The Inverse Variational Problem in Classical Mechanics

The Inverse Variational Problem in Classical Mechanics

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The Inverse Variational Problem in Classical Mechanics

Jan Lopuszanski Institute of Theoretical Physics University of Wroclaw, Poland

World Scientific Singapo re* New Jersey • L ondon • Hong Kong

Published by World Scientific Publishing Co. Pie. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

THE INVERSE VARIATIONAL PROBLEM IN CLASSICAL MECHANICS Copyright © 1999 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-02-4178-X

Printed in Singapore by Regal Press (S) Pte Ltd.

To my friends physicists.

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Foreword

This book arose from my informal discussions with P.C. Stichel and J. Cislo in which we tried to probe into the fascinating problem, the inverse variational problem in classical mechanics. It contains mostly our own reflections and ideas developed during studying the vast literature of the subject. The ideas - I am afraid - are not all new and are probably well known to a small circle of experts, as the problem under consideration is over hundred years old. My notes do not intend to present all results obtained in this domain of research. Their aim is to give a concise picture of present state of affairs, imbued with our own, personal flavor. No advanced methods of contempo­ rary differential geometry were used. All function we are going to use in this lecture notes are assumed to be piecewise continuous and sufficiently many times differentiable. This is meant to make things more simple. The contents of this book is well suited to be used as lecture notes in a university course for physicists. Acknowledgement. I am grateful to Dr. J. Cislo for a careful reading of these notes, for his important critical comments as well as for his deep and fruitful remarks. I express my warm thanks to Dr. C. Juszczak for the painstaking typing of my manuscript.

Vll

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Foreword vii 1 Preliminary notions of Kinematics . Translations. proper rotations SO(3). Galilei group transformations. . . . . . . . . . . . 2 Preliminary notions of Analytical Dynamics. Newton's Equations . 3.1 Constraints . Work . The Principle of Least Action . EulerLagrange Equations . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Constants of motion . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 PoincarB Lemma and its converse. . . . . . . . . . . . . . . . . 4.2 Linear partial differential equations . The method of characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 The Inverse Variational Problem.Helmholtz's Conditions. . . . 5.2 Theorem of Henneaux . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The matrix a . Tr a" is conserved quantity. . . . . . . . . . . . 6.1 Instructive example of Cis10 . . . . . . . . . . . . . . . . . . . . 6.2 Instructive example of Douglas . . . . . . . . . . . . . . . . . . . 6.3 Instructive example of Pardo. . . . . . . . . . . . . . . . . . . . 7.1 Construction of an autonomous one-particle Lagrange function in (3+1) space-time dimensions yielding rotationally covariant Eulcr-Lagrange Equations coinciding with the Newton Equations . 7.2 Canonical variables . Equivalence problem of the Lagrange functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 All Lagrange functions, s-equivalent to the Lagrange function L = i x 2 - U(IxI), in (3+1) space-time dimensions . . . . . . . . 8.2 The case U(lx() # a x 2 -k 0,where a , /3 - constants . . . . . . . 8.3 The case U(lx1) = a x 2 ,6. . . . . . . . . . . . . . . . . . . . .

+

Contents

8.4 The Hamilton formalism for the model investigated in Subsections 8.1 - 8.3. Equivalence sets of Lagrange functions. . . . . . 8.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Example of Henneaux and Shepley. . . . . . . . . . . . . 8.5.2 Example of Stichel. . . . . . . . . . ........ 8.5.3 Example of Raiiada. . . . . . . . . . . . . . . . . . . . . 8.5.4 Second example of Ranlda . . . . . . . . . . . . . . . . . 8.5.5 Example of Cislo. . . . . . . . . . . . . . . . . . . . . . The model of Subsections 8.1 - 8.4 for n # 3 . . . . . . . . . . . 9 10.1 All autonomous s-equivalent one-particle Lagrange functions for (1+1) space-time dimensions. . . . . . . . . . . . . . . . . . . . 10.2 All s-equivalent one-particle Lagrange functions for (1+1) spacetime dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Construction of the most general autonomous one-particle Lagrange function in (3+1) space-time dimensions giving rise to rotationally covariant Euler-Lagrange Equations. . . . . . . . . 11.2 Evaluation of the function Gij . . . . . . . . . . . . . . . . . . . 11.3 Symmetry of Gij and evaluation of the Lagrange function . . . . 11.4 Symmetry properties of the Lagrange function . . . . . . . . . . 12 The largest set of Lagrange functions of one-particle system in a (3+1) dimensional space.time, s-equivalent to a given Lagrange function yielding rotationally forminvariant Equations of Motion (formulation of the problem). . . . . . . . . . . . . . . . . . . . 13.1 Construction of the most general two-particle Lagrange function in (1+1) space-time dimensions giving rise to Euler-Lagrange Equations covariant under Galilei transformation . . . . . . . . . 13.2 Galilei forminvariance of the Euler-Lagrange Equations for two particles in (1 1) space-time dimensions. . . . . . . . . . . . . 14.1 Construction of the most general two-particle Lagrange function in (3+1) space-time dimensions giving rise to Euler-Lagrange Equations covariant under Galilei transformations. . . . . . . . 14.2 Galilei forminvariance of the Euler-Lagrange Equations for two particles in (3 1) space-time dimensions. . . . . . . . . . . . . 15.1 All two-particle Lagrange functions s-equivalent to'a given autonomous Lagrange function yielding Galilei forminvariant E quations of Motion in (1+1) space-time dimensions. . . . . . . 15.2 All Euler-Lagrange Equations, forminvariant under the Galilei transformations . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents

xi

15.2.1 C a s e g # O . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Caseg=O. . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Generalization of the set of Lagrange functions (admission of Euler-Lagrange Equations not covariant under Galilei transformations). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.5 Case of Galilei forminvariant Newton's Equations corresponding to Euler-Lagrange Equations which are not Galilei covariant. (formulation of the problem) . . . . . . . . . . . . . . . . . . . 16 An Outloolc. Application in the Feynman Approach to Quantum Mechanics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

204

Index

220

184 192 198

208 211

1

Preliminary notions of Kinematics. Translations, proper rotations SO(3), Galilei group transformations.

The considerations of Classical Mechanics start with Kinematics, i.e. with investigations of various types of motions of material bodies without paying attention to the cause giving rise to those motions. Our setting will be up to few exceptions, the 3-dimensional Euclidean space. In this space a body is moving. To describe the motion we need still a parameter, the time variable. To have a clear insight into the structure of motions it is convenient to make use of the notion of a rigid body. In a rigid body the distance between two arbitrary points of this body remains unchanged during the motion of the body. The rigid body may change its position with respect to its environment while time flows. flows. The body, as a whole, is displaced from its original position to a different one. The displacement can be: • a rotation around a straight line /; then the only points of the body which do not move are the points lying on I, • a rotation around a point P; then the position of point P does not change. • a parallel translation in a certain direction by an interval /; then the intervals connecting the original and final positions of each point of the body, are parallel to each other, of the same length \I\, as well as of the same orientation in space. If we refrain from the rigid body, the parallel translations have the following properties: i) the translation can be characterized by any one of the intervals of equal length, parallel to each other and of the same direction in space, ii) if AB represents such an interval and AAi AAi, >A%A%,... A%A%, ... AAnnBB (n (n-- natnatural number) represents a broken line, then the operation linked to AB is equivalent to the sum of operations linked to A1A2, ■ ■ ■ AnB. The properties i) and ii) define a quantity called vector. A vector AB can be decomposed into three vectors along the axes of the Cartesian coordinate system, introduced in the Euclidean space, such that the sum of them will give AB again. This vectors we shall call x[ ', x

(AB) i' 2

x

(AB) 3

1

The Inverse Variational Problem in Classical

2

Mechanics.

There is a theorem, due to Euler, which claims that any rotation around a point P is equal to a rotation around an axis passing through P If the rotation around the point P is continuous in time, then the axis can also change continuously its position; the axis at a given time t is the instantaneous axis of rotation. This Euler theorem can be proved as follows. The proper rotation (without reflections) of a point reads, 3

'i = Y^RikXk

1=1,2,3

(1.1)

fc=l

where x = (xi,x-2,xz) and x' can be interpreted as coordinates of some points. The 3x3 matrix R represents an element of the group 50(3) and has the following properties (R-l)ik

= Rki = (RT)tk (orthogonality)

(1.2)

detR = 1 (proper rotations)

(1-3)

[R,RT]=RR~1

(1.4)

Prom -R~1R

=0

follows that R can be diagonalized. We have the following eigenvalues and eigenfunction problem

X > ^ i s ) = A^ (s)

(1.5)

fc=i

As R is a real matrix we have 3

Rikipk

= A

Vi

then, taking into account (1.2) and (1.5), we obtain

£(£^ s , )(E^l s ) ) = |AW|» s#Vi' ) = £^SV? or |A (S, | 2 = 1.

(1.6)

Preliminary

notions of

Kinematics...

3

Since, according to (1.3) 5

det.R=JjA s = l

(1.7)

s

from (1.6) follows that A(S> = 1,

ei{fi,

e~iv,

lp

= Tp.")

(1.8)

Consequently for

4l) = *i

(1-9)

we have for R 6 50(3) /

""ij 3 — i ' 3

The coordinates x in here can be interpreted as the coordinates of P Then for any n £ K 2 J Rij^Xj = fixi.

(1-10)

The axis of rotation is given by {[MX, fj, € K}

If R = R(t) then /xx = jix{t) is the instantaneous axis of rotation. The rotation is characterized by the direction of the axis e(t) and the angle 6(t) of rotation around the axis R = R(e, 6). It is easy to see, taking into account the group properties of 50(3), that two consecutive rotations around a fixed point can be replaced by one rotation. The construction of a displacement is as follows. We translate any properly chosen point P of the rigid body into position P'\ all other points of the rigid body are translated parallelly in a similar way. Now we rotate the rigid body around P'. '1 For the case when all A' s ' are equal to (±1) the proof follows immediately.

4 4

The Inverse Variational Problem in Classical

Mechanics.

We conclude: the most general displacement of a rigid body consists of a translations and a proper rotation around certain axis. If the body moves in time t we may introduce the notion of velocity and acceleration of that body. We have x(t) =( * l ( * ) ,{x X(t)= * 12{t),x ( t ) 2,(t),x !*(«)) 3(t))

(1.11) (l.H)

*(*) = |^xx( (t i))

(1.12) (1-12)

xx ( i ) = ^x( f = * + b + o(b) = (1 + 6F 0 )i + o{b).

(1.27)

The Lie-Cartan relations are (we drop the summation signs as well as the o{)'s) i)

[Llt Lj] - eMXkdiejmnXmdn = ^iklXk^jlndn

= -(SijSkn — XjOi

- ( i H j) =

~ tjklXktilmdn

- Sin6kj)xkSn

(1.28)

=

+ {Sij5kn - 6jn5ki)xk6n

—

XiUj ,

£kij[Li, Lj\ = —2ekjiXjOi = 2Lk , e We exclude the case m = 0.

Preliminary

notions of Analytical Dynamics.

Newton's Equations.

9

numbers rha etc. viz. fh,A

m

= ma = mass of the particle A.

The accelerations V-B^A and &A^B are, following again the stroke of genius of Newton, parallel to each other and in opposite direction. We have m (j4) a B _> A + m (B) a A ->B = 0

(2.2)

(third Newton's Law) and mMa.A =FA

+ FC->A ■■■) + mSAh

= (FB^A

(2.3)

is the resulting force exerted upon the particle A (second Newton's Law). The last equation can be viewed as a definition of force. ¥ A denotes the global force exerted upon the particle A, g denotes the acceleration of gravity. Notice that in case the third Newton's Law is not satisfied, we are not able to define the masses of the particles. Then the second Newton's Law has to be replaced by a relation *A = (A ,

(2.4)

where f^ is a function of the positions and velocities of the particle A and all other particles as well as of the time t. It is not a force — it has the dimension [cm sec - 2 ]. We shall, however, often speak about the relation (2.4) as Newton's Equations or Equations of Motion in normal form. We may introduce in the 3-dimensional Euclidean space the Cartesian coordinate system (see Section 1). Then a particle located at some point A has, say coordinates X

= [X^

,%2

,3^3

) ,

the velocity of it has the coordinates

and the acceleration of it

The variables x'' 4 ) , x'" 4 ' and x ' A ' depend on t as on an independent vari­ able. The (2.4) written in terms of these coordinates reads. a,=f

)

= /f)(xW,xW,xMIiM,.,.,(),

4 = 1 , 2 , 3 . (2.5)

The Inverse Variational Problem in Classical

10

Mechanics.

For more on these topics see e.g. the book of E.T. Whittaker*'

3.1

Constraints. Work. The Principle of Least Action. Euler-Lagrange Equations.

The mechanical system can be subjected to constraints. The constraints can be holonomic, i.e. depending only on the position coordinates of the system and time, or anholonomic, i.e. depending in addition on the ve­ locity coordinates of the system. We restrict ourselves to the holonomic constraints. The difference between the number of position coordinates of the system and the number of holonomic constraints we shall call - the number of degrees of freedom of the system. Using the equations of the constraints we may introduce new coordi­ nates, the number of which, n, is equal to the number of degrees of freedom. Those coordinates are independent from each other. W shall denote them q; we have 4A)="VtA){9i,—

9ntt),

z = 1,2,3, . . .

(3.1)

where A = 1 . . . . TV = number of particles. Of course n < 37V.

(3.2)

We are going now to derive the so called Lagrange Equations, starting from the Newton Equations (2.3) (see also (2.5)). We have m{A)x\A)

= FJ;A),

i= 1,2,3,

A=1,...N.

(3.3)

We assume here that Jo ' do not depend on x's. Let us multiply both sides of (3.3) by

v^' where tp\

, 1 „

is given by (3.1). We get

gM«^V=g^.

(3.4)

*' E.T. Whittaker, A treatise on the analytical dynamics of particles and rigid bodies etc., Cambridge, University Press, 1952.

Constraints.

Work. The Principle of Least Action.

Euler-Lagrange Equations. Equations. 11 11

Notice that

^_±(yd^ dqr r 3