots λ1,2 =

(

1 I iI 1 ω ∓ 2 2

)

4 I 2lP − I 12 ω 2 ,

(23.2.96')

so that the solution of the differential equation (23.2.95') is

z = A1 eλ1t + A2 eλ2t ; expressing the complex integration constants in the form A1 = C 1 eiα1 , A2 = C 2 eiα2 , where C 1 ,C 2 , α1 , α2 are real integration constants, the solution is written in the form z = C 1 eλ1t + iα1 + C 2 eλ2t + iα2 .

If I 12 ω 2 < 4 I 2lP , that is if ω < ω0 = (2 / I 1 ) I 2lP , then the roots (23.2.96') are of the form λ1,2 = ∓a + bi , where a = (1/ 2 I 2 ) 4 I 2lP − I 12 ω 2 > 0 , b = I 1 ω / 2 I 2 > 0 . In this case, the general solution is given by z = C 1 e − αt + i(bt + α1 ) + C 2 eαt + i(bt + α2 ) .

(23.2.97)

One can thus deduce the equations of motion x1 = C 1 e − αt cos(bt + α1 ) + C 2 eαt cos(bt + α2 ), x 2 = C 1 e − αt sin(bt + α1 ) + C 2 eαt sin(bt + α2 ).

(23.2.97')

Observing that in the second term of the vibrations appears an increasing factor ( a > 0 ), this component has an increasing amplitude, so that the motion of rotation of the system is instable. If I 12 ω 2 = 4 I 2lP , hence if ω = ω0 = (2 / I 1 ) I 2lP , then the roots are equal, and the general solution is of the form z = (A1 + A2t )eλ1t = C 1 ei(bt + α1 ) + C 2tei(bt + α2 ) ,

the equations of motion being given by

(23.2.98)


621

x1 = C 1 cos(bt + α1 ) + C 2 cos(bt + α2 ), x 2 = C 1 sin(bt + α1 ) + C 2 sin(bt + α2 ).

(23.2.98')

In this case too, the amplitudes of the second component are increasing, the motion of rotation of the system being also instable. If I 12 ω 2 > 4 I 2lP , that is if ω > ω0 = (2 / I 1 ) I 2lP , then the roots λ1,2 = ip1,2 , with p1,2 = (2 / I 1 )( I 1 ω ∓

I 12 ω 2 − 4 I 2lP ) , are purely imaginary. The general solution

is given by z = C 1 ei( p1t + α1 ) + C 2tei( p2t + α2 ) ,

(23.2.99)

while the equations of motion are written in the form x1 = C 1 cos( p1t + α1 ) + C 2 cos( p2t + α2 ), x 2 = C 1 sin( p1t + α1 ) + C 2 sin( p2t + α2 ).

(23.2.99')

Hence, the two natural modes of vibration are harmonic. The amplitudes of the vibrations remain finite, so that, for ω > ω0 , the motion of rotation is stable. As we have seen, the vertical position of equilibrium of the discrete mechanical system is instable, remaining instable for 0 ≤ ω ≤ ω0 , but becoming stable for ω > ω0 .

23.2.4.4 Vibrations of Axletrees of Negligible Mass Let be an axletree limited by two circular rigid discs, of moments of inertia I 1 and I 2 with respect to the axis of rotation; the axletree A1A2 is of length l , of negligible

mass and of elastic constant k = l /GI p where GI p is the torsional rigidity ( G is the transverse modulus of elasticity, while I p is the polar moment of inertia) (Fig. 23.25). The moments of torsion which act upon the two discs are ± k ( θ2 − θ1 ) , where θ1 and θ2 are the corresponding angles of rotation.

Fig. 23.25 Axletree of negligible mass

We obtain the equations of motion

I 1 θ1 = k ( θ2 − θ1 ), I 2 θ2 = − k ( θ2 − θ1 ) .

(23.2.100)

Searching solutions of the form θ1 = C 1 eλt , θ2 = C 2 eλt , it results the algebraic system


622 ⎛ λ2 + k ⎜ I1 ⎝

⎞ C − k C = 0, − k C + ⎛ λ 2 + k ⎟ 1 I 2 I 1 1 ⎜⎝ I1 ⎠ 1

⎞C = 0 ; ⎟ 2 ⎠

the characteristic equation λ2 +

k I1

k − I2

−

k I1

k λ + I2 2

1 1 = λ 4 + k ⎛⎜ + ⎝ I1 I 2

⎞ λ2 = 0 ⎟ ⎠

has a double root equal to zero and two imaginary roots ω =

λ1,2 = ± ωi ,

k (1/ I 1 + 1/ I 2 ) .

Finally, we obtain the solution

θ1 = A + Bt + C cos( ωt − ϕ ), θ2 = A + Bt − C

I1 cos( ωt − ϕ ) . I2

(23.2.100')

Hence, the motion is formed by a uniform rotation over which is superposed a harmonic oscillatory motion. The ratio of the amplitudes of these oscillations is C /( −CI 1 / I 2 ) = − I p / I 1 ; hence, there exists a point in the interior of the segment of a line A1A2 , determined by this ratio for which the amplitude vanishes during the motion, the respective cross section of the axletree having only a uniform motion.

23.2.4.5 The Dynamical Absorber Let be a discrete mechanical system with one degree of freedom, modelled by a mass M , which is linked to a spring of elastic characteristic K and is in resonance under the action of a perturbing force F = F0 cos ωt ; let us suppose that the proper pulsation p =

K / M in equal to the pulsation ω of the perturbing force. Let us link a mass m

to the mass M by a spring of elastic constant k , so that K / M = k / m ; m and k may be very small with respect to M and K respectively (Fig. 23.26). The discrete mechanical system becomes thus a system with two degrees of freedom, and the mass M , as we will show, does no more vibrate.

Fig. 23.26 Dynamical absorber

The equations of motion of the new discrete mechanical system are Mx1 = − Kx1 + k (x 2 − x1 ) + F0 cos ωt , Mx 2 = − k (x 2 − x1 ),

(23.2.101)


623

or, in a matric form, ⎡M ⎢ ⎣0

0 ⎤ ⎡ x1 ⎤ ⎡ K + k +⎢ m ⎦⎥ ⎣⎢ x 2 ⎦⎥ ⎣⎢ − k

− k ⎤ ⎡ x1 ⎤ ⎡ F0 ⎤ = ⎢ ⎥ cos ωt , ⎥ k ⎦⎥ ⎣⎢ x 2 ⎦⎥ ⎣ 0 ⎦

(23.2.101')

− k ⎤ ⎡ z1 ⎤ ⎡ F0 ⎤ = ⎢ ⎥ eiωt . ⎥ k ⎦⎥ ⎢⎣ z 2 ⎥⎦ ⎣ 0 ⎦

(23.2.101'')

as a real part of the matric equation ⎡M ⎢0 ⎣

0 ⎤ ⎡ z1 ⎤ ⎡ K + k + m ⎥⎦ ⎢⎣ z 2 ⎥⎦ ⎣⎢⎢ − k

Searching solutions of the form z = Ae( ωt − ϕ )i , for which z + ω 2 z = 0 , one obtains, for the real part, x1 =

(k − m ω 2 )F0 cos ωt , { Mm ω 4 − [ Km + k ( M + m ) ] ω 2 + Kk }

kF0 cos ωt x2 = ; 4 { Mm ω − [ Km + k ( M + m ) ] ω 2 + Kk }

(23.2.102)

if one takes into account the conditions imposed to the masses and to the elastic constants, then it results x 1 = 0, x 2 = −

F0 cos ωt , k

(23.2.102')

the effect of resonance being thus avoided (the amplitude of the mass M vanishes). One has thus obtained a dynamical absorber without damping. We notice that the amplitude of the vibrations of the additional mass can be very great if the ratio F0 / k is also very great. On the other hand, the vibrations of the additional mass are in phase opposition (due to the sign minus) with the perturbing force; thus, it results the relation of equilibrium between the tension in the additional spring and the perturbing force ( kx 2 + F0 cos ωt = 0 ), which justifies the performance of the dynamical absorber. If the pulsation ω of the perturbing force is not maintained constant, then the phenomenon of resonance can no more be avoided. Thus, let us consider the denominator of the expressions (23.2.102), divided by the product Mm of the masses; it results

(

f ( ω ) = ω 4 − 2 p2 +

)

k ω2 + p 4 , M

(23.2.103)

where we have introduced the proper pulsation given by p 2 = K / M = k / m . We notice that f ( ±∞ ) = ∞ , f ( p ) = ( − k / M ) p 2 < 0 ; it results that the equation f ( ω ) = 0 has two real roots, contained in the intervals ( −∞, p ),( p , ∞ ) . If the ratio

624


k / m is small, then the roots are close to ω = p , while for ω = p we have x1 = 0 ;

the diagram of the displacement x1 = x1 ( ω ) has two vertical asymptotes (Fig. 23.27). If

k / M = 0 , then one has a double root ω = p , remaining only one vertical

asymptote, as in the classical phenomenon of resonance. One obtains the static displacement x1 = F / K for ω = 0 .

Fig. 23.27 Displacement diagram x1 vs ω

To avoid the phenomenon of resonance mentioned above, we must introduce – beside the additional mass – a damper too. The model of the mechanical damper considered above (Fig. 23.26) can be completed by a viscous damper of constant k ′ (Fig. 23.28). The corresponding system of differential equations is of the form

Fig. 23.28 Mechanical and viscous damper

Mx1 = − Kx1 + k (x 2 − x1 ) + k ′( x 2 − x1 ) + F0 cos ωt , mx 2 = − k ( x 2 − x1 ) − k ′( x 2 − x1 ),

as a real part of the system ( z1 = iωz1 , z1 = − ω 2 z1 , z 2 = iωz 2 , z 2 = − ω 2 z 2 )

(23.2.104)


625

−(k + ik ′ω ) ⎤ ⎡ z1 ⎤ ⎡ F0 ⎤ ⎡ K + k − M ω 2 + ik ′ω ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ eiωt , 2 ′ ′ − ( k + ik ω ) k − m ω + ik ω ⎥⎦ ⎣ z 2 ⎦ ⎣ 0 ⎦ ⎢⎣

(23.2.104')

written in a matric form. We get thus 1 ( k − m ω2 + ik ′ω ) F0 eiωt , N 1 z 2 = ( k + ik ′ω ) F0 eiωt , N N = ( K + k − M ω 2 )( k − m ω 2 ) − k 2 + ⎡⎣ K − ( M + m ) ω 2 ⎤⎦ ik ′ω ; z1 =

(23.2.104'')

the amplitude of the motion of the mass M will be 1 N

x1max = z1 = N =

⎡⎣ ( K + k − M ω 2

)( k

(k

− m ω2

)2

+ k ′2 ω 2 F0 ,

− m ω 2 ) − k 2 ⎤⎦ + ⎡⎣ K − ( M + m ) ω 2 ⎤⎦ k ′2 ω 2 . (23.2.104''') 2

Under the action of the constant force F0 x1st

2

raises the static deformation

= F0 / K ; to study the dynamical influence, we introduce the non-dimensional

ratio ψ = x1max / x1st , given by ψ =

1 K N

(k

− m ω2

)2

+ k ′2 ω 2 .

(23.2.105)

Putting in evidence the terms which contain the damping, we may write ψ2 =

A1 + A2 k ′2 ω 2 , B1 + B2 k ′2 ω 2

(23.2.105')

with A1 = K 2 ( k − m ω 2 B1 = ⎡⎣ ( K + k − M ω 2

)2 , A2

)( k

= K2,

− m ω 2 ) − k 2 ⎤⎦ , 2

(23.2.105'')

2

B2 = ⎣⎡ K − ( M + m ) ω 2 ⎦⎤ .

We can write the relation (23.2.105') in the form B1 ψ2 − A1 + ( B2 ψ2 − A2 ) k ′2 ω 2 = 0 ;

hence, the family of curves (23.2.104') passes through the points of piercing of the curves B1 ψ2 − A1 = 0, B2 ψ2 − A2 = 0 , hence of the curves


626 ψ1 =

A1 = B1 ψ2 =

K k − m ω2

(K

+ k − M ω2

)( k

− mω2 ) − k 2

,

(23.2.105''')

A2 K = . B2 K − ( M + m ) ω2

The curve ψ1 = ψ1 ( ω ) corresponds to the dynamical absorber without damping ( k ′ = 0 ) and the curve ψ2 = ψ2 ( ω ) corresponds to the discrete mechanical system with only one degree of freedom, of mass M + m and constant K (the masses are rigidly linked, corresponding k ′ = ∞ , ω3 = K /(M + m ) ); the two curves are piercing at the points P and Q , through which passes any curve ψ = ψ ( ω ) (Fig. 23.29).

Fig. 23.29 The curves ψ ( ω ) vs ω

Starting from these data, we must search the best additional mass m , the best elastic coefficient k and the best damping coefficient k ′ , so that the dynamic absorber with damping does work in the best conditions. If the ordinate of the point P is greater than the ordinate of the point Q , it is optional that this one be the maximal ordinate, the tangent at P being horizontal; on the other hand, it is recommended that the ordinate of the point P be as small as possible. To realize such an optimum, one must choose conveniently the parameters which are involved. We denote m K k k′ = μ, = p12 , = p22 , kcr′ = 2 mk , χ = , M M m kcr′

(23.2.106)

where kcr′ in a critical damping coefficient, while χ is a non-dimensional damping factor (see Sect. 8.2.1.3, formulae (8.2.14), (8.2.14')); after fastidious calculations, one can show that an optimum corresponding to


627

p2 1 3 , χ2 = ,ψ = = p1 1+μ 8(1 + μ )2

1+

2 , μ

(23.2.106')

function of the non-dimensional coefficient μ , which specifies the mass m .

23.2.4.6 Oscillations of Vehicles Let us model a vehicle of mass M by a rigid bar of length l , elastically supported at A1 and A2 , the corresponding elastic constants being k1 and k2 , respectively; the centre of gravity C is at the distances a1 and a2 from the support A1 and A2 , respectively (Fig. 23.30). We denote by IC the moment of inertia with respect to an axis normal to the plane of the figure at the point C .

Fig. 23.30 Modelling of the oscillations of vehicles

Let x1 = A1A1′ , x 2 = A2 A2′ , ξ = CC ′ be the displacements of the extremities of the bar and of the mass centre, respectively; we denote by θ the angle of rotation of the bar A1A2 , supposed to be very small. We notice that ξ =

1 1 ( a x + a1x 2 ) , θ = ( x 2 − x1 ) , l = a1 + a2 . l 2 1 l

(23.2.107)

The kinetic energy is expressed in the form (we apply Koenig’s theorem) T = =

1 1 M ξ 2 + IC θ 2 2 2

1 ⎡⎣ ( Ma22 + IC ) x12 + 2 ( Ma1a2 − IC ) x1x 2 + ( Ma12 + IC ) x 22 ⎤⎦, 2l 2

(23.2.108)

while the potential energy is given by V =

1 1 k1x12 + k2 x 22 ; 2 2

(23.2.108')

calculating the kinetic potential L = T − V , we get Lagrange’s equations in the form m11x 1 + m12 x 2 + k11x1 = 0, m21x1 + m22 x 2 + k22 x 2 = 0 ,

(23.2.109)


628 where

1 1 Ma22 + IC ) , m12 = m21 = 2 ( Ma1a2 + IC ) , 2 ( l l 1 2 = 2 ( Ma1 + IC ) , k11 = k1 , k12 = k21 = 0, k22 = k2 . l

m11 = m22

(23.2.109')

We notice that the considered discrete mechanical system is dynamically coupled ( k12 = 0, m12 ≠ 0 ). But the maximum comfort is obtained when the mechanical system is entirely uncoupled, hence if we have m12 = 0 too; imposing this condition, we must have Ma1a2 = IC . Introducing the central radius of gyration ( iC2 = IC / M ), we get the condition iC2 = a1a2 ;

(23.2.110)

hence, the radius of gyration iC must be the geometric (proportional) mean of the distances a1 and a2 from the centre of gravity of the vehicle to its points of suspension.

Chapter 24 Dynamical Systems. Catastrophes and Chaos In 1776, ninety years after the apparition of the fundamental treatise of Newton, Laplace enounces his famous principle of the determinism, stating that: “The actual stage of the system of nature is, obviously, a consequence of that it was at the preceding moment and, if we imagine an intelligence, which – at a given moment – knows all the relations between entities of this universe, then it could establish the respective positions and the motions of all these entities, at any moment in the past or in the future” This determinism is – in fact – a mechanistic determinism. But Laplace continues: ... “there exist things which are uncertain for us, things which are more or less probable and we try to counter-balance the impossibility to know them, determining various degrees of probability”. We are thus obliged, at a certain level of knowledge, to accept also a probabilistic principle, which – by Laplace – depends on the accuracy of the instruments of measure. Hundred thirty years later, in 1903, Henri Poincaré observes that: “A very little cause, which escapes from our observation, can lead to a sensible effect and then we say that the effect is due to the chance. It can happen that small differences at the initial conditions do produce an enormous error in what will be later. The prediction becomes thus impossible and we have to do with unforeseeable phenomena”. As an example of the sensibility of the differential equations to initial conditions, E.N. Lorenz, professor of meteorology, says: “If a butterfly which stays today on a flower flaps or not its wings, that has not a great influence on the weather in the following days, but – in exchange – can have a great influence on the weather some years after.” This fact is known today as the Lorenz’s butterfly effect. The uncertainty principle of Heisenberg according to which the position and the momentum of an elementary particle cannot be determined simultaneously with a precision as great as we wish, the Brownian motion, characterized by a great number of collisions between the particles of a very fine solid suspension in a liquid and its molecules, and many other phenomena put in evidence the necessity to introduce notions of the theory of probability as well as the aleatory variables. There appears thus the notion of chaos; and if the chaotic motions are produced in deterministic conditions, then there appears the notion of deterministic chaos, introduced forty years ago by D. Ruelle and F. Takens, by describing some phenomena of turbulent flow. The study of the causes which produce this paradoxical phenomenon introduces the notion of attractor in various forms: punctual attractor, periodic attractor and chaotic strange attractor (Arnold, V.I., 1984, 1988). The geometric representation of the critical (ramification) points led René Thom, in 1972, to the notion of catastrophe, thus being developed the theory of catastrophes (Thom, R., 1972).


629


630

The evolution of the systems in time is modelled, obviously, by non-linear differential equations, for which solutions in analytical closed form can be only very seldom obtained. Even if these systems have been – at the beginning – only mechanical ones, they involved afterwards all the chapters of physics, chemistry, biology etc. To integrate these systems, one uses – in general – numerical algorithms; but all the problems mentioned above are put. Thus appeared the theory of dynamical systems, which has been very much developed last years; we can say that it is a qualitative theory of ordinary differential equations. Obviously, a dynamical system has a more general significance than that of mechanical system, containing also electromagnetical, biological, economical, social, political systems etc. We present in this chapter continuous and discrete dynamical systems, for which we put in evidence periodical solutions and global ramifications. We are thus led to introduce some elements of catastrophe theory. In the theory of chaos (especially deterministic chaos) we will use the notions of strange attractor, fractal etc.

24.1 Continuous and Discrete Dynamical Systems The time has, in general, a continuous variation in the study of a dynamical system; but one can consider also cases in which the results are obtained for discrete values of the time variable. We will deal with both situations. As well, we make distinction between linear and non-linear systems.

24.1.1 Continuous Linear Dynamical Systems We introduce, in what follows, the most important notion necessary to the study of dynamical systems, especially the notion of attractor. We mention also the study of non-autonomous linear systems of differential equations with a control function or with periodic coefficients. 24.1.1.1 Fixed Points. Attractors Let be the non-autonomous system of differential equations with initial conditions (of Cauchy type) dx = X( x; t ), x(t0 ) = x0 , dt

(24.1.1)

where the column vectors ⎡ x1 ⎢ x2 x = ⎢ ⎢ ⎢ ⎣⎢ x n

⎡ X1 ⎤ ⎤ ⎢ ⎥ ⎥ X2 ⎥ ⎥, X = ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ Xn ⎥ ⎦⎥ ⎣ ⎦

(24.1.1')

Dynamical Systems. Catastrophes and Chaos

631

belong to the space n , while t ∈ + (we suppose that t is non-negative). If the time does not appear explicitly, then the system is called autonomous or dynamical, having the form dx = X( x ), x(t0 ) = x0 . dt

(24.1.1'')

If, in this case, all the roots of the characteristic equation (see Sects. 23.1.1.5 and 23.1.1.6) have their real part non-zero (real or complex roots), then the dynamical system is hyperbolic; if the characteristic equation has at least a root or a pair of roots with a null real part (zero root or purely imaginary roots), then the dynamical system is degenerate. As well, if X( x; t ) = X( x; t + T ) ,

(24.1.1''')

then the system is periodical of period T . By a change of variable θ = 2π

t , T

(24.1.2)

the periodic non-autonomous system of order n is transformed in the autonomous system of order n + 1 t dx dθ 2 π , x(t0 ) = x0 , θ (t0 ) = 2 π 0 . = X( x; θ ), = dt dt T T

Hence, f : + ×

n

the →

(24.1.2')

dynamical system thus considered in the application n , definite by the solution x(t ) ; a field of vectors is thus definite in

. We assume that the phase space is orthogonal, the co-ordinates corresponding to the variables necessary to specify the state of a system at a given moment. For instance, in case of a particle in motion in one dimension, the phase space is two-dimensional (the canonical co-ordinates correspond to the position and to the velocity of the particle); in general, Gibb’s space Γ 2s is a phase space. n

Let us consider the case of the non-damped linear oscillator (the harmonic oscillator) (23.1.2), with the initial conditions q (0) = q 0 , q (0) = q 0 . We denote q = x1 , q = x 2 , being led to the system of differential equations dx 1 dx = x 2 , 2 = − ω 2 x1 , x1 (0) = x10 , x 2 (0) = x 20 ; dt dt

the solution of this system is

(24.1.3)


632

x 20 sin ωt , ω x 2 (t ) = x 20 cos ωt − x10 ω sin ωt , x1 (t ) = x10 cos ωt +

(24.1.3')

and we can write the application f ( t ; x10 , x 20

with f : space

+

+

×

× 2

⎛

) = ⎜ x10 cos ωt ⎝

2

→

2

+

x 20 ⎞ sin ωt , x 20 cos ωt − x10 ω sin ωt ⎟ , ω ⎠

(24.1.3'')

. The corresponding representation in the three-dimensional

is given in Fig. 24.1.

Fig. 24.1 Representation of the motion of a non-damped linear oscillator in the three-dimensional space + × 2

The equations (24.1.3) lead to dx 2 / dx1 = − ω 2 x1 / x 2 , wherefrom ω 2 x 12 + x 22 = c 2 , c 2 = ω 2 [ x1 (0) ]2 + [ x 2 (0) ]2 = const ,

obtaining a family of ellipses in the phase space

2

(24.1.3''')

(Fig. 24.2).

x2

c O

c/ω

x1

Fig. 24.2 Motion of the representative point of a non-damped linear oscillator in the phase plane

Let be now the autonomous system (24.1.1'') written in a scalar form dx i = Xi (x1 , x 2 ,..., x n ), i = 1, 2,..., n , dt

(24.1.4)


633

where Xi are smooth functions (of class C 1 ); we can write this system also in the form of a system of n − 1 equations dx j X j ( x1 , x 2 ,..., x n ) , j = 2, 3,..., n , = dx 1 X1 (x1 , x 2 ,..., x n )

(24.1.4')

where we supposed that X1 ( x ) ≠ 0 . The solutions of this system are orbits which do not pierce one the other. If X1 ( x ) = 0 and X2 (x ) ≠ 0 , then we choose x 2 as independent variable. The point x∗ ∈ n for which X ( x∗ ) = 0

(24.1.5)

is called fixed point of the system (24.1.1''); obviously, it is also a solution of the system (24.1.4). This point is called critical point too or – in case of mechanical systems – point of equilibrium. We notice that the system (24.1.1'') cannot be written in the form (24.1.4') in the neighbourhood of such a point. If, in the neighbourhood A ∈ n of a fixed point x∗ of the autonomous system (24.1.1''), the condition x(t0 ) ∈ A implies lim x(t ) = x∗ ,

t →∞

(24.1.6)

then this point is called attractor; if this property takes place for t → −∞ , then the point is a repeller. If a solution f (t ; x0 ) of the system of equations (24.1.1) verifies the relation f (t + T ; x0 ) = f (t ; x0 ) for t ∈ + , then this one is periodical, corresponding a closed orbit (which can be a limit cycle too) in the phase space. If, for t → ∞ , the point x(t ) becomes asymptotical on a limit cycle, then this one is a (stable) attractor, while if this property takes place for t → −∞ , then one obtains a (instable) repulsive limit cycle. In the frame of the above considerations, we can take again the Theorem 23.1.22 and state (Poincaré, H., 1952). Theorem 24.1.1 (Poincaré–Bendixon). If the integral curve f (t ; x0 ) = 0, x0 ∈ 2 , t ≥ 0 , of the system of equations (24.1.1) with n = 2 is contained in a bounded domain in the phase space, then a possible attractor is a fixed point or a limit cycle. In Sect. 23.1.1.4, it has been made a study of the stability of equilibrium of an autonomous discrete mechanical system with a single degree of freedom in linear approximation. One obtains thus portraits of phase in the neighbourhood of the fixed points, i. e.: nodes (which can be attractors (Figs. 23.6 and 23.9b) or repellers (Figs. 23.5 and 23.9a)), linear attractors (Fig. 23.8b), repulsive lines (Fig. 23.8a), saddle points (Fig. 23.7), foci (which can be attractors (Fig. 23.10b) or repellers (Fig. 23.10a)) or centres (Fig. 23.11). The case in which the autonomous discrete mechanical system has n degrees of freedom has been considered in Sect. 23.1.1.5.


634 Let be the differential equation

mx = − kx − k ′x ,

(24.1.7)

corresponding to the motion of a particle acted upon by an elastic force − kx and by a force of viscous damping k ′x ; we can write in the phase plane x1 = x 2 , x 2 = − p 2 x1 − 2λx 2 , p 2 =

k k′ , , 2λ = m m

(24.1.7')

with the solutions

x1 = A1 e− λt cos( αt − ϕ1 ), x 2 = A1 e − λt cos( αt − ϕ2 ) ,

(24.1.7'')

where the constants A1 , A2 , α, ϕ1 and ϕ2 are determined as functions of m , k , k ′ and of the initial conditions. The trajectory in the phase plane has the form of a spiral; we notice that x1 → 0 and x 2 → 0 for t → ∞ and for any initial conditions (Fig. 24.3). The origin O is a fixed point, i. e. an attractor, its existence being a characteristic of the damped motions. In this case, the whole phase is the basin of the attractor. x2

P0

O

x1

Fig. 24.3 Motion of the representative point of a damped linear oscillator in the phase plane

Adding a perturbing force, we obtain the equation mx = − kx − k ′x + F0 cos ωt ,

(24.1.8)

which leads, in the phase space, to the system of differential equations x1 = x 2 , x 2 = − p 2 x1 − 2λx 2 + q cos ωt , p 2 =

F k k′ , 2λ = , q = 0 , m m m

(24.1.8')

with the solutions ( λ < p ) x1 = A1 e− λt cos( αt − ϕ1 ) + x 2 = A1 e− λt cos( αt − ϕ2 ) −

q

(ω

2

( ω2

−p

)

2 2

+ 4λ2 ω 2

qω −p

)

2 2

+ 4λ2 ω 2

cos( ωt − ψ ), sin( ωt − ψ ),

(24.1.8'')


635

where the constants A1 , A2 , α, ϕ1 , ϕ2 , ψ are determined as functions of m , k , k ′, F0 and of the initial conditions. The periodic particular solutions a1 =

q

( ω2

−p

)

2 2

+ 4 λ2 ω 2

qω

, a2 =

( ω2

− p2

)2

+ 4λ2 ω 2

(24.1.9)

lead, by eliminating the temporal variable, to a closed trajectory (an ellipse)

x12 x 22 + = 1. a12 a22

(24.1.9')

We notice that for t → ∞ the solutions (24.1.8'') tend to the solutions (24.1.9); hence, starting from an initial point P0 , interior or exterior to the ellipse, the trajectory tends to the latter one. In this case, the attractor is of limit cycle type, its basin being the whole phase plane too (Fig. 24.4). x2 P0 P0 O

x1

Fig. 24.4 The attractor of limit cycle type of a damped linear oscillator acted upon by a perturbing force in the phase plane

As we know (see Sect. 8.2.2.4), by the superposition of two harmonic vibrations (periodic motions) with periods which have not a common multiple, one obtains a non-periodic (pseudoperiodic) motion, which – after H. Poincaré – admits a representation in a three-dimensional phase space; starting from an arbitrary initial point P0 , the trajectory tends to be an envelope on a torus in 3 for t → ∞ . In this case, the attractor is a dense curve on the torus. In general, in the case of the system of differential equations (24.1.1), we assume that the initial point P0 = P ( x0 ) belongs to a set of points D ⊂ n and that for t → ∞ the solution x(t ) tends to a set of points A . We say that the set A is an attractor for the solutions which start from the set D ; the maximal set Dmax for which P → A for t → ∞ if P0 ∈ Dmax is called the basin of the attractor A . The set A can be a point, a limit cycle in 2 or in 3 or a hypertorus in n . If P → A for t → −∞ , then the set A is a repeller (a repulsive set). A study of the non-linear systems of differential equations and their connection to linear systems is given in Sects. 23.1.2.3 and 23.1.2.4.


636

Let be the autonomous system of differential equations dx = Ax + f ( x ), x ∈ dt

n

, det A ≠ 0 ,

(24.1.10)

with lim | ( f ( x ) | / | x |= 0 for | x |→ 0 , the fixed point being x = 0 . H. Poincaré showed that, if n = 2 , then any singular point of the linearized system of differential equations (with f ( x ) = 0 ) has a corresponding singular point, of the same type and with the same type of stability, for the system of non-linear differential equations; but we mention that to a centre of the linearized system corresponds a centre or a focus for the non-linear one. In the general case (for an arbitrary n ) one can show that x = 0 is an attractor (repeller) for the non-linear system too. As well, if the matrix A has an eigenvalue with a positive real part, then x = 0 is no more attractor for the non-linear system. If the matrix A is singular ( det A = 0 ), then – beside the critical point x∗ – there exist in n also other manifolds which have this property (straight lines, planes etc.). The dimension of these manifolds is equal to the dimension of the nucleus of the matrix A dim ker A = n − r ,

(24.1.10')

where n is the order of the matrix, while r is its rank. We can say that these manifolds correspond to some positions of indifferent equilibrium in the case in which the dynamical system is a mechanical one. A mechanical system can remain in equilibrium an indefinite time at a critical point x∗ if no perturbation intervenes; such perturbations cannot be avoided, because – practically – one cannot take into account, in the mathematical modelling, all the factors which determine the behaviour of the mechanical system. Obviously, a good mathematical model is that in which one takes into account the most important factors; the factors which we neglect lead, in general, to very small perturbations, which may be internal perturbations (called noise) or external perturbations (resulting from factors external to the dynamical system). The importance of the study effected in the preceding chapter on the stability of the equations of equilibrium is thus put in evidence. Let be, e. g., Helmholtz’s oscillator d2 x = x + x2 , dt 2

(24.1.11)

which leads to the system of differential equations of first order ( x1 = x , x 2 = dx 1 / dt ) dx 1 dx = x 2 , 2 = x1 + x12 , dt dt

(24.1.11')


637

with the fixed points (x1∗ , x 2∗ ) = (0, 0), (x1∗ , x 2∗ ) = ( −1, 0) ; a linear analysis shows that the first fixed point is a saddle point, while the second one is a centre (Fig. 24.5). A first integral (the mechanical energy integral) is of the form

Fig. 24.5 Representation of the motion of Helmholtz’s oscillator in the phase plane

x 22 − x12 −

2 3 x = C , C = const , 3 1

(24.1.11'')

corresponding the orbits in the figure. The curve corresponding to C = 0 passes through the point ( −3 / 2, 0) and is called homoclinic orbit. 24.1.1.2 Non-autonomous Systems of Linear Differential Equations with a Control Function We consider now a non-autonomous system of differential equations (24.1.1), (24.1.1') of the form x = Ax + Bv (t ) ,

(24.1.12)

where A and B are constant matrices and v = v (t ) , with ⎡ a11 a12 ⎢ a21 a22 A = ⎢ ⎢ ⎢ an 1 an 2 ⎣

a1n ⎤ ⎡ b11 b12 ⎢ a2 n ⎥ b21 b22 ⎥, B = ⎢ ⎢ ⎥ ⎢ ann ⎥⎦ ⎢⎣ bn 1 bn 2

b1m ⎤ ⎡ v1 ⎤ ⎥ ⎢ v2 ⎥ b2 m ⎥ ⎢ ⎥; , = v ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ bnm ⎥⎦ ⎣⎢ vm ⎥⎦

(24.1.12')

the matrix A is a square one of order n , the matrix B is a rectangular matrix with n rows and m columns, while the function v(t ) , called control function too, is a column vector with m rows. We write the general solution of the system (24.1.12) in the form

x = e At c(t ) ,

(24.1.13)


638 where we use the formal expansion e At = I +

t t2 A + A2 + ... , 1! 2!

(24.1.13')

I being the unit matrix, and where

c(t ) = [ c1 (t ), c2 (t ),..., cn (t ) ]T

(24.1.13'')

is a column matrix with n rows. Introducing in (24.1.12), we obtain – easily – c(t ) = e− At Bv (t ) , wherefrom

x(t ) = e At x0 +

t

∫0 e

A (t − τ )

Bv ( τ ) dτ ,

(24.1.13''')

with the initial condition x(0) = c(0) = x0 . Often, the system (24.1.12) is encountered in the form

x = Ax + Bv (t ), y = CT x ,

(24.1.14)

where C is a constant rectangular matrix with n rows and m columns ⎡ c11 c12 ⎢ c21 c22 C= ⎢ ⎢ ⎢ cn 1 cn 2 ⎣

c1m ⎤ c2 m ⎥ ⎥, ⎥ cnm ⎥⎦

(24.1.14')

while y = [ y1 , y2 ,..., ym ]T is a column vector with m rows. If the eigenvalues λi of the matrix A are distinct, then the corresponding eigenvectors are ui , i = 1,2,..., n (column vectors with n rows); we have seen in Sect. 23.1.1.5 that one can obtain thus the general solution (the proper mode) of the system of equations (23.1.31'). We pass to Jordan’s canonical form by a change of function x = Tξ ,

(24.1.15)

where T = [ u1 , u2 ,..., un ] is the square matrix formed by juxtaposing the eigenvectors (column vectors) of the matrix A , while ξ = [ ξ1 , ξ2 ,..., ξn ]T is a column matrix with n rows. Introducing in (24.1.14), we get

ξ = Jξ + Pv (t ), y = R T ξ , where

(24.1.16)


639

J = T−1 AT, P = T −1 B, R = TT C

(24.1.16')

are Jordan’s matrix of the system of differential equations, the matrix of modal control and the matrix of modal observability, respectively. The rectangular matrix P with n rows and m columns is called control matrix because any zero element of it corresponds to a certain proper mode which is not controlled by a certain component of the column vector v (t ) ; if all the elements of a row of the matrix P vanish, then a whole proper mode is not controlled. As well, the rectangular matrix R with n rows and m columns is called observability matrix, because any zero element of it indicates that a certain proper mode is not observable in a component of the column vector y ; if all the elements of a row of the matrix R vanish, then a whole proper mode is not observable. Assuming that the characteristic equation of the matrix A has distinct roots λi , the eigenvectors being given by Aui = λi ui , i = 1, 2,..., n , it results AT = A [ u1 , u2 ,..., un ] = [ Au1 , Au2 ,..., Aun ] = [ λ1 u1 , λ2 u2 ,..., λn un ] .

Observing that the matrix T−1 = [ u1∗ , u∗2 ,..., un∗ ]T can be obtained by the superposition of the row vectors u∗j , j = 1,2,..., n , which have the property [ u∗j ui ] = [ δij ] , where δij , i , j = 1, 2,..., n , is Kronecker’ symbol, we may write ∗

⎡ u1 ⎢ ∗ ⎢ u2 T −1 AT = ⎢ ⎢ ⎢ u∗ ⎢⎣ n

⎤ ⎡ λ1 ⎥ ⎢ ⎥ ⎢0 u u u λ , λ ,..., λ = [ ] n n ⎥ 1 1 2 2 ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0 ⎥⎦

0 λ2 0

0 ⎤ ⎥ 0 ⎥ ⎥, ⎥ λn ⎥⎦

the Jordan matrix J being thus a diagonal matrix. The equations (24.1.16) can be separated in the form ξi = λi ξi +

m

∑ pik vk (t ), i

k =1

= 1,2,..., n ,

(24.1.17)

where pik , i = 1,2,..., n , k = 1, 2,..., m , are the elements of the matrix P ; ξi are thus normal co-ordinates. In this case,

e Jt

⎡ eλ1t ⎢ ⎢ 0 ⎢ t t2 2 = I + J + J + ... = ⎢ 0 1! 2! ⎢ ⎢ ⎢ 0 ⎢⎣

0

0

λ2 t

0

0

eλ3t

0

0

e

0 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥, ⎥ ⎥ λn t ⎥ e ⎥⎦


640

and the solutions of the system of Jordan normal (canonical) equations (24.1.17) are written in the form

ξi = eλi t ξi0 +

t λ (t − τ ) m i

∑ pik vk ( τ )dτ , i

∫0 e

k =1

= 1, 2,..., n ,

(24.1.17')

where the initial conditions ξi (0) = ξi0 , i = 1, 2,..., n , have been introduced. If the characteristic equation of the matrix A has multiple roots, then appear Jordan blocks in the matrix of the canonical differential equations; obviously, the calculation becomes more complicated, but can be followed on the same way. Often, the control is realized by a scalar control function v (t ) , the matrix B being reduced to a column vector b = [0, 0,..., 0,1]T with n rows and only one non-zero element, equal to unity, the system (24.1.12) being of the form x = Ax + bv (t ), x ∈

n

.

(24.1.18)

If the matrix A has eigenvalues with a negative real part, then the configuration of equilibrium is asymptotically stable; in this case, by a small perturbation v (t ) , the mechanical system (in case of such a system) tends to a configuration of equilibrium if the perturbation disappears. If the matrix A has also eigenvalues with a non-negative (positive or null) real part, then the mechanical system is instable, moving much away, for very small perturbations, v (t ) , from the position of equilibrium. It is also possible that the matrix A be instable; in this case, the stabilization of the equilibrium is made by control. Replacing the operator d / dt by D , one can write the system (24.1.18) in the form ( A − DI ) x + bv (t ) = 0 ,

(24.1.18')

x = −( A − DI )−1 bv (t ) ;

(24.1.18'')

x i = fi (D)v (t ) ,

(24.1.19)

wherefrom

it results

the functions fi (D), i = 1, 2,..., n , being rational functions of the operator D . We can successively calculate x n = fn (D)v (t ), x n −1 =

fn −1 (D) f (D) v (t ),..., x 1 = 1 v (t ) ; fn (D) f2 (D)

(24.1.19')

one can thus schematize the block schema of the system by circuits with feed-back. By the application method, we choose the control function in the form


641

v (t ) = g T ⋅ x + v (t ) ,

(24.1.20)

where g = [ g1 , g2 ,..., gn ]T is a column vector, while the point corresponds to a scalar product. Replacing in the system (24.1.18), it results

x = Λx + bv (t ), Λ = A + bg T ;

(24.1.20')

in a developed form, the characteristic equation det[ Λ − ρI ] = 0 becomes

ρn + a1 ρn −1 + a2 ρn − 2 + ... + an −1 ρ + an = 0 ,

(24.1.20'')

where the coefficients ai are linear functions of g j , i , j = 1, 2,..., n . Choosing convenient roots (with a negative real part, sufficient great in absolute value), Viète’s relations determine the coefficients ai , i = 1, 2,..., n , hence the column vector g too; the input for the block schema of the system is only the perturbation v (t ) . One can study many interesting problems on this way, e. g., the rolling stability of an airplane or the control stability of the artificial satellites of the Earth.

Fig. 24.6 Motion of an artificial satellite of the Earth

In what concerns the last problem, let us consider an artificial satellite of the Earth, the mass centre O of which moves with the angular velocity ω0 on a circular trajectory situated in the terrestrial equatorial plane (Fig. 24.6). The principal axes of inertia Ox1 and Ox 2 are situated in the equatorial plane, while the principal axis Ox 3 is normal to this plane. Besides the attraction force of the Earth acts also a couple of moment M 3 = 3 ω02 (I 1 − I 2 )θ , where θ is the angle made by the Ox 1 -axis with the straight line O ′O , O ′ being the mass centre of the Earth; this angle specifies the small oscillations of the satellite about the Ox 3 -axis and is given by the theorem of moment of momentum written with respect to this axis θ + p 2 θ = 0, p 2 = 3 ω02

I 2 − I1 , I3

(24.1.21)


642

where I 1 < I 2 < I 3 (condition of stability) are the principal moments of inertia. Hence, the angle has a harmonic oscillatory motion of period T =

2π 2π = ω0 p

I3 ≅ 0.57736T0 3(I 2 − I 1 )

I3 , I 2 − I1

(24.1.21')

where T0 = 2 π / ω0 is the period of motion of the mass centre O on the trajectory. The stability characteristics of the Ox1 -axis can be improved (e.g., to maintain the direction of a connection radio antenna) by introducing a moment, linear function of θ and θ ; one obtains the differential equation θ + p 2 θ = − g1 θ − g2 θ + v (t ) ,

(24.1.22)

v (t ) being a perturbation. With the notations θ = x1 , θ = x 2 , there results the system

of differential equations x1 = x 2 , x 2 = − ( g1 + p 2 ) x1 − g2 x 2 + v (t ) ;

(24.1.22')

the matrix 0 1 ⎤ ⎡ A = ⎢ ⎥ 2 ⎢⎣ −( g1 + p ) − g ⎥⎦

has the characteristic equation −ρ 1 ⎡ ⎤ ⎢ ⎥ = ρ 2 + g 2 ρ + g1 + p 2 = 0 . 2 ⎢⎣ −(g1 + p ) −(g2 + ρ ) ⎥⎦

Viète’s relations lead to

g1 = ρ1 ρ2 − p 2 , g2 = −( ρ1 + ρ2 ) ,

(24.1.22'')

so that we can choose, correspondingly, the vector [ g1 , g2 ]T so as the roots have a negative real part ( g1 + p 2 > 0, g2 > 0 ).

24.1.1.3 Non-autonomous Linear Systems of Differential Equations with Periodic Coefficients. Floquet’s Theory Let be the system of non-autonomous differential equations x = A (t ) x, x ∈

n

,

(24.1.23)


643

where A is a square matrix of order n , the elements of which are continuous and periodical functions in t , of periodic T , so that A (t + T ) = A (t ), t ∈

;

(24.1.23')

we add the initial conditions x(0) = x0 , x0 ∈

n

.

(24.1.23'')

We choose the solution of the equation (24.1.23) in the form x(t ) = Φ (t ) x0 ,

(24.1.24)

where the fundamental matrix Φ (t ) verifies the condition Φ (0) = I , I being the unit matrix; introducing in (24.1.23), we see that Φ (t ) must satisfy the matrix differential equation Φ (t ) = A (t )Φ (t ) ,

(24.1.24')

for any initial conditions. If Φ (t ) is a fundamental matrix of the differential equation (24.1.23), then it is almost evident that Φ (t + T ) is such a matrix too and that Φ (t + T ) = Φ (t )C ,

(24.1.25)

where C is a non-singular square matrix of order n . Floquet showed that the fundamental matrix Φ (t ) can be expressed in the form Φ (t ) = P (t )eBt ,

(24.1.26)

where P(t ) is a periodic square matrix ( P (t + T ) = P (t ) ) of order n , while B is a constant square matrix, of order n too. Making the change of function x = P (t ) y

(24.1.27)

and replacing in (24.1.23), we see that the column vector y = [ y1 , y2 ,..., ym ]T verifies the differential equation y = By , y ∈

n

;

(24.1.27')

thus, starting from a non-autonomous system, we have obtained an autonomous system of differential equations.

644


To study the stability of the solutions, Floquet assumes that Φ (t + T ) = λΦ (t ), t ∈

,

(24.1.28)

λ being a constant scalar; making, successively, t = 0,T ,2T ,..., nT , we get

x(2T ) = Φ (2T ) x0 = λΦ (t ) x0 = λ2 Φ0 (0) = λ2 x0 a.s.o.; by complete induction, we obtain, finally, x(nT ) = λn x0 .

(24.1.28')

If | λ |< 1 , then x(nT ) → 0 for n → ∞ , the position of equilibrium x = 0 being stable; if | λ |> 1 , then x(nT ) → ∞ for n → ∞ , while the position of equilibrium x = 0 is instable. By writing the relation (24.1.28) for t = 0 , we determine the values λ ; we get thus

[ Φ (t ) − λI ] x0 = 0 .

(24.1.29)

This homogeneous system of linear algebraic equations admits non-trivial solutions for x0 if and only if det [ Φ (t ) − λI ] = 0 .

(24.1.29')

The matrix Φ (t ) is called matrix of monodromy and the eigenvalues λi = eρi are characteristic multipliers, the numbers

ρi , i = 1, 2,..., n , being characteristic

exponents. The stability takes place if, in modulus, no one of the characteristic multipliers is supraunitary. A non-autonomous differential equation with periodic coefficients is Mathieu’s equation x + ( γ + 2 δ cos 2t ) x = 0 ,

(24.1.30)

where γ and δ put in evidence the properties of the dynamical system; a mechanical system which leads to such an equation is, e.g., the mathematical pendulum for which the length of the thread is a harmonic function of time ( l = l0 + a cos ωt ). The study of this equation can be made by means of Mathieu’s functions. In the following, we make a study of the stability of the solutions of Mathieu’s equation as functions of the parameters γ and δ . If the real part of the roots of the characteristic equation is negative, then the configuration of equilibrium is stable; while if this part is positive, at least partially, the respective configuration is instable; finally, if the roots are purely imaginary, then the


645

solution is periodical. We search the conditions which must be fulfilled by the parameters γ and δ , so as to have limit solutions between the stable and the instable ones. Representing a periodic solution by a Fourier series

x (t ) =

∞

∑ ( an cos nt + bn sin nt ) ,

n =0

(24.1.31)

replacing in (24.1.30), equating to zero the coefficients of the terms in cos nt and sin nt , n = 1, 2,... , and taking into account simple trigonometric relations as cos(n + 2)t + cos(n − 2)t = 2 cos nt cos 2t , sin(n + 2)t + sin(n − 2)t = 2 sin nt cos 2t ,

we may write γa 0 + δa2 = 0,

( γ − 12 + δ ) a1 + δa 3 = 0, 2 δa 0 + ( γ − 22 ) a2 + δa 4 = 0,

(24.1.32)

................................................ δan − 2 + ( γ − n 2 ) an + δan + 2 = 0,

................................................. ( γ − 12 − δ ) b1 + δb3 = 0,

( γ − 22 ) b2

+ δb4 = 0, ................................................

(24.1.32')

δbn − 2 + ( γ − n 2 ) bn + δbn + 2 = 0,

.................................................

We notice that both homogeneous linear algebraic systems (24.1.32), (24.1.32') in the unknowns ai , bi , i = 0,1, 2,... , are decomposed in two subsystems, as the indices are even or odd. To have non-trivial solutions, the four determinants of the coefficients of each subsystem must vanish γ

δ

0

0

0

…

δ

0

0

…

δ

γ − 42

δ

0

… = 0,

0 …

δ …

γ −6 …

2 δ γ − δ2 Δ1 = 0 0 …

2

δ … … …

(24.1.33)


646 γ − 12 + δ

δ

0

δ

γ − 32

δ

0

δ

0

0

0

…

0

0

…

γ −5

δ

0

… = 0,

0

δ

γ − 72

δ

…

…

…

…

… … …

γ − 12 − δ

δ

0

0

0

…

δ

γ − 32

δ

0

0

…

0

δ

γ −5

δ

0

… = 0,

0

0

δ

γ − 72

δ

…

…

…

…

…

Δ2 =

Δ3 =

Δ4 =

2

2

γ − 22

δ

0

δ

γ − 42

δ

0

δ

0 …

0 …

0

…

0

0

…

γ −6

δ

0

… = 0.

δ

γ − 82 …

δ

…

(24.1.33'')

… …

0

2

(24.1.33')

(24.1.33''')

… … …

Taking a sufficient great number of lines and columns, we obtain, in a ( γ , δ ) -plane, curves which separate the hatched stability regions of those of instability (Fig. 24.7). We notice that, for δ → 0 , the instability takes place for γ = 0,12 ,22 , 32 ,... ; as δ is greater, as a parallel to O γ meets smaller zones of instability. This graphic representation is known as Strutt’s diagram.

Fig. 24.7 Strutt’s diagram for Mathieu’s equation


647

Let us consider now another non-autonomous system of differential equations of the from x = [ A (t ) + B ] x ,

(24.1.34)

where A and B are square matrices of order n , the latter one being constant; as one can notice, if we make B = 0 , then we find again the system (24.1.23). Let us suppose that we have the estimation Φ (t ) ≤ c (t ) +

t

∫t

Φ ( τ )Ψ ( τ )dτ ,

(24.1.35)

0

where c (t ) > 0 is a function of class C 1 , while Φ (t ) > 0, Ψ (t ) > 0 are functions of class C 0 , integrable on the interval [t0 , t0 + a ], a > 0, a = const ; we can state

Lemma 24.1.1 (Gronwall). The inequality t

Ψ ( τ )dτ Φ (t ) ≤ c (t0 ) e ∫t0 +

t

∫t

t

c ′( τ ) e ∫τ Ψ ( τ )dτ dτ

(24.1.35')

0

takes place in the conditions of the estimation (24.1.35). In particular, if c (t ) = δ = const , then the inequality (24.1.35') becomes t

Ψ ( τ )dτ Φ (t ) ≤ δ e ∫t0 .

(24.1.35'')

These inequalities are known as Gronwall’s inequalities. A necessary (but not sufficient too) condition that the solution of the differential equation (24.1.34) tend the solution of the equation x = Bx is

lim A (t ) = 0 .

t →∞

(24.1.36)

Using the Lemma 24.1.1, we can state: Theorem 24.1.2 If Re λk ≤ 0, k = 1, 2,..., n , hence if the real parts of the eigenvalues of the matrix B are non-positive, the purely imaginary eigenvalues being distinct, and if the integral t

∫t

A ( τ ) dτ

(24.1.36')

0

is bounded, then the solution of the matric differential equation (24.1.34) is bounded and the solution x = 0 is stable. Theorem 24.1.3 If Re λk < 0, k = 1, 2,..., n , hence all the eigenvalues of the matrix B are negative and if the condition (24.1.36) holds, then the configuration of equilibrium x = 0 is asymptotically stable.


648

24.1.2 Non-linear Differential Equations and Systems of Non-linear Differential Equations After some general considerations, we present, in what follows, various methods of integration of the non-linear differential equations and of the systems of non-linear differential equations; we will use thus both analytical and numerical methods of calculation.

24.1.2.1 General Considerations The non-linear properties of a material, the existence of geometric non-linearities, the existence of free plays between the elements which form a dynamical system, the existence of some non-linear limit conditions, the existence of a non-linear damping, the existence of elements which contain fluids and many other causes lead – by their mathematical modelling – to non-linear differential equations. Some characteristic phenomena appear in this case. Thus, to periodic inputs – unlike the case of linear systems for which the exits are periodical too, with the same period – the exits can be periodical, having another period, can appear jumps (sudden variations) of amplitude or of frequency, known as catastrophes, as well as some phenomena of sudden loss or gain of stability. Let be a non-linear differential equation of first order written in an implicit form F (t ; x , x ) = 0 .

(24.1.37)

If the conditions imposed by the theory of implicit functions are fulfilled, then we can write x =

dx = f (t ; x ) , dt

where we supposed that the point (t ; x ) belongs to an open subset Ω of

(24.1.37') 2

; to solve

the solution (24.1.37) means to find all its solutions and to study their behaviour. We call solution or integral curve or – shortly – integral of the equation (24.1.37) any continuous and differentiable function x = ϕ (t ) , definite on an open interval I , which satisfies the relation ϕ ′(t ) = f (t ; ϕ (t )), ∀x ∈ I , if – supplementary – the points (t ; ϕ (t )) belong to Ω for any t ∈ I .

The solution of the Cauchy problem associated to the equation (24.1.37') means the determination of the solutions x (t ) for which x (t0 ) = x 0 ,

(24.1.37'')

where (t0 ; x 0 ) is a given point in Ω . In certain convenient hypotheses on f , the Cauchy problem for the equation (24.1.37') admits at least one solution; the uniqueness of the solution is ensured if f satisfies certain supplementary conditions. Let us define, at any point P (t ; x ) of the domain Ω , the angle α by the formula


tan α = f (t ; x ) .

649 (24.1.37''')

The point P (t ; x ) together with the angle α forms a so – called element of contact (or linear element) (Fig. 24.8); the set of all the elements of contact is called field of directions and defines the differential equation (24.1.37'). Hence, a solution or an integral curve of the equation (24.1.37') is a curve which has a tangent at any point of it, with the property that any of its elements of contact belongs to the corresponding field of directions.

Fig. 24.8 Linear element of a non-linear differential equation of first order

For instance, to a geometrical determination of the integral curves of the equation dx = t2 + x 2 , dt

Fig. 24.9 Isoclinal lines for two differential equations

we draw, firstly, the curves for which the inclination is the same, called isoclinal lines. Thus, for x = 0, x = 1/ 2, x = 1 we obtain the centre O and two concentric circles of radii 1/ 2 and 1 as inclinal curves. These curves are drawn in Fig. 24.9a, as the integral curves which pass through the points (0, 0), (0, −1/ 2), ( 2, 0) ; the family of integral curves thus obtained depends on a parameter. In Fig. 24.9b is represented the field of directions corresponding to the equation dx x =2 , dt t

650


formed by the tangents to the parabolas x = Ct 2 , C = const . There are many possibilities to study the existence and uniqueness of the Cauchy problem for the equation (24.1.37'), according to the functional frame in which the calculation is made. It is necessary to introduce the notions of maximal solution and of Lipschitz property to can state the classical theorem of existence and uniqueness. If x = ϕ (t ), t ∈ I , is a solution of the equation (24.1.37), then – obviously – the restriction of ϕ to any subinterval of I is, as well, a solution. This remark allows the introduction of an order relationship on the set of the solutions of (24.1.37); more precisely, if ϕ1 (t ), t ∈ I 1 , and ϕ2 (t ), t ∈ I 2 , are two solutions, then we say that ϕ1 (t ) is “smaller” than ϕ2 (t ) and we write ϕ1 ≺ ϕ2 if I 1 ⊂ I 2 and ϕ1 (t ) = ϕ2 (t ) for any t ∈ I 1 . In fact, ϕ1 ≺ ϕ2 means that ϕ2 is the prolongation of ϕ1 . Any maximal element of the set of solutions is called maximal solution. According to this definition, such a solution can no more be prolonged in Ω ; one can also prove that any solution is “smaller” than a certain maximal solution. We say that the function f (t ; x ) is Lipschitzian with respect to x if one can find a constant K > 0 , called Lipschitz’s constant, such that f (t ; x1 ) − f (t ; x 2 ) < K x1 − x 2 , (t ; x1 ) ∈ Ω, (t ; x 2 ) ∈ Ω .

(24.1.38)

The function f (t ; x ) is called locally Lipschitzian if any point of Ω has a neighbourhood on which f is Lipschitzian. There are large classes of functions with Lipschitz’s property, e. g., the analytic functions and, in general, the functions of bounded derivatives with respect to x are also Lipschitzian. A function f may be Lipschitzian in x without being continuous with respect to (t ; x ) . Indeed, let f (t ; x ) = g (t ) + x ; this function is obviously Lipschitz with respect to x , independently of the continuity of g . Let us also note that a locally Lipschitz function is not necessarily Lipschitz on the whole domain of definition; let us take f (t ; x ) = x 2 , (t ; x ) = Ω = 2 , as counter-example. We can state Theorem 24.1.4 Let f (t ; x ) be defined and continuous on the open set Ω ⊂ 2 and locally Lipschitzian in x . Then there is a unique maximal solution of (24.1.37) passing through any arbitrary point of Ω . In what concerns the Cauchy problem, the (local) existence and the uniqueness of the solutions are ensured by Theorem 24.1.5 (Cauchy, Picard, Lipschitz). If the function f (t ; x ) is continuous with respect to the variables t and x in a rectangle D , centred at (t0 ; x 0 ) , and is Lipschitzian in D , then it exists only one integral curve of the equation (24.1.37) which passes through the point (t0 ; x 0 ) . If f is only continuous in D , then one can ensure only the existence of the solution (the Cauchy–Peano theorem), but the uniqueness may fail.


651

The proof of this theorem is constructive, being based on the method of successive approximations, also called the Picard-Lindelöff method (see Sect. 24.1.2.2). Among the differential equations which can be easily integrated, we mention: the differential equations with separate (or separable) variables, the homogeneous differential equations, the total differential equations, as well as the equations which may be brought to such a form by means of an integrant factor. We mention also some non-linear differential equations of a special form, i.e.: Clairaut’s equation x = tx + ϕ ( x ) ,

(24.1.39)

A( x )x + B (x )t + C (x ) = 0 ,

(24.1.39')

x + P (t )x + Q (t )x 2 = 0

(24.1.39'')

x = P (t )x + Q (t )x 2 + R (t ) ;

(24.1.39''')

Lagrange’s equation

Bernoulli’s equation

and Riccati’s equation

there exist specific methods of integration for all these equations. The general form of a non-linear differential equation of second order is F (t ; x , x , x ) = 0 ;

(24.1.40)

if ∂F / ∂x ≠ 0 on the domain of definition of F and if F is sufficiently regular, then, by the implicit function theorem, we can obtain x explicitly, thus getting the normal (canonic) form

x = f (t ; x , x ), f : I × Dx × Dx →

2

.

(24.1.40')

One can associate to the equation (24.1.40'), in a natural form, initial conditions of the form x (t0 ) = x 0 , x (t0 ) = x 0 , x 0 ∈ I ⊂

,

(24.1.41)

which are conditions of Cauchy type; in this case too, one can prove a theorem of existence and uniqueness, similar to the Theorem 24.1.5. We have seen that, in Cauchy’s problem (24.1.40), (24.1.40'), the values of the unknown function and of its derivative are supposed to be known for a same moment t0 . But there exist problems which need another mathematical model, e. g., those in which one knows the values of the function at two moments t1 and t2 ; the most simple conditions of this type are


652

x (t1 ) = a1 , x (t2 ) = a2 , t1 , t2 ∈ I , a1 , a2 ∈

,

(24.1.42)

which, associated to the equation (24.1.40'), form the bilocal (two-point) problem. We encountered such a situation, e.g., in the case of the fundamental problem of ballistics (see Sect. 7.1.2.1). The first difficulty in talking this problem is to get appropriate hypotheses ensuring the existence and uniqueness of the solution, as in this case we have no more the benefit of such a powerful tool as the Cauchy–Picard theorem. We shall suppose that f ∈ C 0 ([t1 , t2 ]) for any x , x ∈ . Obviously, there are infinitely many integral curves passing through the point (t1 , a ) . But it is possible, even in simple cases, that none of these integral curves does reach the point (t2 , a2 ) . In other words, it is possible that the solution of the two-point problem not even exist. Hereafter, we give some of the most common conditions, each of them ensuring the existence and uniqueness of the solution of the two-point problem (24.1.40), (24.1.42): (i) f (t ; x , x ) is bounded; (ii) | f |< C | x |, C < 3 π 3 /(t2 − t1 )2 , for sufficiently great values of | x | ; (iii) f is Lipschitzian with respect to x , x on any finite interval and f (t ; x , x )/(| x | + | x |) tends to zero, uniformly, on [t1 , t2 ] if | x | + | x |→ ∞ . (iv) f is Lipschitzian with respect to x , x on any finite interval and has the form f (t ; x , x ) ≡ ϕ (t ; x ) + ψ (t ; x , x ) ,

where ψ (t ; x , x )/(| x | + | x |) tends to zero uniformly on | x | + | x |→ ∞ ; (v) f allows continuous partial derivatives with respect to x , x and

[t1 , t2 ]

if

∂f ∂f < α, < β , α + β < 1 , with ∂f / ∂x ≥ 0 . ∂x ∂x

A particular case of interest is that of the two-point problem x = f (t ; x ), x (0) = 0, x (t2 ) = 0 .

One can prove the existence and uniqueness of its solution provided that f ∈ C 0 ([0, t2 ] × ) and that there exist two numbers c0 ≥ 0, c1 > 0 such that x

∫0 f (t ; τ ) dτ ≥ −c1x

2

π ⎞ ⎛ − c0 , t2 ∈ ⎜ 0, . 2c1 ⎟⎠ ⎝

In some particular cases, the solution of the non-linear differential equations of second order can be simplified by reducing their order with a unity. Thus, if the differential equation is of the form


f (t ; x , x ) ,

653 (24.1.43)

hence if it does depend explicitly on the unknown function x , then, by a change of function x = p , one obtains an equation of the form F (t ; p , p ) = 0 .

(24.1.43')

If the equation (24.1.40) does not depend explicitly on t , hence if F (x , x , x ) = 0 ,

(24.1.44)

we make the same change of function x = p , resulting the equation

(

F x ; p,

)

dp = 0. dx

(24.1.44')

As well, if the function F (t ; x , x , x ) is homogeneous of degree m in the variables

x , x , x , i. e. F (t ; sx , sx , sx ) = s m F (t ; x , x , x ) ,

(24.1.45)

then one can make the substitution u = x / x , obtaining a differential equation of the form

F (t ;1, u , u + u 2 ) = 0 .

(24.1.45')

A system of n non-linear differential equations of first order can be written in the implicit form Fi (t ; x1 , x 2 ,..., x n , x1 , x 2 ,..., x n ) = 0, i = 1, 2,..., n ,

(24.1.46)

where Fi are considered definite on the same (2n + 1) -dimensional domain and sufficiently regular. If the hypotheses of the theorem of implicit functions – extended to systems of functions – hold, then we can obtain the canonical form of this system of differential equations x i = fi (t ; x1 , x 2 ,..., x n ), i = 1, 2,..., n .

(24.1.46')

We notice that any differential equation of order n , written in the normal form (explicitly with respect to x ( n ) ) x ( n ) = f ( t , x , x , x ,..., x ( n −1) ) ,

(24.1.47)


654

may be written also in the form of the canonical system of differential equations x1 = x 2 , x 2 = x 3 , ..., x n −1 = x n , x n = f (t ; x1 , x 2 ,..., x n ) ,

(24.1.47')

using the notations x1 = x , x 2 = x , x 3 = x ,..., x n = x ( n −1) . Conversely, one can state that a canonical system (in the normal form) of n differential equations of first order is, in general, reducible to only one differential equation of order n . As in case of a non-linear differential equation, we wish to determine that solution of the system (24.2.46') that satisfies the initial conditions x i (t0 ) = x i 0 , i = 1, 2,..., n ,

(24.1.46'')

of Cauchy type, where the point (t0 ; x10 , x 20 ,..., x n 0 ) belongs to the (n + 1) -dimensional domain on which the system (24.1.46') has sense. We can thus state Theorem 24.1.6 (Cauchy–Picard–Lipschitz). If the functions fi , i = 1, 2,..., n , are continuous with respect to all the variable on the (n + 1) -dimensional interval

D = {(t ; x1 , x 2 ,..., x n ) | t − t0 < α, xi − x i 0 < b , i = 1, 2,..., n } , and satisfy Lipschitz’s conditions fi (t ; X1 , X 2 ,..., Xn ) − fi (t ; X1 , X 2 ,..., X n ) n

< K ∑ X j − X j , i = 1,2,..., n , j =1

for any pair of points (t ; X1 , X 2 ,..., Xn ), (t ; X1 , X 2 ,..., Xn ) in D , where K is Lipschitz’s constant, then the solution of Cauchy’s problem (24.1.46'), (24.1.46''), exists and is unique on the real interval [t0 − T , t0 + T ] , where T = min{a , b / M } , while

⎧ ⎫ M = max ⎨ sup fi (t ; x1 , x 2 ,..., x n ) ⎬ . 1 ≤ i ≤ n ⎩ ( t ,x ,x ,...,x n )∈D ⎭ 1 2 The general solution of the non-linear system of differential equations (24.1.46') depends on n arbitrary constants and can be written in the form x i = ϕi (t ;C 1 ,C 2 ,...,C n ), i = 1, 2,.., n ,

(24.1.46''')

wherefrom we can write ϕi = (t ; x1 , x 2 ,..., x n ) = C i , i = 1,2,.., n .

(24.1.46iv)

These constants are determined by initial conditions in the case of Cauchy’s problem.


655

The Theorem 24.1.6 has been applied for the equations of motion of the discrete mechanical systems (see Sect. 11.1.1.4), for the equation of notion of the rigid solid (see Sect. 14.1.1.8), for Lagrange’s equations (see Sect. 18.2.2.4) and for Hamilton’s canonical equations (see Sect. 19.1.1.4), which are systems of differential equations of first order. In connection with the last two mentioned systems of equations, various methods of integration (first integrals, brackets, multipliers etc.) have been studied. We notice that the limit problem which we wish to solve must be well put in the sense of Hadamard, in other words: (i) it admits a unique solution; (ii) the solution has a continuous dependence on the data of the problem (e.g., initial conditions, bilocal conditions etc.). In some particular cases, the solutions of the differential equations can be obtained in an analytical way and can be expressed by means of elementary and special functions etc. But, frequently, the solutions cannot be obtained in this way and we are led to very intricate calculations. We are obliged, in this case, to use approximate methods of calculation, which can be analytical methods, leading to solutions expressed in an approximate analytical form (e.g., the method of successive approximations, the method of power series expansions, the linear equivalence method etc.) or numerical methods, where the solution is obtained in the form of a sequence of numerical values, e. g., starting from the initial conditions, in case of a Cauchy type problem, approximating thus the integral curves. The numerical methods of calculation can be methods with separated steps (one-step methods) or methods with linked steps (multi-step methods). Let be the equation (24.1.37'), with t ∈ I ⊂ . We assume that the interval I is divided by a net t1 , t2 ,..., tn , of step hi = ti + 1 − ti , i = 0,1,2,..., n − 1 , which can be constant or variable. The solution at the point ti is x i = x (ti ) . In a one-step method, if one knows (ti , x i ) , then one can calculate (ti + 1 , x i + 1 ) ; we mention thus the method of expansion into a Taylor series, Euler’s method and the Runge–Kutta method. In a multi-step method, to determine (ti + 1 , x i + 1 ) it is necessary to know (ti , x i ), (ti −1 , x i −1 ),...,(t1 , x1 ) ; we mention thus the Adams–Moulton method and Milne’s method. An interesting semi-analytical method is the method of spline functions. In both methods of calculation (one-step and multi-step) one can use explicit or implicit algorithms. In the case of a one-step method, e. g., an explicit algorithm is of the form x ip+ 1 = x i + hϕ (ti , x i , h ), i = 0,1,2,..., n − 1 ,

(24.1.48)

while an implicit algorithm is given by x ic+ 1 = x i + h ψ (ti , x i , x ip+1 ), i = 0,1, 2,..., n − 1 ,

(24.1.48')

assuming that the step is constant. A method to improve the results thus obtained is the predictor–corrector method; thus the upper index p corresponds to the predictor value, while the upper index c corresponds to the corrector value.


656

In the case of bilocal problems, especially the “shooting” method, the finite difference method or the method of collocation by cubic alpine functions are used. The considerations made above can be extended to the case of systems of non-linear differential equations. The study of the stability of the systems of non-linear differential equations plays a very important rôle, as it has been seen in the previous considerations too. In this order of ideas, we may state Theorem 24.1.7 (Poincaré, Lyapunov). If x = A (t ) x + Bx + f (t ; x )

(24.1.49)

is a matric differential equation, where A (t ) is a square matrix of order n , continuous with respect to time and which verifies the condition

lim A (t ) = 0 ,

(24.1.49')

t →∞

B is a constant square matrix of order n , the eigenvalues of which have a negative real part, while f (t ; x ) is a column vector, Lipschitzian in x , which verifies the

condition lim f (t ; x )

t →∞

x

(24.1.49'')

= 0,

uniform in t and for which f (t ; 0 ) = 0 , then x = 0 is a stable asymptotic solution. General criteria of stability have been given in Sect. 23.1.

24.1.2.2 Method of Successive Approximations Let us consider the non-linear differential equation of first order (24.1.37'). Let be the rectangle D = {(t ; x ) | | t − t0 |≤ a ,| x − x 0 |≤ b } . For f (t ; x ) continuous and Lipschitzian on

D a recurrent sequence formed by the functions

x j (t ) = x 0 +

∫t f ( τ , x j −1 ) dτ , j t

= 1, 2,..., n ,

(24.1.50)

0

is set up. One can show that the sequence {x j (t ), j ∈

} converges uniformly to the

unique solution of the Cauchy problem (24.1.37'), (24.1.37''), on an interval centred at t0 . More precisely, the inequality

x n (t ) − x (t ) ≤

M K

∞

Kj t − t0 j , t − t0 < h , j ! j =n +1

∑

(24.1.50')

takes place, where K is Lipschitz’s constant corresponding to f , while h is defined by


{ Mb }, M =

h = min a ,

657 sup

( t ,x )∈D

f (t , x ) .

(24.1.50'')

The inequality (24.1.50') allows a sufficiently fine evaluation of the distance between the approximate solution and the exact one. In the case of Cauchy’s problem (24.1.46'), (24.1.46''), let us suppose, as well, that the conditions of existence and uniqueness of the solution hold. One can prove that the sequence of approximate solutions x ij (t ) = x i 0 +

j −1 j −1 j −1 ∫t fi ( τ , x1 , x 2 ,..., xn ) dτ , t

0

x i0 = x i 0 , i = 1,2,..., n , j = 1,2,..., m ...

(24.1.50''')

is uniform convergent on the interval [t0 − h , t0 + h ] , where h is determined by the theorem of existence and uniqueness. The construction of the n recurrent sequences which tend to the unique solution of the Cauchy problem is a direct generalization of what has been said previously for the case n = 1 . More precisely, the functions x ij (t ) , definite by recurrence by the relation (24.1.50'''), satisfy the inequality x ij (t ) − x i (t )
1 , then the equilibrium is instable, the succession of the points being in one sense of the Ox -axis, namely the positive one, moving away from O .

24.1.3.3 Discretization in Time. Poincaré’s Transformation To pass from a continuous in time dynamical process to a discrete one, one considers the vector x(t ) at equal intervals τ of time (eventually, sufficiently small); the continuous function

x(t ) ∈

n

is replaced by the discrete function

x(t0 ) ,

x(t1 ), x(t2 ),... where tk = t0 + k τ , k = 1, 2,... To study such a problem, we use the

stroboscopic method, by a recurrence relation of the form x(tk ) = f ( x(tk −1 ) ) , k = 1, 2,... .

(24.1.103)

As an example, let us consider a forced vibration which, after a transitory period, tends to a harmonic vibration x (t ) = a cos ωt ; in Fig. 24.12a is indicated a discretization, while in Fig. 24.12b one obtains x (t + τ ) as a function of

x (t ) ,

joining by a continuous line the points x (tk −1 ), x (tk ) given by the stroboscopic method. Denoting

Fig. 24.12 The stroboscopic method: forced vibrations tending to harmonic vibrations (a); x (t + τ ) vs x (t ) diagram (b)

y (t ) = a cos ω (t + τ ) = a (cos τ cos ωt − sin τ sin ωt )

and eliminating the time t between x (t ) and y (t ) , we find the equation of an ellipse x 2 (x cos τ − y )2 + = 1; a2 a 2 sin2 τ

hence, the mentioned continuous line tends to this ellipse, which is a stable limit cycle. We notice that, using the stroboscopic method, by passing to the discrete system of recurrence equations, we remained in a space with the same number of dimensions.


679

H. Poincaré has introduced another method of discretization in which the dimension of the system thus obtained is with a unit smaller than of the initial system. Let be a trajectory C in a n -dimensional space of the dynamical system and a ( n − 1 )-dimensional hypersurface Σ g (x 1 , x 2 ,..., x n ) = 0 ,

(24.1.104)

so that, at the points C ∩ Σ , the trajectory be transverse (not orthogonal) to it. In case of the dynamical system x = f ( x ), x ∈

n

,

(24.1.105)

f ( x ) is, obviously, a vector tangent to the trajectory at each point of it; the transversality condition will be written, by means of the scalar product, in the form f (x ) ⋅ gradg ≠ 0 ,

(24.1.105')

the components of the gradient being ∂g / ∂x i , i = 1,2,..., n .

Fig. 24.13 Intersection of an orbit C with a hypersurface Σ: periodic case (a,b); non-periodic case (c)

If the orbit is periodical, then the trajectory C returns to the piercing point p with the hypersurface Σ , crossing it in the same sense (Fig 24.13a); as well, in the case of a periodic orbit, it is possible that the trajectory, which starts from the piercing point p , does cross a second time the surface at the point P ( p ) ≠ p , returning then to p a.s.o. (Fig. 24.13b). Obviously, the surface can be pierced several times too. Such solutions are called subharmonics. If the orbit is not periodical, starting from the point p one can reach the point q , then the point P (q ) a.s.o., never returning at p (Fig. 24.13c). To make some considerations concerning the orbital stability of a periodic solution, we will admit that the motion takes place in a three-dimensional space, the surface Σ being pierced at a single point p . A perturbed motion is no more – in general – periodical, the surface Σ being pierced at the points q1 , q2 ,... ; if these points tend to p , then the periodic motion is orbital stable (asymptotically stable), otherwise being orbital instable. If the points q1 , q2 ,... remain in the neighbourhood of the point p , without tending to it, then the periodic motion is orbital simply stable. Thus, Poincaré


680

has reduced the problem of the orbital stability of a trajectory in 3 to a simpler one, i.e. to the problem of the stability of a position of equilibrium, in fact of a fixed point of an application in 2 .

Fig. 24.14 Poincaré transformation

Let be a neighbourhood U ⊆ Σ of the point p at which the trajectory C pierces the surface Σ , transverse to it. The first returning q of the trajectory in U , is called the Poincaré transformation P : U → Σ , defined by the relation q = P ( p ) = Φτ ( p ), q ∈ U ,

(24.1.106)

where τ = τ (q ) is the time necessary to the point which, starting from p attains Σ ; if q ≡ p , then the motion is periodical, of period τ . We notice that p is a fixed point of the transformation, for which p = P (p) .

(24.1.106')

It is convenient to choose a surface Σ which divides the space in two regions, so that the trajectory does pierce it in a great number of points (Fig. 24.14); the set of these points forms the Poincaré transformation.

Fig. 24.15 Poincaré transformation of a one-dimensional non-autonomous system


681

If the system of n differential equations is non-autonomous, we can choose the change of variable (24.1.2), obtaining an autonomous system of n + 1 differential n × S 1 ; let be the surface equations in the cylindrical space Σ = {(x , θ ) ∈ n × S 1 ; θ = θ0 } . After each period T , the orbit x(t ) pierces Σ ; the resulting transformation P : Σ → Σ ( n → n ) , hence the transformation x(t ) → x(t + T ) is a Poincaré transformation. In Fig. 24.15 we put in evidence a Poincaré transformation of a one-dimensional non-autonomous system.

Fig. 24.16 Graphic analysis of a Poincaré application: attractor (a); repeller (b)

Returning to the fixed points which verify the relation (24.1.98), we have seen in the preceding subsection the conditions which must be verified by f ′( x∗ ) so as to have an attractor or a repeller.

Fig. 24.17 The quadric logistic equation: attractor ( 1 < a < 3 ) (a); bifurcation ( 3 ≤ a < 1 + 6 ) (b); a increasing (c); chaotic regime (d)

682


Taking into account the above considerations on the Poincaré application, we can make a graphic analysis in this direction; thus, Fig. 24.16a corresponds to an attractor and Fig. 24.16b to a repeller. One of the mathematical models often used to study the increasing of the population leads to the quadratic logistic equation x n = ax n −1 (1 − x n −1 ), a = const ,

(24.1.107)

the fixed point being x ∗ = 1 − 1/ a ; we notice that f (x ) = ax (1 − x ) , so that f ′( x ∗ ) = 2 − a . Hence, for 1 < a < 3 the fixed point is an attractor (Fig. 24.17a). For a = 3 it results a bifurcation which leads to a cycle of period 2 (Fig. 24.17b), which is stable for 3 < a < 1 + 6 . If a increases, then will result cycles of periods 4 (Fig. 24.17c), 8,16,..., 2n ,... Finally, if a increases very much, then one obtains a chaotic regime, in which the trajectories are as thus corresponding to an aleatory process (Fig. 24.17d).

24.2 Elements of the Theory of Catastrophes From the oldest times, one of the basic wishes of men has to understand the intimate mechanisms of the motion, of the change and of the permanent development, hence of the evolution of the systems which surround us. After modelling the world on the basis of faith in gods or starting from the observations on the motion of stars, men arrived to scientific models of the evolution, models which have been continuously perfected. The first models have had a deterministic character, e.g., the determinism of Newtonian mechanics, representing thus a great achievement in the knowledge of the evolution of phenomena of the nature. But, besides the phenomena with a continuous character, one must mention the phenomena with a discontinuous or aleatory character too, which – as a matter of fact – are much more numerous and the knowledge and understanding of which is more and more important. In this order of ideas, at the beginning of the eighth decade of twentieth century, there have been set up new theories concerning the evolution of systems, i.e.: the theory of dissipative systems (Prigogine, 1971), the synergetics (Haken, 1971) and the theory of catastrophes (Thom, 1972). The theory of dissipative systems deals with the behaviour of the system in the vicinity of the equilibrium or far from it, putting in evidence the differences of state between the two states (Prigogine, I., 1980). The synergetics studies, especially, the causes which produce the changes of phase, showing why some of the variables lead to such changes, while other ones not (Haken. H., 1982). The mathematical support of these two theories is given by the theory of catastrophes, which shows how the phase transitions take place, of how many kinds they are and of how many parameters they are influenced. Unlike the first two theories, which have a qualitative character, the latter one is – essentially – quantitative. In fact, the three theories are completing themselves. Putting an accent on the qualitative transformations produced by jumps, which – obviously – are discontinuous, René Thom elaborates the mathematical model of evolutive processes (Thom, R., 1972). The corresponding theory is – one can say – a mathematical theory of singularities, representing a true revolution in the differential topology.


683

In this order of ideas, we will present – in what follows – phenomena known as ramifications and then we will pass in review the elementary catastrophes (Gilmore, R., 1981; Gioncu, V., 2005; Poston, T. and Stewart, I., 1978; Zeeman, E. C., 1981).

24.2.1 Ramifications The differential equation which models the behaviour of a differential system can depend on one or on several parameters; for some values of these parameters can appear what are called ramifications, the simplest one being the bifurcation (Chow, S.N. and Hale, J.K., 1982; Hale, J.K. and Koçak, H., 1991; Iooss, G. and Joseph, D.D., 1980; Seydel, R., 1988; Wiggins, S., 1988). After some general considerations, we present various types of ramification: fold, fork, hysteresis, multiply ramification, generic ramification etc. 24.2.1.1 General Considerations Let be the differential equation x = f ( x, λ ), x ∈

n

,

(24.2.1)

where λ is a real parameter λ ∈ ; we can consider also the situation in which f depends on an arbitrary number of parameters λ1 , λ2 ,..., λp . Obviously, the critical points are specified by the equation f ( x, λ ) = 0 .

(24.2.1')

A study of this nature has been made in Sect. 7.2.3.4, after Poincaré, for the positions of equilibrium of a particle which moves according to the law q = f (q , λ ) , where q is a generalized co-ordinate, obtaining a two-dimensional diagram, function of the parameter λ . To can use further such a diagram, assuming the existence of only one parameter λ , one introduces a representative scalar quantity [ x ] for the n -dimensional vector x , which can be: one of the components of the vector, eventually the most important one [ x ] = x k , the greatest component in absolute value [ x ] = max{| x1 |,| x 2 |,...,| x n |} or the Euclidean norm [ x ] =| x | . Obviously, each

of these choices for [ x ] can give only partial informations about the phenomena to study. In the case of differential equations with periodic solutions we can choose the amplitude or the frequency for [ x ] . Let us consider the graphic of the function implicitly definite in the form f ([ x ], λ ) = 0 ,

(24.2.1'')

the smooth curves which are put in evidence being called branches. The particular solution ([ x ], λ0 ) of the equation (24.2.1'') for which the number of solution changes when λ passes through λ0 is called ramification point with respect to the parameter λ . For


684

instance, in the situation in Fig. 24.18) we have six ramification points λi , i = 1, 2,..., 6 , resulting: one solution for λ < λ1 and λ = λ4 ; two solutions for λ1 ≤ λ < λ2 , λ3 < λ < λ4 , λ4 < λ < λ5 , λ6 < λ ; three solutions for λ = λ2 , λ = λ3 , λ = λ5 , λ = λ6 ; and four solutions for λ2 < λ < λ3 , λ5 < λ < λ6 .

Fig. 24.18 Ramification points of the equation (24.2.1'')

The ramifications can be simple ramifications: the fold, the fork (supercritic, subcritic or transcritic), the hysteresis or the multiple ramifications. If the equation (24.2.1') has real roots only for discrete real values of the parameter λ , then the respective points are called isolated critical points. For instance, the differential equation

x = λ2 + x 2 , x ∈

,

(24.2.2)

leads to λ2 + x 2 = 0 ,

(24.2.2')

having only the real root λ = 0 for which x = 0 ; hence, the origin O is an isolated critical point. We say that a ramification is structurally stable if, introducing a small perturbation of the form α + βx , with α and β small parameters (one of which can vanish), that one preserves its structural properties. In the case of the isolated critical point of the equation (24.2.2'), introducing the mentioned perturbation, we may write

( x + α2 )

2

+ λ2 + β −

α2 = 0. 4

(24.2.2'')

If β < α2 / 4 , then the equation (24.2.2'') represents a circle with the centre at the point x = − α / 2, λ = 0 and of radius

α2 / 4 − β ; if

β = α2 / 4 , then the circle is

reduced to its centre, at which is shifted the isolated point, while if β > α2 / 4 the isolated point disappears. Hence, the isolated point is structural instable.


685

If, after the perturbation by the mentioned linear term, the connection between the branches and the point of ramification, with small displacements and eventual deformations, are maintained, then the ramification is called generic ramification. 24.2.1.2 Fold The simple ramification corresponding to the differential equation x = λ − x2 , x ∈

,

(24.2.3)

is called fold (cuspidal point), the critical points being given by the algebraic equation f (x ) = λ − x 2 = 0 .

(24.2.3')

There result the real roots λ ≥ 0, x = ± λ , hence the diagram of ramification (Fig. 24.19a), having two branches which meet at the point O ′ of ramification. We notice that f ′(x ) = −2 x , so that f ′( ± λ ) = ∓2 λ ; the branch x = λ is stable, being represented by a continuous line, while the branch x = − λ is instable and is represented by a broken line.

Fig. 24.19 Fold: diagram of ramification (a); initial (broken line) and perturbed (continuous line) fold (b)

Perturbing the equation (24.2.3') by a linear term, we obtain

(

α2 ⎞ α ⎛ ⎜λ + β + 4 ⎟ − x − 2 ⎝ ⎠

)

2

= 0,

(24.2.3'')

hence the same fold, with the point of ramification at O ′ ( − β − α2 / 4, α / 2 ); the initial fold has been drawn by a broken line, while the perturbed one by a continuous line in Fig. 24.19b. Hence, the fold is a structural stable ramification.


686

We mention that the point of ramification of a fold does not always separate the stable branch of the instable one. 24.2.1.3 Fork To the differential equation x = λx + x 3 , x ∈

,

(24.2.4)

f (x ) = λx + x 3 = 0 ,

(24.2.4')

corresponds the equation

obtaining the branches x = 0 and x = ± − λ for λ < 0 ; the values of the derivative f ′(x ) = λ + 3 x 2 lead to the representation in Fig. 24.20a. We maintain the continuous line for the stable branches and the broken one for the instable branches. The above ramification is called subcritical fork.

Fig. 24.20 Fork: subcritical (a); supercritical (b)

In case of the differential equation x = λx − x 3 , x ∈

,

(24.2.5)

the equation λx − x 3 = 0

(24.2.5')

leads to x = 0 and x = ± λ for λ > 0 ; analogously, we get the ramification diagram in Fig. 24.20b for the supercritical fork. Let be also the differential equation x = λx − x 2 , x ∈

;

(24.2.6)

the critical points are given by the equation λx − x 2 = 0 ,

(24.2.6')


687

obtaining the branches x = 0 and x = λ . This case is called transcritical fork, corresponding the ramification diagram in Fig. 24.21. In this case one can make a study of structural stability too. For instance, in the case of a subcritical fork, adding a small parameter, we can write

Fig. 24.21 Transcritical fork

λx + x 3 + β = 0 .

(24.2.4'')

Representing, further, the non-perturbed ramification by a broken line and the perturbed ramification by a continuous line, we obtain the diagram in Fig. 24.22a for β > 0 and the diagram in Fig. 24.22b for β < 0 . Hence, the subcritical fork ramification is structurally instable, because new connections of branches are obtained.

Fig. 24.22 Subcritical fork ramification: β > 0 (a); β < 0 (b)

24.2.1.4 Hysteresis For the differential equation x = λ − x3, x ∈

,

(24.2.7)

the critical points are given by the algebraic equation λ − x3 = 0 ;

(24.2.7')

this equation has only one real root x = 3 λ , both branches being stable (Fig. 24.23a). The ramification is called hysteresis. Introducing a perturbation αx , α small parameter, one can write λx − x 3 + αx = 0 ;

(24.2.7'')


688

we obtain thus three real roots on the interval ( λ1 , λ2 ) and, with the same graphic notations, the diagrams in Fig. 24.23b. Hence, the hysteresis is a structurally instable ramification too.

Fig. 24.23 Initial (a); perturbed (b) hysteresis

24.2.1.5 Multiple Ramification Another interesting differential equation is x = λ2 x − x 3 , x ∈

;

(24.2.8)

the algebraic equation λ2 x − x 3 = 0

(24.2.8')

leads to the branches x = 0 , x = ± λ . One obtains the branches in Fig. 24.24a, the ramification point O being a point of multiple ramification.

Fig. 24.24 Initial (a); perturbed (b) multiple ramification

Adding a perturbation of constant term β (small parameter), we get the equation λ2 x − x 3 + β = 0 .

(24.2.8'')

With the usual graphic notations, we represent in Fig. 24.24b the non-perturbed and the perturbed ramifications; hence, the multiple ramification is structurally instable.


689

24.2.2 Elementary Catastrophes After some general considerations concerning the critical points and the Morse functions, one introduces, after René Thom, the notion of elementary catastrophe; one presents then, in detail, the potentials of a single variable (the fold, the cusp, the swallow tail and the butterfly) and the potentials in two variables (the elliptic umbilic, the hyperbolic umbilic and the parabolic umbilic), corresponding to the elementary catastrophes. 24.2.2.1 General Considerations Let f (x1 , x 2 ,..., x n ) be a function of n variables. The points (a1 , a2 ,..., an ) at which

df =

∂f ∂f ∂f dx + dx + ... + dx n = 0 ∂x1 1 ∂x 2 2 ∂x n

are called stationary points or critical points; at these points ∂f ∂f ∂f = = ... = = 0. ∂x1 ∂x 2 ∂x n

Let us denote aij =

∂2 f ( a , a ,..., an ) , i , j = 1,2,..., n , ∂x i ∂x j 1 2

(24.2.9)

as well as a11

a12

… a1n

a11 a12 a21 Δ1 = a11 , Δ2 = a Δ ,..., n = … 21 a22

a22 …

… a2 n … … ;

a n 1 an 2

(24.2.9')

… ann

if Δi > 0, i = 1, 2,..., n ,

(24.2.9'')

then the stationary point corresponds to a minimum, while if (−1)i Δi > 0, i = 1, 2,..., n ,

(24.2.9''')

then the point corresponds to a maximum. In the particular case n = 2 , the points ( a , b ) in which ∂f (a , b )/ ∂x = ∂f (a , b )/ ∂y = 0 are stationary points for the function f (x , y ) ; we denote


690

2

Δ =

∂2 f (a , b ) ∂2 f (a , b ) ⎡ ∂2 f (a , b ) ⎤ −⎢ ⎥ . ∂x 2 ∂y 2 ⎣ ∂x ∂y ⎦

(24.2.10)

If Δ < 0 , then will correspond a saddle point, while if Δ > 0 , then we have a local extremum (a minimum if ∂2 f (a , b )/ ∂x 2 > 0 or a maximum if ∂2 f (a , b )/ ∂x 2 < 0 ); if Δ = 0 , then we cannot say anything from this point of view. If the determinant of Hesse (the Hessian) Δn , definite by (24.2.9'), is non-zero, then the differential d2 f =

n

n

∂2 f

∑ ∑ ∂xi ∂x j dxi dx j

i =1 j =1

(24.2.11)

is degenerate at the critical points; in this case, the function f (x1 , x 2 ,..., x n ) is called a Morse function.

Fig. 24.25 Critical points of functions of two variables: maximum (a); minimum (b); saddle (c); monkey saddle (d); channel (e); crossing channels (f)

We consider now some functions of two variables for which the origin is a critical point. Thus, for f (x , y ) = −(x 2 + y 2 ) the origin is a maximum (Fig. 24.25a), for the function f (x , y ) = x 2 + y 2 the origin is a minimum (Fig. 24.25b), while for the function f (x , y ) = x 2 − y 2 the origin is a saddle point (Fig. 24.25c); the Hessian being non-zero, these functions are Morse functions. For the function f (x , y ) = x 3 − 3 xy 2 the origin is a monkey saddle point (Fig. 24.25d), for the function f (x , y ) = x 2 the origin is a channel point (Fig. 24.25e), while for the function f (x , y ) = x 2 y 2 the origin is a crossing channels point (Fig. 24.25f); these functions are not Morse functions. The critical points of the first four functions are isolated critical points.


691

Adding a small linear perturbation, on can show that a Morse function is structurally stable. One can easy see that the function f (x ) = x 2 is structurally stable, the functions f (x ) = x n , n ≥ 3 , being structurally instable; in this last case, one can add terms of a smaller degree to obtain a structurally stable function. We notice that, in a polynomial a 0 x n + a1x n −1 + a2 x n − 2 + ... + an −1x + an , by two substitutions – which represent two translations from a structural point of view – we can make to disappear the term a1x n −1 and the free term an ; thus, after x n we add only terms beginning with a2 x n − 2 and without any constant. One obtains thus – after René Thom – elementary catastrophes in one variable; analogously, one can introduce elementary catastrophes in two variables. The corresponding functions, called potentials, will be of the form V (x ; a , b ,...) or V (x , y ; a , b ,...) , where a , b ,... are parameters. Eliminating x between the equations which specify the singularities ∂V / ∂x = 0 (which we represent in the phase space) and ∂2V / ∂x 2 = 0 or eliminating x and y between ∂V / ∂x = ∂V / ∂y = 0 and

( ∂2V / ∂x 2 ) ( ∂2V / ∂y 2 ) − ( ∂2V / ∂x ∂y )2 = 0 , one obtains a relation of the form f (a , b ,...) = 0 ,

(24.2.12)

which represents a surface in the control space of the parameters a , b ,... One can thus appreciate the number of possible positions of equilibrium of the considered dynamical system, as well as their stability. 24.2.2.2 Elementary Catastrophes in One Variable The most simple elementary catastrophe is the fold, the potential of which is V (x ) =

1 3 x + ax , a = const ; 3

(24.2.13)

Fig. 24.26 Representation of the potential of a fold: phase space (a); control space (b)

the first two derivatives are dV d2V = x 2 + a , 2 = 2x . dx dx

We deduce the equations

(24.2.13')


692

x 2 + a = 0, 2 x = 0 .

(24.2.13'')

The first equation represents a parabola in the phase space, for a < 0 (Fig. 24.26a); by eliminating the variable x , we obtain a = 0 in the control space (Fig. 24.26b). In conclusion, for a > 0 we have no position of equilibrium, for a < 0 we have two positions of equilibrium: one stable x = −a (for which d2V / dx 2 = 2 −a > 0 ) and one instable x = − −a (for which d2V / dx 2 = −2 −a < 0 ), while for a = 0 we have a double root x = 0 , to which corresponds a cuspidal point.

Fig. 24.27 Representation of the potential of a cusp in the control space: the surface S (a); the semi-cubical parabola (b); section by a plane a = const ; a < 0 (c); a > 0 (d)

An elementary catastrophe of fourth degree is the cusp of potential V (x ) =

1 4 1 2 x + ax + bx , a , b = const ; 4 2

(24.2.14)

its derivatives allow to write the equations

x 3 + ax + b = 0, 3 x 2 + a = 0 .

(24.2.14')

The first equation leads, in the phase space, to the surface S in Fig. 24.27a. By eliminating the variable x between the equations (24.2.14'), we obtain the equation 4a 3 + 27b 2 = 0 ,

(24.2.14'')


693

corresponding to a semi-cubical parabola (the cusp) b =±

2 3 3

( −a )3 / 2 , a < 0 ,

(24.2.14''')

symmetric with respect to the Oa -axis (Fig. 24.27b). If the point (a , b ) belongs to the hatched domain, then the first equation (24.2.14') has three real roots, while if it is outside that zone, then it has only one real root; finally, if the point (a , b ) is on the cusp, then the equation has all the roots real (one simple and one double). A section by a plane a = const, a < 0 , parallel the plane to Oxb , pierces the surface S along the curve in Fig.24.27c; hence, the trace of the plane Oab is mn . If we start from A and b is increasing, one travels through the path ABCD , having a jump BC ; starting from D , with b decreasing, the path DEFA with the jump EF is travelled through. Hence, the branch BOE is instable. If a > 0 , then the trace on the plane Oab is pq , the intersection with the surface S being the curve in Fig.24.27d.

Fig. 24.28 Representation of the potential of a swallow tail in the control space

The potential of fifth degree V (x ) =

1 5 1 3 1 2 x + ax + bx + cx , a , b , c = const , 5 3 2

(24.2.15)

corresponds to the elementary catastrophe called swallow tail. The equations a 4 + ax 2 + bx + c = 0, 4 x 3 + 2ax + b = 0

(24.2.15')

put in evidence the position of equilibrium and the critical points, respectively. By eliminating the variable x , one obtains a relation of the form f (a , b , c ) = 0 . which represents a surface S in the control space. To make a study of this surface, we calculate


694

db dc = − ( 12 x 2 + 2a ) , = 12 x 3 + 2ax , dx dx

wherefrom dc / db = − x . One obtains the representation in Fig. 24.28 of the surface S , which has four critical points: the point of maximum O , the cuspidal points A and B and the point of self-intersection C ; the name given to this elementary catastrophe is justified by the form of the cross section for a < 0 . The potential of sixth degree V (x ) =

1 6 1 4 1 3 1 2 x + ax + bx + cx + dx , a , b , c , d = const , 6 4 3 2

(24.2.16)

corresponds to the elementary catastrophe called butterfly. One obtains the equations x 5 + ax 3 + bx 2 + cx + d = 0, 5 x 4 + 3ax 2 + 2bx + c = 0 ;

(24.2.16')

in this case, the phase space is five-dimensional, while the control space is four-dimensional. To can make a study of the hypersurface f (a , b , c , d ) = 0 in the control space, we start from

c (x ) = −5 x 4 − 3ax 2 − 2bx , d (x ) = 4 x 5 + 2ax 3 + bx 2 . For instance, for b = 0 and for the butterfly factor a < 0 , one obtains the representation in Fig. 24.29, which justifies the denomination of butterfly given to this elementary catastrophe. The points A and B have the co-ordinates c = (9 / 20)a 2 and

d = ∓(6 / 25)a 2 −(3 /10)a , respectively, the points D and E have the co-ordinates c =0

and

d = ± (6 / 25)a 2

−(3 / 5)a ,

respectively,

while

the

point

of

self-intersection C has the co-ordinates c = (1/ 4)a and d = 0 . Obviously, one can build up also other curves, corresponding to other values of a and b . 2

Fig. 24.29 Representation of the potential of a butterfly in the control space for b = 0 and a < 0

One can consider also potentials of higher degree, but which are less useful in practical applications.


695

24.2.2.3 Elementary Catastrophes in Two Variables The most important elementary catastrophe in two variables is the umbilic. Thus, the potential V (x , y ) = x 3 − xy 2 + ax + by + c ( x 2 + y 2 ) , a , b , c = const

(24.2.17)

leads to the elliptic umbilic. The phase space and the critical points are given by the equations 3 x 2 − y 2 + a + 2cx = 0, − 2 xy + b + 2cy = 0, 6 x + 2c

−2 y

−2 y

−2 x + 2c

= −4 ( 3 x 2 + y 2 − 2cx − c 2

) = 0.

(24.2.17')

Fig. 24.30 Representation of the potential of an elliptic umbilic in the control space (a); a section c = const (b)

We notice that the last of these equations can be written also in the form

( x − c3 )

2

4 2 c 9

+

y2 = 1, 4 2 c 3

(24.2.18)

the denomination of this elementary catastrophe being thus justified. We can use the parametric representation x =

1 2 c (1 + 2 cos θ ), y = c sin θ ; 3 3

(24.2.18')

the first two equations (24.2.17') lead then to 1 4 2 a = − c 2 (3 + 8 cos θ + 4 cos 2 θ ), b = c (sin 2 θ − 2 sin θ ) . 3 3 3

(24.2.18'')


696

The surface is reduced to the point a = b = 0 for c = 0 . The sections made in the surface f (a , b , c ) = 0 by the plane c = const have three cuspidal points for θ = 0 and θ = ±2 π / 3 ; to these values of the parameter θ correspond the points A( −5a 2 , 0), B (c 2 , −2c 2 ) and C (c 2 , 2c 2 ) . The plane Oab is a plane of symmetry for the surface in the control space, because the co-ordinates of these points remain the same if one replaces c by −c . This surface is drawn in Fig. 24.30a, while in Fig. 24.30b is drawn a section c = const in it. The hyperbolic umbilic has the potential V (x , y ) = x 3 + y 3 − ax − by + cxy , a , b , c = const .

(24.2.19)

Fig. 24.31 Representation of the potential of an hyperbolic umbilic in the control space (a); a section c = const (b)

The equations 3 x 2 − ax + cy = 0, 3y 2 − b + cx = 0, 6x

c

c

6y

= 36 xy − c 2 = 0

(24.2.19')

specify the five-dimensional phase space and the critical points; the control space S is three-dimensional. The last of these equations can be written in thus form y =

1 c2 36 x

(24.2. 20)

too, which justifies the name given to this elementary catastrophe; choosing x as a parameter, we also find a = 3x 2 +

1 c3 1 c4 ,b = + cx . 36 x 432 x 2

(24.2.20')

Proceeding analogously, we make sections by planes c = const . If c = 0 , then a = b = 0 ; hence, the axes Oa and Ob belong to this surface. In Fig. 24.31a one presents the surface S , while in Fig. 24.31b is given a section by a plane c = const .


697

The potential

V (x , y ) = y 4 + x 2 y + ax + by + cx 2 + dy 2 , a , b , c , d = const ,

(24.2.21)

Fig. 24.32 Representation of the potential of a parabolic umbilic in the control space: lips (a); plane sections (b)

corresponds to a parabolic umbilic for which the six-dimensional phase space and the critical points are specified by the equations 2 xy + a + 2cx = 0, 4 y 3 + x 2 + b + 2dy = 0, 2y + 2c

2x

2x

12y + 2d 2

= 4 ( 6y 3 + 6cy 2 + dy − x 2 + cd ) = 0. 2

(24.2.21')

Fig. 24.33 Representation of the potential of a parabolic umbilic in the control space: beak to beak (a); plane sections (b)

Eliminating x and y , we obtain the surface f (a , b , c , d ) = 0 in the four-dimensional control space. Assuming that the coefficients c and d have constant values, one obtains figures characteristic also for other catastrophes, but also other figures, i.e.: lips (Fig. 24.32a), which leads to the curves in Fig. 24.32b, as well as beak to beak (Fig. 24.33a), which leads to the curves in Fig. 24.33b.

24.3 Periodic Solutions. Global Bifurcations In what follows, we treat also other interesting aspects concerning the differential equations: periodic solutions and global bifurcations; obviously, we will take in evidence the stability problems which are put.


698

24.3.1 Periodic Solutions We have defined, in Sect. 24.1.1.1, the periodic solutions, in the general case of non-autonomous differential systems. We shall consider now only autonomous systems of the form x = f ( x ), x ∈

n

,

(24.3.1)

for which the solution x(t ) = Φ (t ) x0 , Φ (t ) = I ,

(24.3.1')

called periodic solution, has the property Φ (t + T ) = Φ (t ), ∀t ∈

.

(24.3.1'')

The motion is repeated after intervals of time nT , n ∈ ; we call period the smallest interval of time T after which the motion is repeated. After some considerations with a general character, one introduces the Hopf bifurcation, which allows to generate some periodic solutions; one presents then the Lindstedt method of integration and one makes a study of stability of the periodic solutions. 24.3.1.1 General Considerations In the phase space, the image of the periodic motion specified by the equation (24.3.1) is a closed curve (orbit or cycle). In the case of an autonomous system of differential equations, to a closed orbit without critical points – in the phase space – corresponds always a periodic solution; if the system of differential equations is a non-autonomous one, then this property does no more hold. The study of the periodic motions plays an important rôle due to the fact that many phenomena in the nature have such properties; even in the case of a chaotic behaviour can appear such periodic motions, in certain conditions. On the other hand, as we have seen in §1, the periodic motions separate the stable configurations of a dynamical system from the instable ones. In particular, let be the two-dimensional system x1 = f1 (x1 , x 2 ), x 2 = f2 (x1 , x 2 ) ,

definite in a simply connected domain D ⊂

2

(24.3.2)

, the functions f1 and f2 being of class

C 1 . If C is the cycle corresponding to a periodic solution, then we can write ⎛ ∂f1 + ∂f2 ∂x 2

∫ ∫D ⎜⎝ ∂x1

⎞dx dx = ⎟ 1 2 ⎠

∫ C ( − f2 dx1 + f1 dx 2 ) = ∫ C ( − x 2 dx1 + x1 dx 2 ) = 0 ;

hence, the expression ∂f1 / ∂x1 + ∂f2 / ∂x 2 cannot maintain a constant sign in the interior of the domain D . We can thus state Bendixon’s criterion: The system of


699

differential equations (24.3.2) admits periodic solutions only if the expression ∂f1 / ∂x1 + ∂f2 / ∂x 2 changes its sign or equates zero. Let us consider, e.g., Van der Pol’s differential equation x − μ (1 − x2 )x + x = 0 ,

(24.3.3)

equivalent to the linear system x 1 = x 2 , x 2 = − x 1 + μ ( 1 − x12 ) x 2 ;

(24.3.3')

we calculate the expression ∂f1 ∂f + 2 = μ (1 − x2 ) , ∂x1 ∂x 2

(24.3.3'')

which vanishes for x = ±1 . Hence, the equation does not admit a periodic solution situated entirely in the half-space x < −1 or in the half-space x > 1 or in the strip −1 < x < 1 ; if it has a periodic solution, then this one will intercept one of the straight lines x = ±1 or both of them. Bendixon’s theorem 23.1.22 (called, sometimes, the Poincaré–Bendixon theorem too) ensures the existence of a closed trajectory, hence of a periodic solution in the phase space. Let us tackle again the system of differential equations (23.1.77), considered in Sect. 23.1.2.7, for which the origin x1 = x 2 = 0 is the only critical point; the linearized system is dx 1 dx = R2 x1 − ωx 2 , 2 = ωx 1 + R2 x 2 , dt dt

(24.3.4)

with the characteristic equation R2 − λ

−ω

ω

R2 − λ

= λ2 − 2 R2λ + R 4 + ω 2 = 0 ,

of roots λ1,2 = R2 ± ωi . The critical point is thus an instable focus. We calculate now the scalar product of the vector r (x1 , x 2 ) and v (x1 , x 2 ) , which make the angle α , in the form r ⋅ v = r v cos α = x1x1 + x 2 x 2 = ( x12 + x 22

)( R2

− x12 − x 22 ) .

Let us consider the circles of radii r1 and r2 and the circular annulus for which r1 < R and

r2 > R ;

along

r1 ⋅ v1 = r1v1 cos α =

r12 (R2

the

circumference

−

> 0 , hence cos α > 0, 0 < α < π / 2 , while along

r12 )

of

radius

r1

one

has


700

the circumference of radius r2 one has r2 ⋅ v2 = r2 v2 cos α = r22 (R2 − r22 ) < 0 , hence cos α < 0, π / 2 < α < π , so that the trajectories enter inside the circular annulus.

Hence, the circular annulus is a bounded domain which does not contain critical points of the system of differential equations, and the trajectories ones entered in this domain, remain there; according to the Poincarè–Bendixon theorem, the system of differential equations (23.1.77) admits at least a closed trajectory entirely situated in the interior of the circular annulus, hence at least a periodic solution of the form x1 = R cos( ωt − ϕ ), x 2 = R sin( ωt − ϕ ) ,

(24.3.4')

as it can be easily shown. Let be Liénard’s differential equation x + f ( x )x + x = 0 ;

(24.3.5)

one can show that this equation admits a periodic solution if: (i) the function f (x ) is Lipschitz continuous in ; (ii) F (x ) =

x

∫0 f ( ξ )dξ

is an odd function;

(iii) there exists a constant α > 0 so that, for 0 < x < α , one has F (x ) < 0 ; (iv) there exists a constant β > 0 so that, for x > β , F (x ) > 0 is monotone increasing and lim F (x ) = ∞ . x →∞

As well, the differential equation x + f (x ) = 0

admits a periodic solution if f (x ) is an odd function of class C 0 , definite in verifies the condition f (x ) > 0 for 0 < x ≤ α, α > 0 .

(24.3.6) , which

Fig. 24.34 The system of differential equations (24.3.7); the diagram f (r , λ ) = 0

Let be the system of differential equations (24.2.1), depending on the real parameter λ ∈ ; a study of the equation has been made in Sect. 24.2.1.1, obtaining a statistical ramification diagram and a certain number of configurations of equilibrium. We notice that, for certain values of the parameter λ , it is possible to have not such a motion; in this case too, one can build up a diagram of ramifications, there appear points of ramification etc.


Let be, e.g., the system of differential equations in polar co-ordinates, in r = f (r , λ ), θ = ω0 , ω0 = const ;

701 2

, (24.3.7)

the diagram f (r , λ ) = 0 which – obviously – corresponds to some circular periodic motions with the angular velocity ω = ω0 , can have the form in Fig. 24.34. In this case, for λ = λ3 , λ = λ4 , λ = λ5 there exists a periodic motion, for λ < λ1 , λ2 < λ < λ3 , λ4 < λ < λ5 there exist two periodic motions, for λ = λ1 , λ = λ4

there exist three periodic motions, while for λ1 < λ < λ2 there exist four periodic motions; for λ3 < λ < λ4 and λ > λ5 there do not exist periodic motions. Sometimes, it is quite difficult to get the number of the periodic solutions. 24.3.1.2 Hopf’s Bifurcation The local bifurcations govern, in general, the mutation of the configurations of equilibrium or of the limit cycles. We mention thus Hopf’s bifurcation, which transforms a configuration of equilibrium into a limit cycle (Marsden, J.E. and McCracken, M., 1950). Let be a damped non-linear free oscillator, the motion of which is described by the differential equation x + kx + ω 2 x + f ( x , x ) = 0 ,

(24.3.8)

where f (x , x ) contains the non-linear terms; the equivalent system of differential equations is of the form

x1 = x 2 , x 2 = − kx 2 − ω 2 x1 − f (x1 , x 2 ) .

(24.3.8')

It is easily seen that to a change of sign of the constant k , from + to –, the eigenvalues of the linearized system pierce the imaginary axis at λ , λ = ± − ω 2 . Hence, the Hopf bifurcation can be associated with the vanishing of the linear damping in the motion of an oscillator. Let us consider also the Van der Pol equation x + α ( x 2 − 1 ) x + ω 2 x = 0, α > 0 ,

(24.3.9)

which represents the motion of a non-linear oscillator after a Hopf bifurcation. If x is small, then we can neglect the term x 2 in the bracket, obtaining an oscillation of increasing amplitude. If x increases, then the term of linear damping x 2 x becomes dominant, but the motion must remain bounded. In the phase portrait, the system leads to a postcritic limit cycle. One obtains the same conclusion using the term f (x ) = x 3 , which corresponds to the equation


702

x − kx + ω 2 x + x 3 = 0 ,

(24.3.10)

which has a fixed point at the origin; there appears a Hopf bifurcation when k passes from negative to positive values. Some phase portraits for various values of k are given in Fig. 24.35. If k < −2 , then the origin is an asymptotically stable node (Fig. 24.35a), obtaining a stable spiral for k = −2 (Fig. 24.35b). For k = 0 (Fig. 24.35c) we obtain the bifurcation, the convergence being very weak, while for k > 0 we get an instable spiral, the flux tending to the limit circle to which it gives rise (Fig. 24.35d). If k > 2 , then the origin is an instable node and the limit circle continues to increase (Fig. 24.35e). Changing the sense of the flux, we pass from a phase portrait of a Hopf supercritical bifurcation to a phase portrait of a Hopf subcritical bifurcation.

Fig. 24.35 Phase portraits for the non-linear differential equation (24.3.10): k < −2 , asymptotically stable node (a); k = −2 , stable spiral (b); k = 0 , bifurcation (c); k > 0 , instable spiral (d); k > 2 , instable node (e)

In connection with the Hopf bifurcation can be put the Neimark bifurcation (second bifurcation of Hopf), which describes the continuous increase of a second mode of oscillation, starting from a limit circle. The possibility of appearance or of disappearance of a periodic solution of the differential equation x = f ( x, λ ), x ∈ D ⊂

n

, α < c, c > 0 ,

(24.3.11)

where f is an analytic real function definite on D × [ −c , c ] , is put in evidence by Theorem 24.3.1 (Hopf). Let be an analytic real vector function g , so that f ( g (λ ), λ ) = 0 ; we express f ( x, λ ) in the form f ( x, λ ) = Lλ x + f ∗ ( x, λ ), x = x − g (λ ) ,

(24.3.12)


703

where Lλ is an n -dimensional square real matrix, which depends only on λ , while f ∗ ( x , λ ) is the non-linear part of

f ( x, λ ) . We assume that there exist only two

complex conjugate eigenvalues α (λ ), α (λ ) of Lλ with the properties Re α (0) = 0 and Re[dα (0)/ dα ] ≠ 0 . In this case, the system of differential equations (24.3.11) has a periodic solution

x(t , ε )

of period T ( ε )

with λ = λ ( ε ) , so that

λ (0) = λ , x(t , 0) = g (0) and x(t , ε ) ≠ g (λ ( ε )) for any parameter ε ≠ 0 sufficiently

small; moreover, λ ( ε ) ,

x(t , ε ) and T ( ε ) are analytical in ε = 0 , while

T (0) = 2 π / | Im α (0) | . These “small” periodic solutions exist only for one of the

three cases: for λ < 0 or for λ > 0 or for λ = 0 . We mention that the theorem does not specify how “small” must be ε . The Hopf bifurcation allows – as well – the generation of periodic solutions. Thus, if a position of equilibrium is stable for λ < λ0 and become instable for λ > λ0 , then can appear a stable position of equilibrium; the orbit is developed starting from a point in the phase plane (for λ = λ0 ) to closed curves, by continuous deformations, as λ is greater than λ = λ0 . This is the soft generation.

Fig. 24.36 Hard generation of a periodic motion

If the generation of the periodic motion takes place by a jump, in the sense that the amplitudes of the periodic motion have – from the very beginning – relative great finite values (not increasing continuously from zero to a finite value, as far as λ is increasing), then we have to do with a hard generation. In Fig. 24.36 is represented the variation of a variable [ x ] , definite in Sect. 24.2.1.1, as function of λ ; one can see that there are possible periodic motions for λ < λ0 too. On the semi-straight line AA′′ , for which λ < λ0 , the solution [ x ] = 0 is stable, while on the semi-straight line AA′ , for which λ > λ0 , this one is instable. One observes that for λ > λ1 a stable periodic motion is possible (the curve BCD or the curve B ′C ′D ′ ), as well as an instable periodic motion (the curve BA or the curve B ′A′ ). We see – as well – that for λ somewhat smaller than λ0 , the system being in a stable equilibrium, a periodic motion with a sufficiently great amplitude can be suddenly


704

generated (the segments of a line AC or AC ′ ). It is possible that, for λ > λ1 , for sufficiently great perturbations, the position of stable equilibrium does disappear, a periodic motion of a sufficient great amplitude taking place, especially if λ approaches λ0 , even if λ < λ0 . If λ corresponds to a point a , for a perturbation less than ab , then the system returns to the position of equilibrium, while if the perturbation is greater than ab , then the system becomes – by a jump – a stable periodic motion and does no more return to the position of equilibrium; hence, the curve BAB ′ , which – practically – is never realized, represents the limit between the domain of attraction of the stable position of equilibrium EA and the domain of attraction of the periodic motion BCD (or B ′C ′D ′ ). 24.3.1.3 Lindstedt’s Method Lindstedt has elaborated a numerical method to determine the periodic solutions of a non-linear differential equation, using – in fact – the perturbations method, but eliminating, at each stage of calculation, the secular terms which appear. Let be thus the differential equation

x + ω 2 x = ε f (x , x ) ,

(24.3.13)

where ε is a small parameter and f (x , x ) is a non-linear function of x and x . In the linear case ( ε = 0 ) the period of the motion is T = 2 π / ω ; in the non-linear case ( ε ≠ 0 ) we have, obviously, T = 2 π / ω + O ( ε ) , hence the pulsation Ω of the motion is of the form Ω = ω0 + εω1 + ε2 ω2 + ... ,

(24.3.14)

with Ω = 2 π /T as unknown. It is convenient to make a change of independent variable τ = Ωt ,

(24.3.15)

which – by replacing t and 2 π / Ω – leads to τ = 2 π , hence to a known period. Calculating x =

dx dτ dx d τ = Ωx ′, x = = Ω 2 x ′′ , dτ dt dτ dt

where we have denoted by x ′ and x ′′ the derivatives of the function x with respect to the variable τ , we may write the differential equation (24.3.13) in the form

Ω 2 x ′′ + ω 2 x = ε f (x , Ωx ′) .

(24.3.13')

Assuming an expression x = x 0 + εx1 + ε2 x 2 + ... ,

(24.3.16)


705

where x i = x i ( τ ), i = 0,1,2,... , we can write f ( x , Ωx ′ ) = f ( x 0 + εx 1 + ε2 x 2 + ..., ( ω0 + εω1 + ε2 ω2 + ... ) ( x 0′ + εx 1′ + ε2 x 2′ + ... ) ) ;

introducing in (24.3.13') and taking into account (24.3.14), (24.3.16), one obtains ω02 x 0′′ + ω 2 x 0 + ε ⎣⎡ ω02 x1′′ + ω 2 x1 − f1 ( x 0 , ω0 , ω1 ) ⎦⎤

+ ε2 ⎡⎣ ω02 x 2′′ + ω 2 x 2 − f2 ( x 0 , x1 , ω0 , ω1 , ω2 ) ⎤⎦ + ε3 ⎡⎣ ω02 x 3′′ + ω 2 x 3 − f3 ( x 0 , x1 , x 2 , ω0 , ω1 , ω2 , ω3 ) ⎤⎦ + ... = 0;

equating to zero the coefficients of various powers of the small parameter, it results the system ω02 x 0′′ + ω 2 x 0 = 0, ω02 x1′′ + ω 2 x1 = f1 ( x 0 , ω0 , ω1 ) , ω02 x 2′′ + ω 2 x 2 = f2 ( x 0 , x1 , ω0 , ω1 , ω2 ) ,

(24.3.17)

ω02 x 3′′ + ω 2 x 3 = f3 ( x 0 , x1 , x 2 , ω0 , ω1 , ω2 , ω3 ) , ......................................................................

determining thus the functions x 0 ( τ ), x1 ( τ ), x 2 ( τ ),... , as well as the pulsations ω0 , ω1 , ω2 ,... Integrating the first equation (24.3.17), one obtains a periodic solution of period 2π ; imposing now the initial condition x 0 (0) = x (0) = x 0 as well as the condition x 0′ (0) = (1/ Ω )x 0 = v0 / Ω , there result x 0 ( τ ) and ω0 . Replacing in the second equation we obtain – by integration – x1 ( τ ) and ω1 , with homogeneous initial conditions x1 (0) = 0, x1′ (0) = 0 (because the non-zero initial conditions have been previously verified); in the expression of x1 ( τ ) appears a secular term, the coefficient of which is equated to zero. The procedure is repeated till the ( n + 1 )th equation, resulting x (t ) = x 0 (Ωt ) + εx1 (Ωt ) + ... + εn x n (Ωt ) ,

(24.3.18)

Ω = ω0 + εω1 + ε2 ω2 + ... + εn ωn .

(24.3.18')

where

For instance, if use Lindstedt’s method for the differential equation

x + x + εx 3 = 0 ,

(24.3.19)

we get ( n = 3 )

(

)

1 3 23 5 2 a ε+ a ε cos Ωt 32 1024 1 3 3 5 2 1 5 2 + a ε+ a ε cos 3Ωt + a ε cos 5Ωt , 32 128 1024 x (t ) = a −

(

)

(24.3.19')


706 where Ω =1+

3 2 21 4 2 a ε− a ε , x (0) = a , x (0) = 0 . 8 256

(24.3.19'')

24.3.1.4 Stability of Periodic Solutions Let be the system of non-linear differential equations x = f ( x; t ), x ∈

n

,

(24.3.20)

and let Φ (t ) be a solution, in general non-periodical, of this system; if y is a small perturbation, let us take x = Φ (t ) + y ,

(24.3.21)

replacing then in (24.3.20). We get

⎡ Df ( x; t ) ⎤ y + ... , Φ (t ) + y = f ( Φ (t ) + y ; t ) = f ( Φ (t ); t ) + ⎢ ⎣ D x ⎥⎦ x = Φ ( t ) wherefrom, taking into account that Φ (t ) is a solution of the system (24.3.20), it results (we neglect the terms in higher powers of y ) ⎡ D f ( x; t ) ⎤ y = ⎢ y = A (t ) y , ⎣ D x ⎥⎦ x = Φ ( t )

(24.3.20')

hence the linearized system of the initial one; formally, we can write t

y (t ) = y 0 e ∫0 A ( τ )dτ .

(24.3.22)

If, for t → ∞ , the exponential in (24.3.22) tends to infinite, then the solution x = Φ (t ) is instable, while, if this exponential tends to zero, then the solution is asymptotically stable; a special study must be make in other cases. In the case in which the system (24.3.20) is autonomous and the solutions are periodical, the matrix A (t ) is periodical too and one can use Floquet’s theory (see Sect. 24.1.1.3). In this order of ideas, one can state: Theorem 24.3.2. Let be the linear system (24.3.20') with a periodic matrix A (t ) , of period T . The sum of the characteristic exponents is given by λ1 + λ2 + ... + λn =

1 T

where trA is the trace of the matrix A .

t

∫0 trA( τ )dτ ,

(24.3.23)


707

Theorem 24.3.3. Let be the autonomous non-linear system (24.3.1) which admits a periodic solution Φ (t ) , of period T , and for which the vector f ( x ) is of class C 1 in a domain D ⊂

n

, which contains the periodic solution too. The linearized system ⎡ Df ( x ) ⎤ y = ⎢ y ⎣ D x ⎥⎦ x = Φ ( t )

(24.3.24)

has one null characteristic exponent; if the other n − 1 characteristic exponents have a negative real part, then the solution Φ (t ) is stable. The stability considered above is an orbital stability, in the Poincaré sense (a solution x(t ) , which starts from a neighbourhood of the system Φ (t ) remains during the motion in the same neighbourhood).

24.3.2 Global Bifurcations The global bifurcations are these qualitative changes which take place in the structure of orbits in an extended domain of the phase space, by a variation of the parameters which intervene in the system of differential equations. After some general considerations, one presents the problem of heteroclinic and homoclinic trajectories, giving also some examples.

24.3.2.1 General Considerations We call phase current the operator Φt which, applied to the vector of state x0 = x(0) ∈ n , allows to obtain the solution x(t ) of the system of non-autonomous equations (24.3.20) by the relation x(t ) = Φt x0 .

(24.3.25)

If the system of differential equations is autonomous (we consider only such systems in what follows), then the phase current verifies the property of translation Φt + τ = Φt Φτ .

(24.3.26)

For the point specified by the vector x = [ x1 , x 2 ,..., x n ]T ∈ n , solution of the system of autonomous equations (24.3.1), we introduce the ω -limit point, defined by

lim Φt ( x ) = ω( x ) ,

t →∞

(24.3.27)

and the α -limit point, defined by

lim Φt ( x ) = α ( x ) .

t →−∞

(24.3.27')

In the case of the system of recurrence equations (24.1.77'), we – analogously – obtain


708 n →∞

lim f ( xn −1 ) = ω ( x ) ,

(24.3.28)

lim f ( xn −1 ) = α ( x ) ,

(24.3.28')

n →−∞

tending to infinite by integers.

Fig. 24.37 Phase plane: critical point O, stable limit cycle C 1 and instable limit cycle C 2

As an example, let be in the phase plane (for the sake of simplicity we are situated in ) the critical point O , the stable limit cycle C 1 and the instable limit cycle C 2 . We denote by D1 the domain interior to the limit cycle C 1 and by D2 the domain contained between the two cycles (Fig. 24.37). One can notice that for all the points situated in the domain D1 \ O , the point O is an α -limit point, while the limit cycle C 1 is an ω -limit set, while the limit cycle C 2 is an α -limit set (as well as for the points external to C 2 ). If 2

x ∈ U → Φτ ( x ) ∈ U , ∀τ ∈

,U ⊂

n

,

(24.3.29)

then U is an invariant set for the phase current Φτ . In case of the system of recurrence equations (24.2.97'), the relation of definition is x ∈ U → f ( xn −1 ) ∈ U , ∀n ∈

.

(24.3.29')

Referring to the preceding example, the critical point O and the limit cycles C 1 and C 2 are invariant sets. An ω -limit point is “non-wandering” for the phase current Φt if, for any neighbourhood U of it, there exists an arbitrary great moment t , so that Φt (U ) ∩ U ≠ 0 .

(24.3.30)

An analogous definition can be given for the recurrence equations (24.1.77'). The fixed points and the points which form the limit cycles are, obviously, non-wandering. We mention that not all the invariant sets are formed by non-wandering


709

points. In general, the wandering points correspond to a transitory behaviour; a behaviour “on long term” (asymptotical) corresponds to the orbits of non-wandering points. A simply connected closed set D ⊂ n , for which Φt ( D ) ⊂ D , ∀t > 0 ,

(24.3.31)

is called trap-region; the mentioned condition is verified if the vector f ( x ) is directed towards the interior of the domain D at any point of its frontier. One can give an analogous definition for the recurrence equations (24.1.77'). The set A =

∩ Φt (D ) ,

(24.3.32)

t ≥0

where D is a trap-region, is called attraction set; hence, a set A ⊂ n is an attraction set if there exists a neighbourhood U of A so that Φt ( x ) ∈ U for t ≥ 0 and Φt ( x ) → A for t → ∞, ∀x ∈ U . The union of all the points which, indifferently of the moment in the past ( t ≤ 0 ) reach the neighbourhood U of the attraction set A at the moment t = 0

∪ Φt (U )

(24.3.33)

t ≤0

is called domain of attraction of the attraction set A . The domains of attraction of the attraction sets do not intersect, being separated by stable variations of the non-attraction sets. Such a domain is called basin of attraction. If an attraction set has the property of invariance Φt ( A ) = A, ∀t ∈

,

(24.3.34)

too, then this one is an attractor. For instance, a stable focus or a stable node are attractors. The structure of an attraction set can be, sometimes, very complicated. Let be thus a one-dimensional phase current, defined by D. Ruelle’s differential equation x + x 4 sin

π = 0, x ∈ x

,

(24.3.35)

which has an infinite but numerable set of fixed points x = 0, x = ±1/ n , n ∈ interval [ −1,1] being an attraction set. Calculating

, the

df ( x ) π π = −4 x 3 sin + πx 2 cos , x x dx

we see that df (1/ n )/ dx = ( π / n 2 ) cos n π ; hence, the derivative is positive for an even n , the set of fixed points being a set of repeller (instable equilibrium), while for

710


n odd the derivative is negative, the fixed points forming an attraction set (stable equilibrium). For the fixed point x = 0 one can nothing state. Couley proposed, in 1978, that this attractor be called quasi-attractor.

24.3.2.2 Heteroclinic and Homoclinic Trajectories A trajectory which unites two distinct critical points O1 and O2 in the phase space is called heteroclinic trajectory; we assume that O1 and O2 are hyperbolic critical points and denote by V s and V i their stable and instable variants, respectively (Fig. 24.38a). For the critical point O1 , the heteroclinic line is a part of its instable invariant variety, O1 being an ω -limit point (for t → ∞ ); these two points do not belong to the heteroclinic line (being critical points, they cannot belong to other trajectories in the phase space).

Fig. 24.38 Heteroclinic (a) and homoclinic (b) trajectories

A trajectory in the phase space which unites a critical point O with itself is called homoclinic trajectory (Fig. 24.38b); for this point, supposed to be hyperbolic, which is – at the same time – ω -limit point and α -limit point and which does not belong to the homoclinic line, the respective trajectory belongs both to the stable and to the instable invariant variety. The heteroclinic and homoclinic lines are both global bifurcations. It has been shown in Sect. 24.1.1.1 that Helmholtz’s oscillator leads to a homoclinic orbit of equation (24.2.11'') for C = 0 . We considered in Sect. 7.2.3.4 the equation of the mathematical pendulum in a non-linear case, which led us to the phase portrait in Fig. 7.22. The orbits corresponding to h = 1 pass through the points ( 0, ±2 ω ) and unite the points ( ± π , 0 ); these latter points are hyperbolic critical points, so that the corresponding orbits seem to be heteroclinic orbits. Taking in view the periodic character of the motion, we can consider as phase space a cylinder the points ( − π, 0 ) and ( π , 0 ) being coincident; hence, the orbit (unique) is – in fact – homoclinic. Let us consider the motion of a heavy rigid solid, fixed at its mass centre (the Euler–Poinsot case of integrability, see Sect. 15.1.2). If we denote ξ1 = I 1 ω1 , ξ2 = I 2 ω2 , ξ3 = I 3 ω3 , where I 1 , I 2 , I 3 are the central principal moments of inertia, while


711

ω1 , ω2 , ω3 are the components of the rotation velocity vector ω with respect to the central principal axes of inertia, then we can write the first integrals (15.1.47) in the form

ξ12 + ξ22 + ξ32 = (I Ω )2 , ξ12 ξ22 ξ32 + + = 1, ( I 1I Ω )2 ( I 2 I Ω )2 ( I 3 I Ω )2

(24.3.36)

where I and Ω are constant quantities of the nature of a moment of inertia and of a rotation velocity, respectively. It result thus a sphere and an ellipsoid at the intersection of which one obtains the searches orbits. Assuming that I 1 ≥ I 2 ≥ I 3 , one obtains the centres ( ± IΩ , 0, 0) , (0, 0, ± IΩ ) , corresponding to the intersection of the sphere with the stable central principal axes of inertia, and the saddle points (0, ± IΩ , 0) , corresponding to the intersections of the sphere with the instable central principal axis of inertia (Fig. 24.39). One of the saddle points is an ω -limit point and the other one is an α -limit point. The centres are surrounded by closed curves, while the saddle points are united by heteroclinic trajectories.

Fig. 24.39 Representation of the motion of a heavy rigid solid fixed at its mass centre: intersection of a sphere with an ellipsoid

The heteroclinic and homoclinic trajectories are – in general – separating domains in the phase space with different qualitative motions; e.g., in the case of the mathematical pendulum previously considered, the homoclinic line h = 1 separates the domain in which the motions are periodical from the domain in which these ones are rotational.

712


If, instead of the system of differential equations x = f ( x ), x ∈ n , one considers the system of differential equations x = f ( x ), + εg ( x ), ε positive small parameter, the heteroclinic and homoclinic lines of the first system are fundamentally changed from a structural point of view, being thus structurally instable.

24.4 Fractals. Chaotic Motions In the previously made studies we have encountered the notion of attractor, in the frame of deterministic motion; this one can be a point, a limit cycle or even a dense n curve on a torus (eventually, in spaces, n > 3 , on tori of higher order). Corresponding to chaotic motions, a characteristic attractor has been put in evidence, i.e. “the strange attractor”; this one has the structure of a fractal, so that a preliminary study of this motion becomes necessary.

24.4.1 Fractals The notion of fractal (in Latin “fractus”, irregular) has been introduced by Benoit B. Mandelbrot, in 1967, and developed by him in the monograph “Les objects fractals: form, hasard et dimensions”, published in 1975. This denomination can be applied both to some fractal mathematical sets and to natural fractals (natural forms, which can be represented by such sets). Mandelbrot put the bases of the fractal geometry, but many fractals appeared – even if not with this denomination – in the work of great mathematicians as Georg Cantor, Giuseppe Peano, David Hilbert, Helge von Koch, Wacław Sierpinski, Gaston Julia and Felix Hausdorff. Mandelbrot showed that such “mathematical monsters” (Peano’s curve, Hilbert’s curve, Koch’s curve, Menger’s sponge, Hausdorff’s dimension etc.) are usually encountered in the nature and do not present simple, classical forms, but forms with a great level of complexity (Mandelbrot, B.B., 1975). In what follows, after some general considerations concerning the notions of distance and dimension, we present some methods to generate fractals, as well as the Julia and Mandelbrot sets.

24.4.1.1 General Considerations From a descriptive point of view, a fractal is – after Mandelbrot – a set which presents the same irregularities at any scale they would be seen; from a geometrical point of view, it is a quantity the parts of which are – in a great measure – identical with the entire set. This mathematical property is called similarity. We give some classical examples, in the frame of this definition (Barnsley, M.F., 1988; Barnsley, M.F. and Demko, S.G., 1989; Falconer, K.J., 1990; Peitgen, H.O., et al., 1992; Smale, S., 1980). Let be a segment of a line of length equal to unity, from which we eliminate the middle third part; further we eliminate the middle third parts from the two segments of a line which have remained a.s.o., obtaining Cantor’s set. One observes that, reducing the scale to 1/ 3,(1/ 3)2 ,...,(1/ 3)n ,... , one obtains always the same image (Fig. 24.40). Let us consider a segment of a line (Fig. 24.41a); we replace the middle third part of it by a broken line, formed by two segments of a line of length equal to 1/ 3 of the length of the initial segment of a line (Fig. 24.41b). We proceed, further, analogously


713

with all the segments of a line a.s.o. (Fig. 24.41c–e), obtaining Koch’s curve, which is continuous, but has not tangent at any point of it (it is of class C 0 ), Reducing the scale to 1/ 3,(1/ 3)2 ,...,(1/ 3)n ,... , one obtains always the same image.

Fig. 24.40 Cantor’s set

Fig. 24.41 Koch’s curve

Let be an equilateral triangle, from which we eliminate the median triangle, after dividing it in four equal triangles; there remain three equilateral triangles, equal to the eliminated one, from which we eliminate – as well – the median triangles a.s.o. One obtains thus the Sierpinski sieve (Fig. 24.42); reducing the scale to 1/ 2,(1/ 2)2 ,...,(1/ 2)n ,... , there results always the same image.

Fig. 24.42 The Sierpinski sieve

Fig. 24.43. The Sierpinski carpet

Analogously, we start from a square with the side equal to unity, which we divide in nine equal squares with the side equal to 1/ 3 ; we eliminate the square at the middle, continue the same operation with the other squares a.s.o., obtaining the Sierpinski carpet (Fig. 24.43).

Fig. 24.44 The Menger sponge

Fig. 24.45 The curve 3 / 2


714

Let us start now from a cube of side equal to unity; dividing each edge in three equal parts, one obtains 27 cubes of 1/ 3 edge, from which we eliminate those on the central lines. There remain 20 cubes with which we proceed analogously a.s.o., obtaining thus the Menger sponge (Fig. 24.44). Starting from a segment of a line of length equal to unity, replacing it by segments of a line of length 1/ 4 and continuing the procedure, one obtains the curve 3 / 2 (Fig. 24.45).

Fig. 24.46 Peano’s curve

Fig. 24.47 Hilbert’s curve

If we divide a segment of a line of length equal to unity in three equal parts, build up a broken line with 9 sides, each one of 1/ 3 length, and if we continue the procedure, then we obtain Peano’s curve, which fills a square (Fig. 24.46). Analogously, we can set up Hilbert’s curve in Fig. 24.47, which – as well – fills a square. We notice that all these curves do not pierce themselves.

24.4.1.2 Distances Let be the points A, B ∈ X ; the distance between these points is a function d ( A, B ) , d : X × X → , with the properties: (i) d ( A, B ) ≥ 0 ; (ii) d ( A, B ) = 0 ⇔ A ≡ B ; (iii) d ( A, B ) ≤ d (A,C ) + d (C , B ), ∀A, B ,C ∈ X , the last property corresponding to the inequality between the sides of a triangle. The application d is called metrics and the corresponding space X is a metric space.


715

The points A( x A , yA ), B (x B , yB ) of a plane being given, the Euclidian distance is definite by the relation

d1 =

( x A − x B )2 + ( yA − yB )2 ,

(24.4.1)

the Manhattan distance (the distance between two persons on two orthogonal streets (one to the other) in the Manhattan district of New York) is given by d2 = x A − x B + y A − y B ,

(24.4.1')

while another definition can be d3 = max { x A − x B , yA − yB } .

(24.4.1'')

Let be the sequence of points x1 , x 2 , x 3 ,... in the space X and another point x ∈ X ; we say that x is the limit of this sequence if lim d ( x n , x ) = 0 ,

(24.4.2)

n →∞

the sequence converging to x . We notice that the limit does not belong always to the metric space. A metric space which contains the limit of any sequence of points of this space is a complete metric space. The set of all the points of a straight line or the set of all the points of a plane are examples of complete metric spaces for any of the distances d1 , d2 , d3 definite above.

Fig. 24.48 Compact set A; ε – collar set Aε

Let be a compact set A ; we call ε -collar set, denoted Aε , the set A together with all the points belonging to the complete metric space X , the distance of which to A is at the most equal to ε (Fig. 24.48), i.e.

{

}

Aε = x ∈ X | d ( x , y ) ≤ ε for some y ∈ A .

(24.4.3)

We call Hausdorff distance the distance between the compact sets A and B defined by

{

}

h (A, B ) = inf ε | A ⊂ Bε and B ⊂ Aε .

(24.4.4)

All the subsets of a complete metric space X for which the Hausdorff distance has been introduced form another complete metric space.


716

24.4.1.3 Dimensions The notion of dimension has been one of the preoccupations of many mathematicians. At a certain moment, it has been considered that the dimension of an object is equal to the number of parameters necessary to specify the position of one of its points. Starting from the dimension zero of a point, Poincaré deduced that a segment of a line has the dimension one, because a point – with the dimension zero – divides a line in two parts, a square has the dimension two (because a line divides it in two parts), a cube has the dimension three (because a square divides it in two parts) etc. Such a dimension has been called topological dimension, considering that it cannot be changed if the geometric figure is transformed by a homeomorphism. G. Peano, in 1890, and D. Hilbert, in 1891, have conceived curves which filled a square, but the transformation g : [0,1] × [0,1] was not a homeomorphism. A rigorous definition of the notion of dimension has been given by F. Hausdorff in 1918. Let be a set of points x ∈ A, A ⊂ n , for which we define the Euclidean distance d (x , y ) =

If U ⊂

n

n

∑ (x i

i =1

− yi )2 , x , y ∈ A .

(24.4.5)

, then we call diameter of U

{

}

diam U = sup d ( x , y ) | x , y ∈ U .

(24.4.5')

We suppose that the subset U is open. A family of open subsets U 1 ,U 2 ,... is an open cover of A if A⊂

∞

∪U i .

i =1

(24.4.6)

Being given two positive numbers s > 0 and ε > 0 , we define ⎧ n ⎫ hε0 (A) = inf ⎨ ∑ (diam U i )s ⎬ , ⎩ i =1 ⎭

(24.4.7)

where U 1 ,U 2 ,... is a cover of the set A with diamU i < ε ; hence h s (A) = lim hεs ( A) , ε→0

(24.4.7')

where the limit can be a finite or an infinite number. We call Hausdorff dimension of the set A the number DH (A) for which, if s < DH (A) , then we have h s (A) = ∞ , while if s > DH (A) , then h s (A) = 0 . For instance, if we cover a segment of a line of length L by discs of diameter 2ε , hence by L / 2 ε discs – according to the definition – we have


717

hεs (A) = (2 ε )s

L = L (2 ε )s −1 ; 2ε

hence, h s (A) = ∞ if s < 1 and h s (A) = 0 if s > 1 , the Hausdorff dimension of the segment of a line being thus DH = 1 . Analogously, one can show that DH = 2 for a rectangle and DH = 3 for a parallelepiped. Let us suppose that we reduce a segment of a line of dimension 1 to the scale s ; we notice that there are necessary N (s ) = 1/ s reduced segments of a line to remake the initial one. Analogously, in case of a square or of a cube, there are necessary N (s ) = (1/ s )2 or N (s ) = (1/ s )3 such segments of a line, respectively. If D is the topological dimension, then, in general, N (s ) =

( s1 )

D

,

(24.4.8)

log N (s ) . 1 log s

(24.4.8')

wherefrom D =

One obtains thus the dimension of self-similarity (capacity) introduced by Kolmogorov in 1958. For instance, in case of the Cantor set, we notice that at the scale s = 1/1 the number of the elements is N (s ) = 1 , at the scale s = 1/ 3 we have N (s ) = 2 elements a.s.o.; at the scale s = (1/ 3)k will be N (s ) = 2k elements. Hence, D =

log N (s ) log 2k log 2 = lim = = 0.6309 . 1 k →∞ log 3k N ( s ) →∞ log 3 log s lim

In case of Koch’s curve, we see easily that at the scale s = (1/ 3)k we have

N (s ) = 4k ; it results thus D =

log N (s ) log 4k log 4 = lim = = 1.2618 . k 1 k →∞ N ( s ) →∞ log 3 log 3 log s lim

Sierpinski’s sieve leads to s = (1/ 2)k and N (s ) = 3k , wherefrom D =

log N (s ) log 3k log 3 = lim = = 1.5850 . 1 k →∞ log 2k N ( s ) →∞ log 2 log s lim


718

For the Sierpinski carpet we have N (s ) = 8k

and s = (1/ 3)k , resulting

D = log 8 / log 3 = 1, 8928 , while for the Menger sponge

s = (1/ 3)k

and

N (s ) = 20 , so that D = log 20 / log 3 = 2.7268 . k

In what concerns the curve 3 / 2 , we notice that s = (1/ 4)k and N (s ) = 8k , wherefrom D = log 8 / log 4 = 3 / 2 , its denomination being thus justified. For the Peano curve s = (1/ 3)k and N (s ) = 9k and we get D = log 9 / log 2 = 2 , while for Hilbert’s curve s = (1/ 2)k and N (s ) = 4k −1 − 1 , wherefrom – by passing to limit – D = log 4 / log 2 = 2 , hence the same dimension. If we wish to measure practically the length of a rectifiable curve, then we inscribe in this curve a polygonal line for which we can measure the lengths of the sides; e.g., in the case of a circle of unit radius, by inscribing an equilateral triangle, one obtains the length of the circumference L = 3l3 = 3 3 = 5.1962 , by inscribing a regular hexagon, it results L = 6l6 = 6 ⋅ 1 = 6.0000 , by inscribing a regular dodecagon, we will have L = 12l12 = 12 2 − 3 = 6( 6 − 2) = 6.2117 a.s.o.; if the number of the sides of the regular polygon increases indefinitely, then the length L tends to 2 π . Lewis Fry Richardson, wishing to calculate the length of the west coast of Great Britain, arrived at unexpected conclusions; indeed, assuming that one cannot make measurement along the coast, but only on the map, e.g., at the scales 1 : 1000000 , and – using a compass – by which to 3 cm correspond 30 km, one obtains a certain length L . Taking then smaller lengths or using maps with smaller scales ( 1 : 500000 or 1 : 100000 a.s.o.), one obtains values which increase indefinitely. Indeed, using maps at smaller scales, one finds new gulfs, new peninsulas etc. To can measure such a length, Richardson proposed an empiric formula of the form L ( ε ) = F ε1 − D ,

(24.4.9)

where ε is the length taken by the compass, while F and D are two characteristic constants; he assumes that F is the searched length, but for D he did not find an interesting significance. By a different choice of ε , there can result sufficient great differences, till 20 − 25% ; e.g., the length of the common frontier between Spain and Portugal is considered to be 987 km in the Spanish encyclopedia and 1214 km in the Portuguese one. Mandelbrot considered that the number of sides N ( ε ) is much more important in the measurement of the above length; from (24.4.9) it results that N ( ε ) = F ε − D , because εN ( ε ) = L ( ε ) . Assuming that one has not to do with a length, but with a body of dimension D , we can multiply by εD , so that F ε − D − ε − D = F ; it results that F is just the searched length (as it has been supposed by Richardson), while D is of the nature of a dimension, called compass dimension or fractal dimension. This dimension coincides with that of self-similarity. One can define also the informational dimension DI (which is always smaller or at least equal to the dimension of capacity), the correlation dimension DK , the punctual dimension DP etc.


719

In very complicated cases, e.g., in the case of a “wild fractal”, as in Fig. 24.49, one can use the method of “box counting”, a systematic measurement which can be applied both in the plane and in the tree-dimensional cases. In the plane case, one covers the structure by a net with square eyes of side s and one counts the squares which contain parts of the structure, obtaining N (s ) ; choosing various sides s , we note in the plane the points of co-ordinates log(1/ s ) and log N (s ) , and then one draws a straight line as near as possible to these points. The inclination of this straight line is called the box-counting dimension Db . This method leads practically to the same fractal dimension for Koch’s curve and for the curve 3 / 2 .

Fig. 24.49 Wild fractal

24.4.1.4 Methods to Generate Fractals Starting from the property of similarity, the fractal is a theoretical notion; but the mathematical fractal is a model for the natural fractal. We mention that the fractals intervene not only in statical phenomena, but also in dynamical ones. Studies of this nature have been made both at micro and at macro scales; e.g., after Mandelbrot, the fractal dimension of the Universe is D = 1.23 . We call fractal a set for which the topological dimension DT is less than its compass (fractal) dimension. DT < D .

(24.4.10)

Corresponding to this definition, the Cantor set ( DT = 0 < D = log 2 / log 3 ), Koch’s curve ( DT = 1 < D = log 3 / log 2 ), Peano’s curve ( DT = 1 < D = 2 ), Sierpinski’s sieve ( DT = 1 < D = log 3 / log 2 ) are fractals. We obtain thus a language which allows to build up a geometry of fractals. As simple elements, we consider the affine transformations of the form u = ax + by + e , v = cx + dy + f

(24.4.11)

in the plane, which transform the point (x , y ) into the point (u , v ) , e.g., a rectangle into a parallelogram.

720


In particular, if a = d = 1, b = c = 0, e , f ≠ 0 , that is if u = x + e, v = y + f ,

(24.4.12)

then it results a translation; if a = d = cos ϕ , b = −c = − sin ϕ , e = f = 0 , hence if u = x cos ϕ − y sin ϕ , v = x sin ϕ + y cos ϕ ,

(24.4.12')

then one obtains a rotation; if 0 < a = d = s < 1, b = c = e = f = 0 , that is if u = sx , v = sy , 0 < s < 1 ,

(24.4.12'')

then one gets a reduction by similitude; if 0 < a ≠ d < 1, b = c = e = f = 0 , hence if u = ax , v = dy , 0 < a ≠ d < 1 ,

(24.4.12''')

then it results an affine reduction; if a = −d = −1, b = c = e = f = 0 , that is if u = −x , v = y ,

then one obtains a mirror symmetry (with respect to the a = d = 1, b ≠ 0, c = e = f = 0 , hence if u = x + b, v = y ,

(24.4.12IV) Oy -axis); if

(24.4.12V)

then one gets a shearing (in the direction of the Ox -axis). The transformations mentioned above are considered being operators: W1 ,W2 ,...,Wn ; assuming that the operators are applied on a set A , we may write W (A) = W1 (A) ∪ W2 (A) ∪ ... ∪ Wn ( A) ,

(24.4.13)

obtaining Hutchinson’s operator. Let us image an iterative application of this operator in the form W (A0 ) = A1 ,W (A1 ) = A2 ,...,W (An ) = An + 1 ,... ;

(24.4.14)

the succession of images thus obtained can tend to A∞ , for which takes place the relation W (A∞ ) = A∞ .

(24.4.14')

In general, the set A∞ is a fractal, which appears as a fixed point of Hutchinson’s operator. One has thus a convenient method to generate fractals. For instance, let us consider operators functions of only one variable


W1 =

x x 2 ,W = + , 3 2 3 3

721 (24.4.15)

respectively, the first one corresponding to a contraction in the ratio 1/ 3 and the second one corresponding to the same contraction, followed by a translation of 2 / 3 . Using Hutchinson’s operator W = W1 ∪ W2 a.s.o., we obtain Cantor’s set (Fig. 24.50).

Fig. 24.50 Cantor’s set, obtained using Hutchinson’s operator

Analogously, one can generate both Koch’s curve and Sierpinski’s sieve. Using Hutchinson’s operator defined by ⎧⎪ u = 0.85 x cos(2 30 ′ ) + 0.85y sin(2 30 ′), W1 ( x , y ) ⇒ ⎨ ⎪⎩ v = −0.85 x sin(2 30 ′) + 0.85 y cos(2 30 ′) + 1.6, ⎪⎧ u = 0.3 x cos(49 ) − 0.34 y sin(49 ), W2 (x , y ) ⇒ ⎨ ⎪⎩ v = 0.3 x sin(49 ) + 0.34 y cos(49 ) + 1.6, ⎧⎪ u = 0.3 x cos(120 ) + 0.37 y sin(50 ), W3 (x , y ) ⇒ ⎨ ⎪⎩ v = 0.3 x sin(120 ) + 0.37 y cos(50 ) + 0.44, ⎧⎪ u = 0, W4 (x , y ) ⇒ ⎨ ⎪⎩ v = 16y ,

(24.4.16)

we can start from a rectangle as initial image, obtaining Barnsley’s fern (Fig. 24.51), hence a sufficiently sophisticated natural fractal. The geometry of fractals can thus be a method of investigation in the science of complexity. B. Mandelbrot showed in 1977 that the Julia set, defined by the recurrence relation

zn + 1 = zn2 + c ,

(24.4.17)

where zn and c are complex numbers, introduced by G. Julia in 1918, is – in fact – a fractal. It has been thus shown that the attraction basins of the three cubic roots of the unity: 1, ( −1 ± 3i)/ 2 are limited by fractals (not by curves, as believed Cayley).

722


A. Douady showed that one can obtain a great number of Julia sets: “Some of them appear as thick clouds, other ones as a dry thorn, other – further – as sparks winding as fireworks; someones seem to be as a hare, while many of them have a tail as a sea horse”.

Fig. 24.51 Barnsley’s fern, obtained using Hutchinson’s operator

The Mandelbrot set generalized the Julia set; the latter one is in the plane of the initial values z 0 , but the Mandelbrot set belongs to the plane of the complex constant c . An image of this set is given in Fig. 24.52; one may notice a great complexity: an interior without any structure and frontier with an infinity of various forms with a fractal structure, of fractal dimension close to 2 . It seems that this fractal is one of the most beautiful and complex objects of modern mathematics, leading to numerous researches.

Fig. 24.52 An image of the Mandelbrot set


723

24.4.2 Chaotic Motions After some introductory considerations and some elements of the theory of signals, Lyapunov’s exponents and Mel’nikov’s method are presented. A special attention is given to the strange attractors and to the evolution towards chaos. As examples, one makes a study of Duffing’s and of Van der Pol’s equation. As well, the mappings of Feigenbaum and of Hénon are put in evidence.

24.4.2.1 Introductory Considerations It is very difficult to show what is regular and normal, ordered or chaotic. The notion of “chaos” appeared long time ago by Anaxagoras (Vth century A.C.), but it became various forms, various modes to be understood along the time. If e.g., the sequence of natural numbers 1, 2, 3,... or the rational ones (e.g., 2 / 3 = 0.666... ) present a certain regularity, then the irrational numbers (e.g., 2 = 1.4142135... ) or the transcendent numbers (e.g., π = 3.1415926... , e = 2.7182818... ) have not any regularity; the Champernowne number C = 0.123456789101112... , formed by the succession of the natural numbers put after the point, hence after a certain rule, is a transcendent number too. Even if the chaotic motions have been discovered by Poincaré, a profound knowledge in this direction has been obtained together with the apparition of the electronic computers. In the last 50 years, one can mention the researches made by the meteorologist Edward Lorenz in 1963, taken again by David Ruelle and Floris Takens in 1971, who introduced the notion of strange attractor (after other researches, chaotic attractor). Mitchell Feigenbaum showed that there exist chaotic phenomena interrupted by periodical windows. Benoit Mandelbrot has introduced, as we have seen, the notion of fractal, while Stephen Smale discovered, studying dynamical systems, a mapping which, defined on a square, transforms this one, by stretching in a direction, by contraction in the transverse direction and by folding – in a horseshoe, called Smale’s horseshoe (Smale, S., 1980).

Fig. 24.53 Portrait of the motion in case of a deterministic chaos

We mention also that, in the study of motions depending on parameters, for certain values of them, there can appear the phenomenon of chaos, which is – in this case – a deterministic chaos. To study chaotic phenomena, one must put in evidence a significant variable Z (t ) ; supposing that one can obtain the values of this variable at discrete intervals of time,

724


denoting Z (k τ ) = Zk , k = 0,1,2,... If the dynamical system is deterministic, then we can determine Z k = ϕ (Zk −1 , Zk − 2 ,..., Zk − r ) , where we can have r = 1, 2,..., k ; in the most simple case r = 1 . Thus, a dynamical system can lead to a strange attractor. If Z = Z (t ) corresponds to only one significant variable, then we can represent the motion at a moment t by a point (Z , Z ) in the phase plane, obtaining a portrait of the motion for various values of t ; thus, we obtain simultaneously both the position and the velocity of a representative point. In the case of a periodic motion, this portrait is a closed curve, while in the case of a regular motion it is a curve which tends to a stable fixed point or to a limit cycle, In case of a chaotic motion, one obtains a portrait with a complex aspect, e.g., as that in Fig. 24.53. Let be the points (x (t ), x (t )) in the phase plane for t = 0, τ ,2 τ ,... , the time interval τ being conveniently chosen, e. g., the period of a perturbing force which may intervene in the system of differential equations of a dynamical system. The set of these points constitutes the stroboscopic image of the motion of the dynamical system, hence a Poincaré mapping; thus, a three-dimensional study is reduced to a two-dimensional one.

Fig. 24.54 A Poincaré section corresponding to one of Ueda’s strange attractors

One obtains thus a Poincaré section. A periodic motion is thus reduced to a point, subharmonic periodic motions to a finite number of points and a motion which tends to a limit cycle to an infinite number of points which have as limit point the stroboscopic image of the motion on this cycle. In the case of a chaotic motion, the Poincaré section has a much more complex aspect, e. g., that in Fig. 24.54, corresponding to one of Ueda’s strange attractors. Let us try to study a deterministic chaotic system by means of a statistical system. If the system is formed of N elements, one of them being produced with a probability pi , i = 1, 2,..., N , then the entropy of the system is defined by the relation


725

N

S = − ∑ pi log pi .

(24.4.18)

i =1

This quantity can be considered as a measure of “the quantity of information” or, for the dynamical system, as a measure of “the quantity of disorder”. If the system is of equal probability, then we have p = 1/ N , so that S = log N , corresponding the maximal entropy; if the probability is equal to the unity, then it vanishes for all the other events (indeed, the sum of all the probabilities is equal to one), so that S = 0 , attaining its minimal value.

24.4.2.2 Lyapunov’s Exponents. Mel’nikov’s Method The divergence of the trajectories which start from very near initial positions leads, usually, to chaotic motions. Let be the equation x = f ( x ), x ∈ , x = x (t ; x 0 ) , with the initial condition x (0; x 0 ) = x 0 . If Δt is a sufficiently small interval of time, then let us denote x k = x (k Δt ; x 0 ), k = 1, 2,..., n . A neighbouring motion starts from the point x 0 + δ , δ sufficiently small; let us denote δ1 = x ( Δt , x 0 + δ ) − x ( Δt , x 0 ) .

(24.4.19)

If we express the relation between δ and δ1 in the form

δ1 = δ eL1 ,

(24.4.19')

then the exponent L1 can be chosen as a measure of the divergence between the two trajectories. Starting from the initial positions x1 and x1 + δ , we find, analogously,

δ2 = x ( Δt , x1 + δ ) − x ( Δt , x1 ) = δ eL2

(24.4.20)

a.s.o.; in general, δn = x ( Δt , x n −1 + δ ) − x ( Δt , x n −1 ) = δ eLn ,

(24.4.21)

wherefrom Ln = ln

δn . δ

(24.4.21')

Taking the mean of the exponents thus introduced, till N , we obtain L =

making δ → 0 , we get

1 N

N

∑ Ln

n=1

=

1 N

N

∑ ln

n=1

δn ; δ

(24.4.22)


726 L =

1 N

N

∑ ln

n= 0

dx ( Δt , ξ ) , dξ ξ = xn

(24.4.23)

a useful formula if Δt is sufficiently small and if the order N of iteration is sufficiently great. The number L is a Lyapunov exponent. If we have to do with a system of n > 1 differential equations, then one calculates the Lyapunov exponents L1 , L2 ,..., Ln , the most important of them being the greatest one; if this exponent is positive, then the motion is chaotic. If the maximal Lyapunov exponent is negative, then appear the so-called “periodic windows”. One can thus see that the Lyapunov exponents play an important rôle in emphasizing the chaotic character of a motion. It is known that the heteroclinic and homoclinic trajectories have an instable character. A homoclinic trajectory is, at the same time, a stable and an instable variety for the same critical point; as a consequence of a perturbation, these varieties do not coincide any more, being distinct, These varieties can pierce each other, and if the intersection is a transverse one – very complicated trajectories are generated, the motion becoming chaotic. In connection with these problems one can mention the Lambda theorem given by Palis in 1969, the theorems of Smale and Birkhoff, as well as Smale’s horseshoe; V.K. Mel’nikov made studies in this direction too. Let be a time periodic system of the form x = f ( x ) + εg ( x; t ), x ∈ U ⊂

2

,

(24.4.24)

where f ( x ) is a Hamiltonian vector field f1 ( x1 , x 2 ) =

∂H ∂H , f2 (x1 , x 2 ) = − , ∂x 2 ∂x1

while ε is a small parameter. We suppose that this system of two differential equations admits a homoclinic orbit, for ε = 0 , at a saddle point P0 and that, for ε ≠ 0 , it has a point Pε with two distinct varieties: a stable variety and an instable one. If q εi (t , t0 ) and q εs (t , t0 ) are the positions of two points situated one the instable or on the stable variety of the critical point Pε at the moment t = t0 , respectively, then – by an expansion into a power series – we can write q εs (t , t0 ) = q 0s (t , t0 ) + εq1s (t , t0 ) + O ( ε2 ), t ∈ ( −∞, t0 ], q εi (t , t0 ) = q 0i (t , t0 ) + εq1i (t , t0 ) + O ( ε2 ), t ∈ [ t0 , ∞ ).

(24.4.25)

We notice that q 0s (t , t0 ) = q 0i (t , t0 ) = q 0 (t , t0 ) corresponds to the same point on the homoclinic trajectory. Defining Mel’nikov’s function in the form M (t0 ) =

∞

∫ −∞ f ( q

0

(t , t0 ) ) ∧ g ( q 0 (t , t0 ), t ) dt ,

(24.4.26)


727

we can state Theorem 24.4.1 (Mel’nikov). If Mel’nikov’s function has simply zeros and is independent on ε , then – for ε > 0 , sufficiently small – the instable variety and the stable one have a transverse intersection. If M (t0 ) does not equate to zero, then these varieties do not pierce one the other. The possibility of apparition of chaotic motions in case of Hamiltonian systems definite on a domain in 2 is thus put in evidence.

24.4.2.3 Strange Attractors. Smale’s Horseshoe. Routes to Chaos We have met, in the first paragraph of this chapter, the notion of simple attractor in the form of stable fixed points, stable limit cycles or in the form of tori of various orders. The strange attractors (or chaotic attractors) are characterized by a fractal structure or by a fractal structure of the frontier of the attractor’s basin and have a great sensibility to the initial conditions; thus, it is very difficult to foresee a chaotic motion, especially if the fractal dimension of the attractor is great. Using a conjecture stated by James Kaplan and James Zorke in 1978, one can foresee the dimension of a strange attractor, in some cases, if one knows Lyapunov’s exponents (Hénon, M. and Pomeau, Y., 1976; Ruelle, D., 1989).

Fig. 24.55 The graphic of γ (k ) vs k; Lyapunov’s dimension DL

One starts from the exponents L1 ≥ L2 ≥ ... ≥ Ln defining the function γ (k ) = L1 + L2 + ... + Lk , k ≤ n , γ (0) = 0 ;

(24.4.27)

we draw the graphic of this function, assuming that between two successive natural numbers ( k and k + 1 ) one has a straight line (Fig. 24.55). Lyapunov’s dimension of the strange attractor is given by DL = m +

1 Lm + 1

m

∑ Lj , j =1

(24.4.28)

where m is the greatest integer with γ (m ) ≥ 0 ; if L1 < 0 or DL = 0 , then the attractor is not a strange one.


728

An important rôle in understanding the fractal structure of a strange attractor is played by Smale’s horseshoe, introduced by the application

Fig. 24.56 Smale’s horseshoe: application f

f :D →

2

, D = {( x , y ) ∈

2

0 ≤ x , y ≤ 1} ;

(24.4.29)

Fig. 24.57 Smale’s horseshoe: application f −1

the square has a contraction of λ < 1/ 2 in the direction of the Ox -axis, a stretching of μ > 2 in the direction of the Oy -axis and then a folding (Fig. 24.56). The inverse mapping f −1 acts on the domain D by a stretching of 1/ λ along the Ox -axis, a contraction of 1/ μ along the Oy -axis, followed by a folding (Fig. 24.57). The application f (D ) ∩ D leads to two vertical rectangles (Fig. 24.58a) while the application f −1 (D ) ∩ D leads to two horizontal rectangle (Fig. 24.58b).

Fig. 24.58 Smale’s horseshoe: two vertical (a) and horizontal (b) rectangles

Applying several times the mapping f of Smale, one can define the mappings f , f 3 ,..., f n ; analogously, starting from the inverse mapping f −1 , we obtain the mappings f −2 , f −3 ,..., f − n . Applying the mapping f to the set 2


A=

∞

∩

729

f n (D ) ,

(24.4.30)

n = −∞

we see that f (A) = A , obtaining thus a strange attractor; there have thus been put in evidence the character of self-similarity and that of fractal of “two-dimensional Cantor set” type. The Hausdorff dimension of this attractor is equal to the double of the dimension of a Cantor set, hence DH = 2 log 2 / log 3 = 1.2618 . The operations corresponding to Smale’s operator f can be compared to those made by a pastry cook when he is preparing a paste: a succession of contractions and stretchings in two orthogonal directions, followed by a folding of it. In general, we can consider a system of differential equations of the form x = f ( x, a ), x ∈

n

,a ∈

p

,

(24.4.31)

or the system of recurrence equations xm = f ( xm −1 , a ), xm ∈

n

,a ∈

p

.

(24.4.31')

The column vector parameter a = [a1 , a2 ,..., a p ]T plays an important rôle; its components can lead to a regular or to a chaotic behaviour of the system of equations. Let be a system which depends on only one parameter a . If the solutions of the system have a regular (stationary, periodic or pseudoperiodic solutions) behaviour for a < a 0 and if the system begins to have a chaotic behaviour for a > a 0 , then one has an evolution towards chaos for a = a 0 ; but, for a > a 0 , there can exist periodic windows, usually narrow, for which the system has once more a regular behaviour. One can identify three principal routes to chaos: (i) by three bifurcations (the Ruelle-Takens theory); (ii) by doubling the period (Feigenbaum’s theory); (iii) by intermittency (Devaney, R., 1986; Gutzwiller, M.C., 1990; Kapitaniak, T., 1990, 1991, 1998; Schuster, H.G., 1984). Thus, in the first scenario, starting from a stationary state, a first bifurcation leads the system to a periodic state, when appears only one pulsation ω1 ; a second bifurcation leads to a pseudoperiodic state, when intervenes a second pulsation ω2 , so that ω1 / ω2 ≠ m / n , m , n ∈ . Finally, small perturbations of the quasi-periodic trajectories on a torus (the third bifurcation) lead to a strange (chaotic) attractor. Thus, from a laminar flow one is led to a turbulent flow (structurally stable), an infinity of bifurcations – as believed L. Landau – being not necessary. The experiment brought a confirmation of the theoretical results. Feigenbaum’s scenario will be presented in Sect. 24.4.2.6. The route to chaos by intermittency is obtained by a scenario imagined by Yves Pomeau and Paul Manneville; thus, there can appear “abnormal fluctuations” if a control parameter leaves a value called intermittency threshold. If the displacement is small, then the regular oscillations can be interrupted at aleatory intervals of time by so-called “attacks of turbulence”.


730

24.4.2.4 The Strange Attractors of Lorenz and of Rössler Trying to model much better the meteorological phenomena, E.N. Lorenz dealt with the study of the Navier-Stokes equations, written in a simplified form x = − σ (x − y ), y = ( R − z )x − y , z = − Bz + xy ,

(24.4.32)

where σ is Prandtl’s number, R = Ra / Rc ( Ra – Rayleigh’s number, Rc – its critical value) and B is a parameter. We remark that the function x (t ) is in direct proportion with the intensity of the motion of convection, while y (t ) is in direct proportion with the difference of temperature between an ascendent current and a descendent one. We mention that there exists a symmetry with respect to the Oz -axis, hence to the solution ( x (t ), y (t ), z (t ) ) corresponds the solution ( − x (t ), − y (t ), − z (t ) ) too. As well, the solution x = y = 0, z = z 0 c− Bt , z 0 ≠ 0 , of the system tends to the origin (0, 0, 0) for t → ∞ . The equations − σ (x − y ) = 0,(R − z )x − y = 0, − Bz + xy = 0

(24.4.33)

give the solutions of equilibrium; there exists only one solution x = y = z = 0 for R ≤ 1 , while for R > 1 there exist three solutions, i. e. x = y = z = 0, x = y = ± B ( R − 1), z = R − 1 .

(24.4.33')

One can show that the origin (0, 0, 0) is a stable position of equilibrium for 0 < R < 1 and an instable one for R > 1 ; the other solutions (24.3.33') are stable positions only for 1 < R < RH , where RH =

σ ( σ + B + 3) , σ > B + 1. σ − (B + 1)

(24.4.34)

As well, one can show that all the solutions of the system (24.4.12) are bounded. One sees that there do not exist stable limit cycles, which could lead to periodic motions, hence to regular motions. Two of the eigenvalues of the Jacobi matrix of the system (24.4.32) are purely imaginary for R = RH , being thus fulfilled the conditions imposed by a Hopf bifurcation, which is subcritical; indeed, for R < RH there exist two instable periodic solutions around the points corresponding to the stable positions of equilibrium, while for R > RH these solutions disappear. Lorenz adopted the values σ = 10 and B = 8 / 3 for the parameters of the equation, being thus led to RH = 24.74 ; as well, he took R = 28 too. In this case, the set of solutions contains very complicated orbits, corresponding to a strange attractor


731

(Lorenz’s strange attractor). Simulations on computer put in evidence the presence of two strata, in which the trajectories have the form of spirals which go away from their centres; if this distance attains a certain limit, then the solution is ejected from a stratum and is attracted by the other one, where it describes another spiral a.s.o. (Fig. 24.59).

Fig. 24.59 Lorenz’s strange attractor

The sensibility of the solution to initial conditions can be, as well, put in evidence. For a deviation ε = 1/100000 , an interval of time Δt = 1/ 400 and 100000 steps of the computer one obtains the maximal Lyapunov exponent L1 = 0.93572 (practically L1 = 0.9 ); from the study of the volume variation, we get L2 = 0 , L3 = −14.57 . The presence of the positive exponent L1 puts in evidence the fact the

motion is chaotic; as well, the dynamical system is dissipative, because the sum of Lyapunov’s exponents is negative. The dynamical system being dissipative, we can state that the topological dimension of the strange attractor vanishes. Using the Kaplan–Yorke conjecture, we obtain Lyapunov’s dimension of the strange attractor DL = 2.062 . Otto Rössler considered, in 1976, a system of differential equations of the form x = −(y + z ), y = x + ay , z = b + (x − c )z , a , b , c = const ,

(24.4.35)

which constitutes – perhaps – the most simple model of building up of chaos; we notice that, in this system, only one of the equations – the third – is non-linear. Let us suppose, firstly, that z is very small and can be neglected in the first equation; we may – as well – take not in consideration the third equation. The system becomes x = − y , y = x + ay ;

(24.4.36)

eliminating the co-ordinate y , we can write the equation x − ax + x = 0 .

(24.4.36')

732


We obtain thus an oscillator with negative damping for a > 0 , the origin being an instable focus for a ∈ (0, 2) . We can thus imagine that the trajectories of the complete system of equations will present spirals which go away from the origin in the neighbourhood of the Oxy -plane; the constant c can thus contribute to the braking or accenting of this tendency of increasing.

Fig. 24.60 The Rőssler strange attractor: case of a periodic motion (a); chaotic motion (b)

It is possible that, for certain values of the constants a , b , c the motion be even periodical (Fig. 24.60a). But, in general, the motion is chaotic, corresponding to the Rössler strange attractor; for a = 0.2, b = 0, c = 5.7 one obtains the image in Fig. 24.60b.

24.4.2.5 Duffing’s and Van der Pol’s Equations In the study of the “hardening” effect of a spring, G. Duffing considered, in 1918, a non-linear oscillator with a cubic rigidity, being led to the differential equation (Duffing’s equation)

x + δx − βx + αx 3 = γ cos ωt .

(24.4.37)

The homogeneous equation (without perturbing force) x + δx − βx + αx 3 = 0

(24.4.38)

is equivalent to the system of differential equations of first order

x = y , y = βx − αx 3 − δy ;

(24.4.38')

the critical points are given by the system of algebraic equations y = 0, βx − αx 3 − δy = 0 .

(24.4.38'')


733

If β < 0 , then there exists only the position of equilibrium x = y = 0 . The eigenvalues of the Jacobi matrix are given by the equation λ2 + δλ − β = 0 ; the position of equilibrium is stable (we have δ > 0 ), i.e.: node if δ 2 > −4 β , focus if δ 2 < −4 β and centre if δ = 0 .

Fig. 24.61 Duffing’s equation. Families of curves with centres ( δ = 0 ): for β < 0 (a); for β > 0 (b)

If β > 0 , then we have three positions of equilibrium, i.e.: x = y = 0 ; x = ± β / α , y = 0 . For the first critical point we obtain the same Jacobi matrix, but

this point is instable (a saddle); for the other two points, the eigenvalues of the Jacobi matrix are given by the equation λ2 + δλ + 2 β = 0 , finding that these ones are stable (foci if δ 2 < 8 β , nodes if δ 2 ≥ 8 β and centres if δ = 0 ) .

Fig. 24.62 Duffing’s equation. Families of curves with foci ( δ ≠ 0 ): for β < 0 (a); for β > 0 (b)

Let us consider first of all the case δ = 0 ; the system (24.4.38') becomes x = y , y = βx − αx 3 .

(24.4.39)

By combining the two equations, we get yy − βxx + αx 3 x = 0 , wherefrom, by integration,

734


2 ( y 2 − βx 2 ) + αx 4 = C , C = const ;

(24.4.39')

one obtains thus a family of curves which depends on the parameter C , represented in Fig. 24.61a for β < 0 and in Fig. 24.61b for β > 0 . If δ ≠ 0 , then centres in Fig. 24.61 become stable foci both for β < 0 (Fig. 24.62a) and for β > 0 (Fig. 24.62b).

Fig. 24.63 Duffing’s equation. Passing from the neighbourhood of a focus to the neighbourhood of the other one for some initial condition (without chaos)

Due to the sensibility to initial conditions, the trajectories may tend from the neighbourhood of the left focus to the right focus for t → ∞ and conversely. But the motion is not chaotic. The stable variety separates the Oxy -plane in two regions: one hatched and one non-hatched (Fig. 24.63); the trajectories which remain in the hatched region tend to the focus at the left, while the trajectories in the non-hatched region tend to the focus at the right. A trajectory which starts from a point of a separation line tends to the centre O . In case of the apparition of a perturbing force ( γ ≠ 0 ) one can become a chaotic behaviour of the dynamic system. If we study problems which lead to partial differential equations for which we use the method of separation of variables, then it is possible to have to integrate just systems of equations analogue to those above. We consider now an oscillator with non-linear damping, the energy being generated by small amplitudes and dissipated by great ones. We introduce the Vander Pol equation of the form x + αϕ (x )x + x = β p (t ) ,

(24.4.40)

where ϕ ( x ) is an even function, negative for | x |< 1 and positive for | x |> 1, p (t ) is a periodic function, while α, β ≥ 0 . The Van der Pol equation can be replaced by the autonomous system x = y − Φ (x ), y = − x + βϕ ( θ ), θ = 1, (x , y , θ ) ∈

2

× S1 ,

(24.4.40')


where Φ (x ) =

x

∫0 ϕ ( ξ )dξ

735

is an odd function, while Φ (0) = Φ ( ±a ) = 0 for a certain

a > 0 . For a small β we choose ϕ ( x ) = x 2 − 1, Φ (x ) =

1 3 x − x , p (t ) = cos ωt , 3

(24.4.40'')

while for a great β we choose a partially linear function p (t ) . One makes firstly a study of the homogeneous system for β = 0 ; if a 1 1/ a , e.g., one shows that there exists an annular domain so that, at any point of the frontier, the field vector is directed towards its interior, that one being thus a trap-region. Because critical points do not exist, on the basis of the Poincaré–Bendixon theorem, one can state that the domain contains at least a periodic solution, which – as one cam show – is unique. In the case of the forced oscillator ( β ≠ 0 ) one puts in evidence bifurcations, heteroclinic and homoclinic trajectories, generating chaotic motions. The 1 and for α, β 1. study in made for α, β The equations of Van der Pol type are encountered in many problems of mechanics.

24.4.2.6 Feigenbaum’s and Hénon’s Mappings In 1977–1978, M. Feigenbaum considered a mapping defined by the recurrence equation x n = λx n −1 (1 − x n −1 ), x n ∈ (0,1), λ ∈ (0, 4 ] .

(24.4.41)

The fixed points of this mapping are given by the equation f (x ) = λx (1 − x ) = x ,

(24.4.42)

obtaining x1 = 0, x 2 = 1 − 1/ λ . The condition | f ′(x ) |=| λ (1/ 2 x ) |< 1 corresponds to some stable fixed points. We find thus that the point x1 is stable for 0 < λ < 1 , while the point x 2 is stable for 1 < λ < 3 ; otherwise one has instability. Let us calculate further f 2 (x ) = f ( f (x )) = λ [ λx (1 − x )][1 − λx (1 − x )] ;

the fixed points of this mapping are given by f 2 (x ) = x , resulting the fixed points of the previous mapping ( x1 and x 2 ), as well as x 3,4 =

λ +1±

(λ + 1)(λ − 3) , 2λ

the last two roots being real if λ > 3 . Taking, e.g., λ = 3.1 and imposing the condition | df 2 (x )/ dx |< 1 , we see that x 3 and x 4 are stable fixed points, while x1


736

and x 2 are instable fixed points; hence, for λ = 3 takes place a bifurcation and, instead of 2 fixed points, one obtains 22 = 4 fixed points, two stable fixed points and two instable ones. The equation f 3 (x ) = x is of 8th degree, appearing two bifurcations for λ = 1 + 6 = 3.449490 ; we are thus led to 23 = 8 fixed points, 4 for of which are instable a.s.o. We find thus, successively, the values λ1 = 1 , λ2 = 3 , λ3 = 3.449490 , λ4 = 3.545090 , λ5 = 3.564407 , λ6 = 3.568759 , λ7 = 3.569296 , λ8 = 3.569891 ,

λ9 = 3.5699934 etc. By this numerical experiment we can assume that the values λn

tend to a limit λ∞ , so that λn = λ∞ − cF − ( n −1) ;

(24.4.43)

taking λ∞ = 3.569946 , c = 2.6327 and F = 4.669202 , we obtain the above results. The constant F , which intervenes in many natural phenomena where appear chaotic aspects, is called the Feigenbaum constant. Eliminating the constants λ∞ and c , we obtain F =

λn − λn −1 . λn + 1 − λn

(24.4.44)

Fig. 24.64 Diagram of the entropy (S vs λ)

In general, for the equation f n ( x ) = x we find 2n fixed points; 2n −1 points are stable and 2n −1 points are instable. The number of the fixed points increases indefinitely for n → ∞ , but no one value of the variable x n is repeated, while the set of the values f n (x ) is chaotic. For λ ∈ (λ∞ , 4) the chaos can be interrupted by periodic phenomena, which take place for very small values in the neighbourhood of the parameter λ , called periodic windows. We have defined the entropy S by the relation (24.4.17); we represent the variation of S as function of the parameter λ in Fig. 24.64. We notice that for 1 < λ < 3 we


737

have S = 0 ; for 3 < λ < 4 the entropy is increasing, excepting the vertices directed towards dawn, corresponding the periodic windows mentioned above. In the limit case λ = 4 , the recurrence relation (24.4.41) becomes x n + 1 = 4 x n (1 − x n ) ;

(24.4.45)

after Stanislaw Ulan and John von Neumann, we make the change of variable x n = sin2 ( πyn ), yn > 0 , obtaining sin2 ( πyn + 1 ) = sin2 ( 2 πyn ) ,

wherefrom yn + 1 = 2 πyn + k π , k ∈

(24.4.45')

, or, remaining limited to angles less than π ,

yn + 1 = 2 yn .

(24.4.45'')

Using numbers in the binary system, e.g., x 0 = 0.101101 , with a finite number of decimals, we have, successively (shifting the point at the right and suppressing the significant cipher at the left of the point): x 0 = 0.101101 , x1 = 0.01101 , x 2 = 0.1101 , x 3 = 0.101 , x 4 = 0.01 , x 5 = 0.1 , x 6 = 0 , x 7 = 0,... The precision

of the measurement being limited, we must be content with a finite period of time; for an indefinite time, a “perfect” measurement of the initial conditions is necessary, which is practically impossible. This phenomenon is called “the loss” of the memory of the initial conditions.

Fig. 24.65 Hénon’s mapping, representation in the Oxy – plane, for a = 1.4 and b = 0.3 . After 10000 iterations (a); after 100000 iterations (b); after 1000000 iterations (c)

In 1975, to can study the orbits of celestial bodies, M. Hénon and Y. Pomeau have imagined a mapping given by the recurrence relation

x n + 1 = 1 − ax n2 + yn , yn + 1 = bx n , a > 0, b ∈ (0,1) ;

(24.4.46)

this mapping, known as Hénon’s mapping, is used for the study of the motion of the particles electrically charged in the accelerators of particles too. In certain initial conditions and for certain values of a and b appears a strange attractor.

738


Thus, in a representation in the Oxy -plane, for the parameters a = 1.4 , and b = 0.3 and for the initial conditions x 0 = 0.631 and y 0 = 0.189 , one obtains, after 10000 iterations, the image in Fig. 24.65a. Taking a small rectangle A , increasing the region, we obtain – after 10000 iterations – the image in Fig. 24.65b; continuing for a small rectangle B and increasing the respective region, we find – after 10000 iterations – the image in Fig. 24.65c. One puts thus in evidence the properties of self-similarity and of fractal structure. The strange attractor thus obtained leads to a chaotic behaviour of Hénon’s mapping for great values of n .

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Subject Index

Action, 228 elementary, 228 kinetic, 228 kinetic-potential, 251 Lagrangian, 228 mechanical, 228 potential, 228 Atwood’s machine, 41 Belidor’s crane bridge, 39 Bracket, 138 of Lagrange, 138 of Poisson, 139 Catastrophe, 682 butterfly, 694 in two variables, 695 dissipative systems, 682 elementary, 689 in two variables, 691 swallow tail, 693 synergetics, 682 umbilic, 695 elliptic, 695 hyperbolic, 696 parabolic, 697 Chaotic motion, 723 Duffing’s equation, 732 Lyapunov’s exponent, 725, 726 mapping, 735 Feigenbaum, 735 Hénon, 735 Mel’nikov’s method, 725, 726, 727 Smale’s horseshoe, 727 strange attractor, 727 Lorenz, 730, 731 Rössler, 730, 732 Ueda, 724 Van der Pol’s equation, 732, 734 Constraint, 3 Built-in mounting, 5 by threads, 5 catastatic, 5, 8, 412 first kind, 4 hinge, 5 holonomic, 3, 8, 19, 412 ideal, 3, 5

non-holonomic, 3, 19, 58, 412 rheonomous, 3, 8, 439 scleronomic, 3, 8, 412 simple support, 5 sliding, 5 with friction, 3 Control system, 402 Bolza’s problem, 403 conditions of minimum, 406 optimal trajectory, 407 principle of maximum, 408 Co-ordinate, 7 canonical, 115, 116 cyclic, 134, 166 generalized, 46 hidden, 76, 133 ignorable, 76 material, 260 normal, 126 quasi- , 421, 423 spatial, 260 Descartes’s ovals, 41 Differential principle, 20 d’Alembert, 20 Gauss, 28 Hertz, 28 Jourdain, 27 Newton, 20 virtual work, 24 Displacement, 2 generalized, 7 possible, 7 real, 7 virtual, 7, 49 parallel, 427 possible, 2 real, 2 virtual, 3, 412 Dynamical system, 629 attractor, 630, 633 basin, 635 butterfly effect, 629 centre, 633 differential equations, 648 Bernoulli, 651 Clairaut, 651

759

760 Duffing, 673 Lagrange, 651 non-linear, 648 Riccati, 651 fixed point, 630, 633, 676 Floquet’s theory, 642 Gronwall’s lemma, 647 hyperbolic, 631 indifferent equilibrium, 636 linear, 630 continuous, 630 degenerate, 631 discrete, 675 Mathieu’s function, 644 methods, 656 averaging, 670 expansion in power series, 657 linear equivalence, 660 one- pass, 665 perturbation, 668 successive approximations, 656 node, 633 non-autonomous, 637 periodic coefficients, 642 with a control function, 637 periodical, 631 Poincaré’s transformation, 678 problem, 652 two- point, 652 well put, 655 repeller, 633 saddle point, 633 Strutt’s diagram, 646 theorems, 633 Cauchy–Picard–Lipschitz, 654 Poincaré–Bendixon, 633 Poincaré–Lyapunov, 656 Toma, 663, 664 Van der Pol plane, 670 Enthalpy, 119 Entropy, 119 First integral, 68 Jacobi, 69, 70, 218 Lagrange’s equations, 68 linear, 299 mechanical energy, 69, 71 Painlevé, 69, 70 reduction of the number, 127 Formalism in mechanics, 390 Birkhoffian, 397, 399 Hamiltonian (q,p), 257 inverse problem, 397 Lagrangian (q), 254 space, 390 events (q,t), 390 momentum-energy (p,H), 393 states (q,p,t), 394 states and energy (q,p,t,H), 396

MECHANICAL SYSTEMS, CLASSICAL MODELS Fractal, 712 affine transformation, 719 Barnsley’s fern, 721 box counting, 719 Cantor’s set, 712 curve, 713 Hilbert, 714 Koch, 713 Peano, 714 dimension, 716 Hausdorff, 716 self-similarity, 717 distance, 714 Hausdorff, 715 Manhattan, 715 Hutchinson’s operator, 720 Menger’s sponge, 714 Sierpinski’s carpet, 713 Sierpinski’s sieve, 713 Function, 119 in involution, 296 of state (Gibbs), 119 Functional derivative, 256 Group, 291 Galileo, 323 Galileo–Newton, 331 Lie, 311 one parameter, 311 properties, 291 rotation, 321 Symplectic, 291 translation, 320 time translation, 323 with m parameters, 316 Hamiltonian mechanics, 120 equations, 120 canonical (Hamilton), 120 Euler–Lagrange, 218 Hamilton–Jacobi, 160, 162 of Routh, 130 of Whittaker, 135 sixth form, 121 with separate variables, 172 function, 125 Hamilton, 125 principal Hamilton, 171 Routh, 131 Hamiltonian, 120 Hamilton–Jacobi method, 160, 190 Levi-Civita’s conditions, 188 motion of planets, 202 Poisson–Jacobi identity, 143 Integral invariant, 335 absolute, 338 complete, 339 of order 2s, 339 of first order, 341

Subject Index of order 2s, 335 of Poincaré, 343 of Poincaré–Cartan, 348 relative, 335, 343 universal, 345 Invariance property, 293 Lagrange bracket, 295 Poisson bracket, 295 phase space, 294 Invariantive mechanics, 369 antiMinkowskian universe, 378 dilatation of the Universe, 385 exterior differential calculus, 368 fundamental differential form, 369 Hubble’s law, 385 Lorentz’s generalized force, 377 Minkowski–Einstein universe, 378 momentum-energy quadrivector, 373 perihelion of Mercury, 388 principles, 369 Cartan, 369 inertial motion of a particle, 370 system of two particles, 379 Lagrangian mechanics, 1 basic form, 25 fifth, 90 first, 25 fourth, 58, 84 second, 27 third, 28 collision, 95 electromechanical analogy, 87 energy potential, 14 equations, 24 central, 83, 84 equivalent, of first order, 81 Jacobi, 75 Lagrange, first kind, 24 Lagrange, inertial frame of reference, 93 Lagrange, normal form, 61, 63 Lagrange, reduction, 74 Lagrange, second kind, 57 Lagrange, with multipliers, 59 Liouville type, 84 Mangeron–Deleanu, 91 Nielsen, 88 Tsenov, 90 with separate variables, 86 Euler–Lagrange derivative, 12, 303 force of friction, 44 generalized quantity, 2 acceleration, 48 conservative force, 12 conservative system, 12, 166, 306 constraint, 10 dissipative force, 15 force, 10 gyroscopic force, 15, 55 non-potential force, 15, 55

761 percussion, constraint, 95 percussion, given, 95 quasi-conservative force, 12, 13 velocity, 49 virtual velocity, 49 Lagrangian, 59 Rayleigh’s dissipative function, 73 Mechanical systems, 1 catastatic, 8 continuous, 86, 254 degrees of freedom, 171 two, 172 s, 177, 197 discrete, 1 holonomic, 4 natural, 11 non-catastatic, 8 non-holonomic, 4 of one particle, 16 rheonomous, 4 scleronomic, 4 Mechanics, 1 Hamiltonian, 115 invariantive, 368 Lagrangian, 1 Motion of linear elastic solid, 259 constitutive law, 261 deformation work, 261 displacement, 260 acceleration, 261 velocity, 260 Lamé’s constants, 261 motion, 262 bars, 264 strings, 262 Poisson’s ratio, 261 strain, 260 angular, 260 linear, 260 local rotation of rigid body, 260 stress, 261 Non-holonomic mechanical system, 411 acceleration energy, 478 Appell’s function, 478 collision, 491 applications, 501 basic equations, 491 first integrals, 498 generalized percussion, 492 conditions of holonomy, 421 equations, 430 Boltzmann–Hamel, 440 Chaplygin, 467 Gibbs–Appell, 478 Hamel, 439, 441 Hamilton, 475 Lagrange, 430, 436 Lagrange, in quasi-co-ordinates, 437

762 Lagrange, with multipliers, 436 Maggi, 474 Maggi, canonical form, 475 Volterra, 473 Voronets, 472 heavy circular disc, 485 on a horizontal plane, 485, 489 on an inclined plane, 487 kinematics, 411 motion of a rigid solid, 430 on a fixed surface, 430 on a horizontal plane, 430 motion of a heavy circular disc on a plane, 447 motion of a rigid solid with a fixed point, 483 motion of a skate, 443 motion of a sphere on a rough fixed surface, 434 non-holonomic space, 421, 427 Vrănceanu’s theorems, 427 relations of transposition, 424 representative space, 414 Ricci’s coefficients, 429 pure pivotting, 418 rolling, 413 of a circular disc, 413 of a rigid solid on another one, 418 of a sphere on a horizontal plane, 420, 452 pure, 418 the carriage, 459 four-wheeled, 464 two-wheeled, 459 Operator, 51 space differentiation, 51 time differentiation, 51 total differentiation, 51 Pascal’s limaçon, 41 Periodic motion, 356 action variable, 361 adiabatic invariants, 365, 366 angle variable, 361 criteria, 365 multiply, 357 quasi-, 356, 360 Sommerfeld’s quantification criteria, 365 Perturbation theory, 285 Problem, 1 basic, 1 motion, 99 in an electromagnetic field, 108, 155 of a particle, 99, 151, 191 of a rigid solid, 109, 158, 209 of libration, 186 pendulum, 111 double, 111, 159 sympathetic, 112, 159 plane motion, 250

MECHANICAL SYSTEMS, CLASSICAL MODELS two centres, 106, 208 two particles, 105, 202 Ramification, 683 attractor, 709 multiple, 684 simple, 684 bifurcation, 697 global, 697, 707 Hopf’s, 701 Hopf’s, subcritical, 702 Neĭmark, 702 branch, 689 fold, 685 fork, 685 generation, 703 hard, 703 soft, 703 hysteresis, 687 multiple, 688 Lindstedt’s method, 704 periodic solution, 697, 698, 706 structural stability, 684 trajectory, 710 heteroclinic, 710 homoclinic, 710 Representative space, 2 configurations, 46 kinetic energy, 52 natural system, 52, 58 E3 n , 6

Δs , 47 phase space, 115 potential, 13 quasi-potential, 13 generalized, 54 scalar, 13, 20 simple, 54 vector, 13, 20 velocity, 3 generalized, 7 possible, 3 real, 3 virtual, 3 work, 52 real elementary, 54 Stability of equilibrium, 506 asymptotic, 511 criteria, 529 global, 511 orbital, 512 autonomous discrete mechanical system, 518 n degrees of freedom, 525 one degree of freedom, 518 criteria, 530 geometrical, 532 Liénard–Chipart, 531 Routh–Hurwitz, 530

Subject Index dissipative discrete mechanical system, 535 function of definite sign, 512 equilibrium, 506 indifferent, 507 instable, 506, 533, 535 labile, 507 non-potential generalized force, 516 perturbation, 507 perturbed motion, 510 theorems, 514 Chetaev, 533 Lagrange–Dirichlet, 514, 516 Lyapunov, 533, 534 Torricelli, 514 Stability of motion, 537 in first approximation, 544 limit cycle, 551 Lyapunov’s function, 547 motion in a central field, 562 motion of rotation, 555 of a projectile, 559 of a rigid body, 555 non- linear differential equations, 540 systems of cyclic co-ordinates, 550 theorems, 537 Bendixon, 554 Chetaev, 542 Lyapunov, 537, 539, 541, 543, 544, 546, 547 Routh, 550, 551 Theorems, 23 Carathéodory lemma, 290 conservation of mechanical energy, 23 ergodic, 355 Lagrange’s basic lemma, 217 second basic lemma, 219 of Burger, 368 of Cartan, 370 of differentiability of the solution, 65, 123 of Donkin, 118 of Euler–Ostrogradskiĭ, 265 of existence and uniqueness, 63, 122 of generalized momentum, 78 for percussions, 96 of Gibbs–Hertz, 368 of Hamilton, 232 of Hamilton–Jacobi, 163 of Hölder, 224, 237 of Hwa-Chung Lee, 346 of Jacobi, 248, 279 of Jacobi–Poisson, 145 of kinetic energy, 23 second form, 23 of Lagrange, 139 of Lamé, 262 of Larmor, 254 of Lie, 148, 295, 298 of Liouville, 149, 181, 339, 341 of Livens, 235

763 of Love, 264 of Maupertuis, 245 of Mayer, 157 of Noether, 304, 306, 308 reciprocal, 311 of Peano, 64 of Poincaré, recurrence, 354 of Poisson, 144 of Routh–Helmholtz, 79 of Stäckel, 183 of the least constraint, 31 of the least curvature, 33 of Torricelli, 26 of virtual work, 25 of Voss, 238 of Whittaker, 251 Transformation, 13 canonical, 265, 274 complete, 266, 268 free, 275, 281 generating function, 277 univalent, 275, 281 conditions of canonicity, 265, 270, 272, 274 gauge, 13, 67 identical, 284 infinitesimal canonical, 286 generating function, 289 structure, 289 of contact, 260 of Legendre, 118 point, 65, 275 symmetry, 301, 304 space-time, 319 Variational methods, 213 principles, 225 general integral, 225, 229 of Fermat, 249 of Hamilton, 231 of Hölder, 237 of Larmor, 253 of least action, 246 of Maupertuis, 243 of Voss, 238 transitivity relation, 217 variations, 215 asynchronous, 215, 220, 237 canonical form, 235, 258 synchronous, 215, 231 Vibrations of mechanical systems, 566 axletree of negligible mass, 621 discrete, 572 dynamical absorber, 622 equations, 567 of motion, 567 secular, 570 forces, 597 periodic perturbing, 597 small dissipative, 603 which depend explicitly on time, 601

764 which do not depend explicitly on time, 600 normal co-ordinates, 576 oscillations, 566 free about a stable position of equilibrium, 566 free damped small, 596 of vehicles, 627 small forced, 597 small with one degree of freedom, 585 small with two degrees of freedom, 587 three harmonic, 583 two harmonic, 581 proper pulsations, 589 regulator, 613 centrifugal, 616 of Watt, 613 rotor axletree system, 619 supplementary holonomic constraints, 592

MECHANICAL SYSTEMS, CLASSICAL MODELS theorems, 577 Daniel Bernoulli, 577 Rayleigh, 593 variation of constants, 609 vibrations, 574 non-linear, 606 parametric, 607 proper forms, 574 self-, 611 Work, 3 of constraint forces, 3 real, 3 virtual, 3 of the forces of inertia, 6 real elementary of the generalized forces, 3 constraint, 10 given, 10 virtual of the generalized forces, 3 constraint, 11 given, 11

Name Index

Abraham, R., 739 Abraham, R.H., 739 Adams, 655 Adhémar, R.d’, 739 Agostinelli, Cataldo, 739 Alarcón, E., 742 Alembert, Jean Baptiste leRond d’, 2, 20, 21, 25, 213, 739 Amaldi, Ugo, 748 Ames, J.S., 739 Anastasiei, M., 749 Anaxagoras, 723 Andelić, T., 739 Andonie, George Şt., 739 Andrade, 208 Andreescu, D., 750 Andronov, A., 739 Anosov, D.V., 739 Appell, Paul-Émile, 2, 24, 411, 478, 482, 739 Argyris, J., 739 Arnold, V.I., 115, 335, 629, 739, 740 Aron, I.I., 750 Arya, A., 740 Atanasiu, Mihai, 740, 755, 756 Atkin, R.H., 740 Atwood, George, 41 Avez, A., 740 Babakov, L.M., 740 Babitsky, V.I., 740 Babuška, Ivo, 740 Bahar, L.Y., 399 Bai-Lin, H., 740 Baker, G.L., 740 Baker Jr., R.M.L., 740 Banach, Stephan, 740 Barbashin, 541 Barbu, L., 753 Barbu, Viorel, 740 Barenblatt, G.L., 740 Barger, V., 740 Barnsley, M.F., 712, 721, 740 Bat, M.I., 740 Bălan, Ştefan, 740, 756 Beer, F.P., 740

Beghin, 97 Beju, Iulian, 740 Belenkiĭ, I.M., 740 Belidor, 39 Bellet, D., 740 Bellman, R., 403, 740 Beltrami, Enrico, 740 Bendixon, 554, 633 Berezkin, E.N., 741 Berge, P., 741 Bergman, P.G., 741 Bernoulli, Daniel, 1, 577 Bernoulli, Jacques (Jacob), 1, 651 Bernoulli, Jean, 1, 218 Bernousson J., 741 Béghin, H., 741 Biezeno, C.B., 741 Birkhoff, Georg David, 335, 390, 397, 399, 402, 476, 741 Bishop, A.R., 741 Blitzer, L., 742 Bobylev, D.K., 411 Boerner, H., 741 Bogolyubov (Bogoliuboff), Nikolaĭ Nikolaevich, 607, 741, 747 Bolotin, V.V., 741 Boltzmann, Lothar, 341, 741 Bolza, 403 Bone, J., 741 Born, Max, 741 Borş, Constantin, 741 Boucher, M., 741 Bouligand, Georges, 741 Bourlet, C.E.E., 741 Bradbury, T., 741 Bradistilov, 112 Bratu, Polidor, 741, 757 Brădeanu, D., 741 Brădeanu, Petre, 741 Brelot, M., 741 Bricard, R., 741 Brillouin, L., 741 Brot, M.A., 747 Brousse, Pierre, 741 Brouwer, D., 741

765

766 Brown, Robert, 629 Budó, A., 741 Bukholts, N.N., 741 Bulgakov, B.V., 741 Burgatti, P., 189 Burgers, 365, 368 Burileanu, Ştefan, 741 Burlacu, L., 749 Butenin, N.V., 741 Butterfield, H., 741 Buzdugan, Gheorghe, 741, 745, 755 Cabannes, Henri, 742 Caldonazzo, B., 742 Califano, S., 742 Cantor, Georg, 712 Carathéodory, 422, 430 Cartan, Élie, 239, 341, 348, 350, 368, 369, 371, 742 Cassiday, G., 743 Cauchy, Augustin Louis, 21, 64, 122, 650, 654 Caviglia, 399 Cayley, Arthur, 721, 742 Chaikin, C., 739 Champernowne, 723 Chaplygin, Sergei Alekseevich, 411, 417, 430, 437, 467, 468, 471, 475, 742 Charlier, C.L., 742 Charpit, 144, 326 Chazy, Jean, 742 Chetaev, N.G., 505, 542, 548, 742 Chiroiu, Călin, 742 Chiroiu, Veturia, 742, 749, 754 Chow, S.N., 683, 742 Chow, T.L., 742 Christoffel, E.B., 63, 441 Chipart, 531 Chua, L.O., 751 Ciofoaia, Vasile, 755 Clairaut, Alexis-Claude, 1, 161, 651 Clebsch, Rudolf Friedrich Alfred, 406 Clemence, G.M., 741 Coe, C.J., 742 Collatz, Lothar, 742 Conti, R., 753 Coppel, W.A., 742 Corben, H.C., 742 Coriolis, Gustav Gaspard, 94 Correas, J.M., 742 Couley, 710 Courant, Richard, 742 Crandall, S.H., 742 Crăciun, Eduard Marius, 742 Crede, C.E., 745 Cromer, A.H., 753 Currie, D.G., 399 Curtu, Ioan, 755

MECHANICAL SYSTEMS, CLASSICAL MODELS Curtu, R., 742 Cushing, J.T., 742 Darboux, Gaston Jean, 399, 742 Dautheville, S., 739 Davidson, M., 742 Davis Jr., L., 742 Deciu, Eugen, 752 Delassus, E., 742 Deleanu, 91, 92 Demko, S.G., 712, 740 Desboves, 208 Descartes, René, 41 Destouches, J.L., 742 Devaney, R., 729, 742 Dincă, George, 208 Dincă, Florea, 742 Djanelidze, G.Yu., 740 Doblaré, M., 742 Dobronravov, V.V., 115, 743 Dolapčev, Blagovest, 89, 91, 743 Donescu, Ştefania, 742, 750 Donkin, 118 Douady, A., 722 Douglas, Gregory, R., 743 Douglas, I., 399 Dragnea, Ovidiu, 743 Dragoş, Lazăr, 743 Draper, C.S., 743 Drăganu, Mircea, 743 Drâmbă, Constantin, 743 Duffing, G., 673, 723, 732, 743 Dugas, R., 743 Dvor’nkov, A.L., 742 Eckhaus, W., 743 Eddington, A.S., 743 Edgar, G.A., 743 Einstein, Albert, 373, 378 El Naschie, M.S., 743 Emmerson, J.Mc L., 743 Erdmann, 407 Eringen, A. Cemal, 743 Euler, Leonhard, 1, 218, 220, 245, 303, 437, 438, 566, 667, 743, Falconer, K.J., 712, 743 Falk, G., 743 Faust, G., 739 Feigenbaum, Mitchell, 723, 729, 735, 736 Fermat, Pierre de, 213, 249 Fernandez, M., 743 Fertis, D.G., 743 Feshbach, H., 749 Fetcu, Lucia, 741 Fetter, A.L., 743 Feynman, R.P., 743

Name Index Fillipov, A.P., 743 Finch, J., 745 Finkelshtein, G.M., 743 Finzi, Bruno, 743 Fischer, Ugo, 743 Floquet, 642 Florian, V., 752 Follin, Jr., J.M., 742 Fomin, S.V., 744 Forbat, 190 Fowles, G., 743 Föppl, August, 744 Frank, N.H., 754 Frank, Ph., 743, 744 French, A., 744 Frobenius, 412 Fues, E., 160, 750 Fufaev, N.A., 411, 474, 750 Gabos, Zoltan, 744 Galileo, Galilei, 319, 323, 324, 331, 744 Gallavotti, G., 744 Gantmacher, F., 744 Gauss, Carl Friedrich, 20, 28, 31, 90, 213, 449 Gavete, L., 742 Gelfand, I.M., 744 Georgescu, A., 744 Germain, Paul, 744 Gibbs, Josiah Dixon Willard, 116, 119, 365, 368, 411, 478, 482, 744 Gilbert, 94 Gilmore, R., 683, 744 Gioncu, Victor, 683, 744 Girard, A., 744 Gleick, J., 744 Godbillon, C., 744 Goldstein, Herbert, 2, 115, 744 Gollub, J.P., 740 Golubev, V.V., 744 Golubitsky, M., 744 Gordon, I., 739 Graffi, Dario, 744 Grahn, R., 746 Grammel, Richard, 741, 744, 757 Gray, A., 744 Grebenikov, E.A., 744 Greenhill, A.G., 744 Greenwood, D.T., 745 Griffith, B.A., 755 Gronwall, 647 Grumăzescu, Mircea, 754 Gruner, G., 741 Grübler, M., 753 Guckenheimer, J., 745 Guillemin, V., 744 Gumowski, I., 745 Gurel, O., 745 Gutzwiller, M.C., 729, 745

767 Haase, M., 739 Habets, P., 753 Haikin, S.E., 739, 745 Haken, H., 682, 745 Halanay, Aristide, 745 Hale, J.K., 683, 742, 745 Halliday, D., 745 Halphen, Georges Henri, 745 Hamburger, Leon, 745 Hamel, Georg, 4, 84, 411, 438, 439, 441, 745 Hamilton, William Rowan, 115, 116, 125, 135, 160, 162, 163, 171, 172, 213, 231, 232, 257, 335, 475, 745 Hammermesh, M., 745 Hand, L., 745 Hangan, Sanda, 745 Hao, B.L., 745 Harris, C.M., 745 Hartog Den, J.P., 745 Hassard, B.D., 745 Hasselblatt, B., 747 Hausdorff, 712, 715, 716 Hayashi, C., 745 Heisenberg, Werner Karl, 629 Helleman, R.H.G., 745, 746 Helmholtz, Hermann Ludwig Ferdinand von, 79, 131, 398, 636 Hemming, G., 745 Herglotz, G., 745 Herman, Jacob, 1 Hertz, Heinrich Rudolf, 20, 28, 133, 213, 365, 368, 411, 412, 745 Hesse, 690 Hestenes, D., 745 Hénon, M., 727, 735, 737, 745 Hilbert, David, 712, 714, 716, 742 Hirsch, M.W., 745 Hlavaček, V., 745 Hokjman, S.,399 Holden, A.V., 745 Holmes, P., 745 Hooke, Robert, 261 Hopf, 701, 702 Horovka, J., 743 Hort, W., 746 Hölder, O., 213, 224, 231, 237, 238 Hubble, Edwin Powell, 379, 385, 389 Hunt, G.W., 755 Hunter, G., 746 Hurwitz, Adolf, 529, 530, 532 Huseyin, K., 746 Hutchinson, 720 Huygens, Christian, 566, 746 Hwa-Chung Lee, 341, 345, 346 Iacob, Andrei, 746 Iacob, Caius, 746 Iakovenko, G.N., 752

768 Ille, Vasile, 755 Imshenetskiĭ, V.G., 189 Ioachimescu, Andrei G., 746 Ionescu-Pallas, Nicolae, 746 Iooss, G., 683, 740, 746 Irimiciuc, Nicolae, 749 Irwin, M.C., 746 Isaacson, N., 746 Ishlinskiĭ, A.Yu., 746 Ivanov, I., 740 Jackson, E.A., 746 Jacobi, Carl Gustav Jacob, 69, 74, 75, 100, 115, 143, 145, 160, 162, 163, 205, 213, 240, 245, 248, 279, 437, 498, 746 Jammer, M., 746 Janssens, P., 746 Jansson, P.-A., 746 Jaunzemis, W., 746 Jeffers, S., 746 Jeffreys, B.S., 746 Jeffreys, W., 746 Johnson, 411 Joos, G., 746 Jose, J.V., 746 Joseph, D.D., 683, 740, 746 Jourdain`, 88, 213 Jukovskiĭ, Nikolai Egorovich, 746 Julia, Gaston, 712, 721, 746 Jumarie, G., 746 Jung, G., 746 Jurgens, H., 712, 751 Kahan, Theo, 746 Kamke, E., 746 Kantorovich, L.V., 746 Kapitaniak, T., 729, 746 Kaplan, James, 727 Kaplan, M.H., 731, 747 Kapli, D.M., 750 Kasner, E., 747 Katok, A., 747 Kauderer, H., 747 Kauffman, S.A., 747 Kazarinoff, N.D., 745 Kármán, Theodor von, 747 Kästner, S., 747 Kecs, Wilhelm, 747 Keller, H.B., 746, 747 Kellog, O.D., 747 Kelzon, A.S., 740 Khanukaev, Yu.I., 752 Kilchevski (Kilcevski), N.A., 747 Kirchhoff, Gustav Robert, 747 Kirpichev, Viktor L’vovich, 747 Kittel, Ch., 747 Klein, Felix, 747 Knight, W.D., 747

MECHANICAL SYSTEMS, CLASSICAL MODELS Kobussen, J.A., 399 Koch, Helge von, 712, 713 Kochin, Nikolaĭ Evgrafovich, 747 Koçak, H., 683, 745 Koenig, Samuel, 245, 480 Kolman, B., 753 Kolmogorov, Andrei Nikolaevich, 717 Kolosov, Aleksandr Aleksandrovich, 359 Kooy, J.M.J., 747 Koshlyakov, V.N., 747 Kosmodemyanskiĭ, A.A., 747 Kosyakov, B., 747 Kranz, C., 748 Krasovski, 541 Krbek, F., 748 Kron, G., 748 Kronecker, Leopold, 62 Krylov, Alekseĭ Nicolaevich, 559, 607, 748 Krylov (Kryloff), Vladimir Ivanovich, 747 Kuiper, G.P., 748 Kutta, 655, 667, 668 Kutterer, R.E., 748 Kuzimenkov, L.S., 751 Kuznetsov, Y.A., 748 Kwatny, H.G., 399 Lagrange, Joseph-Louis, 2, 24, 25, 45, 57, 59, 83, 92, 138, 144, 213, 217, 218, 245, 303, 335, 407, 412, 430, 436, 437, 438, 505, 566, 651, 671, 748 Lakhtin, L.M., 748 Laloy, M., 753 Lamb, H., 2, 748 Lamé, Gabriel, 261, 262 Lampariello, G., 748 Landau, Lev Davidovich, 2, 729, 748 Lantsosh, K., 748 Laplace, Pierre Simon de, 629 Larmor, Joseph, 253, 254 La Salle, J.P., 748 La Valée-Poussin, Charles Jean, 2, 748 Lebesgue, Henri, 355 Lecornu, L., 748 Leech, J.W., 748 Lefschetz, S., 748 Legendre, Adrien-Marie, 118, 119, 406 Leimanis, E., 748 Leipholz, H., 748 Lejeune-Dirichlet, Peter Gustav, 505 Leontovich, E., 739 Levi-Civita, Tullio, 188, 368, 427, 748 Lie, Sophus, 145, 148, 295, 298, 311 Liénard, 531 Lifshitz, E., 2, 748 Lindelöf, 411, 651, 657 Lindsay, R.B., 748 Lindstedt, 704 Liouville, Joseph, 84, 145, 149, 181, 202, 339, 341

Name Index Lippmann, Horst, 748 Lipschitz, A.M., 422 Lipschitz, Rudolf, 21, 64, 122, 650, 654, 657 Littlewood, D.E., 748 Livens, G.H., 235 Loft, E.E., 750 Loitsyanski, L.G., 748 Lomont, J.S., 748 Lorentz, Hendrik Anton, 19, 374 Lorenz, Eduard N., 629, 723, 730, 731 Love, A.E.H., 264 Luca-Motoc, D., 755 Lurie, Anatoli I., 2, 115, 748 Lyapunov, Aleksandr Mikhailovich, 505, 510, 511, 534, 535, 537, 539, 541, 543, 544, 546, 547, 566, 607, 656, 723, 725, 726, 748 Lyubarskiĭ, G.Ya., 748 McCracken, M., 701, 749 MacDonald, G., 749 Mach, Ernst, 748 Macke, W., 748 Maclaurin, Colin, 1 Macmillan, W.D., 748 McLachlan, N.W., 749 Macomber, G.R., 743 Madan, R., 749 Maggi, G.A., 411, 474, 475 Magnus, K., 749 Maier, A., 739 Malkin, I.G., 749 Malvern, L.E., 749 Mandel, Jean, 749 Mandelbrot, Benoit B., 712, 719, 723, 749 Mandelstam, S., 757 Mangeron, Dumitru, 91, 92, 744, 749 Manneville, Paul, 729 Marchyuk, G.Z., 749 Marcolongo, Roberto, 749 Marinca, V., 751 Marion, J.B., 749 Marsden, J.Ernst, 701, 739, 749 Mathieu, E., 269, 607, 644, 672 Maunder, L., 740 Maupertuis, Louis-Moreau de, 1, 213, 242, 245 Mawhin, J., 753 Maxwell, James Clerk, 376 Mayer, 157, 403 Meirovich, I., 749 Mel’nikov, V.K., 723, 725, 726, 727 Melo, W. de, 751 Menger, 712, 714 Mercheş, I., 749 Mercier, A., 2, 749 Merkin, D.N., 749 Meshcherskiĭ, Ivan Vselovod, 749 Middlehurst, M.B., 748 Mihăilescu, E., 742

769 Mihăilescu, Mircea, 749 Mikhailov, 532 Miller Jr., W., 749 Milne, E.A., 655, 749 Minkowski, Hermann, 373, 378 Minorsky, N., 749 Mira, C., 744 Miron, Radu, 749 Mises, Richard von, 744 Mitropolskiĭ (Mitropolsky), Yu.A., 607, 741, 749 Mook, D.T., 750 Moon, F.C., 749 Moreau, J.J., 749 Morel, J., 741 Morera, G., 190 Morgenstern, D., 749 Morse, P.M., 690, 749 Mortici, C., 753 Moulton, 655 Mourre, L., 749 Munk, W., 750 Munteanu, Ligia, 742, 750, 754 Munteanu, M., 750 Murnaghan, F.D., 739, 750 Muszyńska, Agnieszka (Agnes), 750 Naimark, M., 750 Naumov, A.L., 750 Nayfeh, A.H., 750 Neĭmark, Yu.I., 411, 474, 702, 750 Nekrasov, A.I., 750 Nelson, W.C., 750 Neumann, John von, 41 Newbult, H.O., 750 Newton, Isaac sir, 20, 21, 213, 319, 331, 379, 397, 750 Nguyen, Quoc-son, 750 Nguyen, Van Hieu, 750 Nicolaenko, B., 741 Nicolis, G., 750 Nicorovici, Nicolae-Alexandru, 755 Nielsen, J., 88, 89, 750 Nikitin, E.M., 750 Nikitin, N.N., 742 Nitecki, Z., 750 Niţă, Mihai M., 750 Noether, Emmy, 300, 304, 306, 308, 311, 320, 327 Noll, Walter, 756 Nordheim, L., 160, 750 Novikov, S.P., 740 Novoselov, V.S., 411, 750 Nowacki, Witold, 750 Nyayapathi, V., 750 Obădeanu, Virgil, 390, 751 Okunev, V.N., 751 Olariu, S., 751

770 Olkhovskiĭ, I.I., 751 Olson, H.F., 751 Olson, M., 740 Onicescu, Octav, 368, 373, 751 Oppenheim, I., 756 Osgood, W.F., 751 Ostrogradskiĭ, Mikhail Vasilievich, 213, 220, 232, 607, 751 Ott, E., 751 Ovsiannikov, V., 751 Painlevé, Paul, 69, 437, 498, 751 Palis, J., 726, 751 Pandrea, Marina, 751 Pandrea, Nicolae, 751, 754 Parker, T., 751 Pars, L., 76, 160, 356, 411, 751 Pascal, Blaise, 41 Pastor, M., 742 Pastori, Maria, 743 Pavlenko, M.G., 751 Peano, Giuseppe, 64, 123, 650, 712, 714, 716 Peitgen, H.O., 712, 751 Peixoto, M.M., 751 Percival, I., 752 Perry, J., 752 Petkevich, V.V., 752 Pérès, J., 752 Pfaff, Johann Friedrich, 2, 135 Picard, 651, 654, 657 Pignedoli, A., 739 Planck, Max Karl Ernst Ludwig, 752 Plăcinţeanu, Ion, 752 Pohl, R.W., 752 Poincaré, Jules Henri, 239, 341, 343, 348, 350, 354, 369, 512, 523, 547, 591, 607, 629, 633, 635, 636, 656, 678, 680, 716, 723, 724, 752 Poisson, Siméon-Denis, 138, 139, 143, 144, 145, 261, 752 Pol, Van der, 607, 670, 673, 723, 732, 734 Polak, L.S., 752 Pomeau, Yves, 727, 729, 737, 741, 745 Pontriaguine (Pontryagin), L.S., 407, 752 Poole, Ch.P., 744 Pop, Ioan, 741 Popescu, D., 756 Popoff, K., 752 Popov, V.M., 752 Posea, Nicolae, 752 Poston, T., 683, 752 Poteraşu, V.F., 756 Pöschl, T., 752, 757 Prager, Milan, 740 Prange, G., 752 Prigogine, Ilya, 682, 750, 752 Pyatnitskiĭ, E.S., 752 Quangel, J., 475

MECHANICAL SYSTEMS, CLASSICAL MODELS Racah, G., 752 Radeş, Mircea, 741 Radu, Anton, 752 Rausenberger, O., 752 Rawlings, A.L., 752 Rădoi, M., 752 Reichl, L.E., 752 Resnick, R., 745 Riccati, 651 Ricci, Gregorio, 429 Richards, D., 751 Richardson, K.I.T., 752 Richter, P.H., 751 Riemann, Georg Friedrich Bernhardt, 355 Ripianu, Andrei, 752, 755 Rocard, Y., 753 Rohrlich, F., 753 Roitenberg, L.N., 753 Roman, P., 752 Rose, N.V., 752 Roseau, Maurice, 753 Rouche, N., 753 Rousseau, 97 Routh, E.J., 79, 115, 131, 254, 529, 530, 550, 551, 566, 753 Roy, A.E., 753 Roy, M., 744, 753 Rössler, E.E., 730, 732, 745 Ruderman, M.A., 747 Ruelle, David, 629, 709, 723, 727, 729, 753 Runge, 655, 667 Ruppel, W., 743 Russel Johnston Jr., E., 740 Ryabov, Yu.A., 744 Sabatier, P.C., 390, 753 Safko, J.L., 744 Saletan, E.J., 399, 746, 753 Samuel, M.A., 750 Sanders, J.A., 753 Sansone, Giorgio, 753 Santilli, R.M., 356, 398, 753 Sarlet, W.I., 399 Sauer, R., 753 Saupe, D., 751 Savarensky, E., 753 Savet, P.H., 753 Sburlan, Silviu, 753 Scarborough, J.B., 753 Schaefer, Cl., 753 Schaeffer, D., 744 Scheck, Florian, 753 Scheifele, G., 755 Schild, A., 755 Schmutzer, E., 753 Schoenfliess, A., 753 Schuster, H.G., 729, 753 Scouten, 411

Name Index Secu, A., 756 Sedov, Leonid I., 753 Seitz, F., 754 Sergiescu, V., 754 Serret, 208 Sestini, G., 754 Seydel, R., 683, 754 Shapiro, I.S., 754 Sharp, R.T., 754 Shaw, C.D., 739 Shaw, R., 754 Shigley, J.E., 754 Shub, M., 754 Siacci, F., 754 Siegel, C.L., 754 Sierpinski, Waclaw, 712, 713 Signorini, A., 754 Silaş, Gheorghe, 754 Simion, F.P., 757 Simpson, 667, 668 Slater, J.C., 754 Slătineanu, Ileana, 745 Smale, Stephen, 712, 723, 726, 727, 728, 745, 754 Smith, D.E., 754 Sneddon, Ian N., 754 Soare, Mircea, V., 660, 754 Sofonea, Liviu, 754 Somigliana, Carlo, 754 Sommerfeld, Arnold, 266, 365, 366, 747, 754 Soós, Eugen, 740, 754 Souriau, J.M., 754 Sparrow, C., 754 Spiegel, M.R., 754 Staicu, C.I., 754 Stan, Aurelian, 754 Stan, Ion, 744 Stănescu, Nicolae-Doru, 751, 754 Stäckel, P., 183, 360, 754 Stehle, P., 742 Stephan, W., 743 Stern, A., 750 Sternberg, S., 754 Stiefel, E.L., 755 Stewart, H.B., 756 Stewart, I., 683, 752 Stoenescu, Alexandru, 754, 755 Stojanović, R., 739 Stoker, J.J., 755 Stora, R., 746 Strelkov, S.P., 755 Stroe, I.V., 756 Strutt, John William (lord Rayleigh), 73, 593, 596, 646, 752 Suslov, Gavriil Konstantinovich, 755 Swamy, V.J., 750 Sylvester, James Joseph, 576, 755 Symon, K.R., 755

771 Synge, J.L., 390, 755 Szabo, I., 749, 753, 755 Szava, I., 755 Szebehely, V., 755 Szemplinska-Stupnika, W., 755 Ştiucă, Petru, 742 Tabor, M., 755 Tait, P.G., 755 Takens, Florens, 629, 729 Talle, V., 752 Targ, S., 755 Tenot, A., 755 Teodorescu, Nicolae, 755 Teodorescu, Petre P., 252, 660, 740, 747, 754, 755 Teodosiu, Cristian, 742, 754 Ter Haar, D., 115, 755 Thirring, W., 755 Thom, René, 629, 682, 755 Thoma, A., 745 Thompson, J.M.T., 755 Thomsen, J.J., 756 Thomson, J.J., 756 Thomson, William (lord Kelvin), 76, 756 Thornton, S.T., 749 Timoshenko, Stepan (Stephen) Prokof’evich, 607, 756 Tisserand, François Félix, 756 Tocaci, Emil, 752, 756 Tokuyama, M., 756 Toma, Ileana, 660, 663, 664, 754, 756 Torricelli, Evangelista, 26, 514 Troger, H., 756 Truesdell, Clifford Ambrose, 756 Trukhan, N.M., 752 Tsenov, I., 90, 91, 401 Tsiolkovskiĭ, Konstantin Eduardovich, 756 Ţăposu, I., 756 Ţiţeica, Gheorghe, 746, 755 Ueda, 724 Ungureanu, S., 756 Urrutia, L.F., 399 Uytenbogaart, J.W.H., 747 Vâlcovici, Victor, 230, 251, 252, 756 Verhulst, V., 753, 756 Vidal, Ch., 741 Vieru, D., 756 Vigier, J.-P., 745 Vilenkin, N.Ya., 756 Vitašek, E., 740 Vitt, A.A., 739 Voiculescu, Dumitru, 752, 756 Voinaroski, Rudolf, 756 Voinea, Radu, 756 Voltaire (Arouet, François Marie dit), 245

772


Volterra, Vito, 473, 474, 757 Voronets, 411, 472 Voronkov, I.M., 757 Voss, A., 213, 231, 237, 238, 757 Vrănceanu, Gheorghe, 411, 427, 429, 430, 473, 757

Wigner, E., 757 Willers, Fr.A., 757 Winkelmann, M., 757 Witner, A., 757 Wittenbauer, 757

Waddington, C.H., 757 Wagner, 411, 430 Walecka, J.D., 743 Walther, H.O., 751 Wan, Y.-H., 745 Wason, W.R., 757 Watt, James, 613 Webster, A.G., 757 Weeney, R.Mc., 757 Weierstrass, Karl, 406, 407, 757 Weizel, W., 757 Weyl, Hermann, 757 Whiteside, D.T., 757 Whittaker, E.T., 135, 137, 251, 399, 757 Wiggins, S., 683, 757

Yano, Kentaro, 757 Yarov-Yarovoi, M.S., 189 Yorke, 731 Yoshizawa, T., 757 Young, D.H., 756, 757 Yourgraum, W., 757 Zeeman, E.C., 683, 757 Zeveleanu, C., 757 Zhukovskiĭ, Nikolaĭ Egorovich, 411 Ziegler, H., 758 Zlătescu, Anca, 112 Zoretti, L., 758 Zorke, James, 727 Zweifel, P.F., 744

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