Hamiltonian Dynamics
Gaetano Vilasi
World Scientific Publishing
Hamiltonian Dynamics
Hamil~onian
ynamics
Gaetano...
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Hamiltonian Dynamics
Gaetano Vilasi
World Scientific Publishing
Hamiltonian Dynamics
Hamil~onian
ynamics
Gaetano Vilasi ~ ~ ~ v e of r sSalerno ~~y
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World Scientific Publishing Co.Pte. Ltd. P 0 Box 128, Farrer Road, Singapore912805 USA o@ce: Suite lB, 1060 Main Street, River Edge, NJ 07661 UK opce: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-~~fication Data A catalogue record for this book is available from the Britjsh Library.
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L DYNAMICS ~ O ~
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Copyright 0 2001 by World Scientific Pubtishing Co. Pte. Ltd. Ail rights reserved. This book, or parts thereoJ m y not be reproduced in anyform or by any means, e~ecironicor ~ c ~ n i ci ~a l~ u, d ~ ~ g p ~ i ~recording o p ~ ~ornany s , i n f o r ~ ~ istorage on and retrievaf system now known or to be invented, without written permisxion from the P ~ b l ~ s ~ r ,
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Preface
There are many books on classical mechanics. They can be roughly divided into two classes. One contains books which, in order to be more accessible, are sometimes leas transparent with respect to the underlying theoretical structures; the other contains books giving the general, analytical and geometrical, structures of classical mechanics, In the latter, due to greater complexity of the mathematical tools involved, it is however difficult to find books suitable for teaching the subject to graduate students, often because they do not contain a teaching proposal but rather they appear to be written by authors for their colleagues. This book is intended to belong to the second class, but without the shortcoming that WFLSjust mentioned. Part I, Part I1 and, partially, Part I11 are intended to be a teaching proposal suitable for graduate students. Thus, they are written from the point of view of a student but with the aim of giving a general ~ d e r s t a n d i n gof the theory. Part IV, instead, is concerned with the current research topic of completely integrable field theories and could be even used independently of the others. This part is not written with the same pedagogic spirit that animates the previous chapters and probably it would have required additional chapters concerning the Lagrangian and the Harniltonian formulation of field theory. However, a pedagogic treatment of the last subject would have taken too much space-time. I am grateful to my friends: Giuseppe Marmo, for the invaluable help in reading the manuscript, criticism and important suggestions and for the very many years of common efforts toward an understand~ngof complete integrabil~tyin field theory. V
Preface
vi
Giovanni Sparano and Alexandre M. Vinogradov, for criticism and several remarks in special topics. Vladimir S. Guerdjikov, Mario Rasetti and Geoffrey L. Sewell who strongly encouraged me to write this book. I also wish to thank my friends: Sergio De Filippo and Gianni Landi for many years of collaboration. Finally, I wish to thank Roberto De Luca whose expertise, both in Physics and English, allows me to offer a readable final version. Of course, I am the only one responsible of remaining mistakes. G.Vilasi Salerno, April 1998
Introduction
A large amount of scientific activity has been devoted to the asymptotic and geometrical analysis of dynamical systems. This interest was born towards the close of the nineteenth century after the publication of Les Me‘thodes Nouvelles de la Mkcanique Ce‘leste in 1892, by Henri Poincarh. The proposed new methods insist on interpreting differential equations as integral curves of vector fields on manifolds, and to analyze the problems concerning long term stability of a dynamical system, for instance, the solar system, by studying their topological properties. Henri Poincarh was the first to recognize the extraordinarily complicated behavior (today known as chaos) of orbits in the vicinity of a separatrix, whose analysis needed the introduction of entirely new mathematics. Poincarh’s suggestion lies in the origin of modern topology, with its powerful tools consisting of tangent and cotangent bundles, differential forms, exterior algebra and calculus, homology and cohomology. All such notions are usually associated with general relativity, string theories, or gauge theories, and not with their main source, Classical Mechanics. On the other hand, in the last few decades there has been a renewed interest in completely integrable Hamiltonian systems, whose concept goes back to the last century, and which, loosely speaking, are dynamical systems admitting a Hamiltonian description and possessing sufficiently many constants of motion, so that they can be integrated by quadratures. This interest, which previously had considerably weakened, for the really exiguous number of physically prominent examples of completely integrable dynamics with finitely many degrees of freedom, revived with the discovery of vii
viii
Introduction
k ~ e ~ r e s e n t a ~and ~ oInverse n Scattering ~ e t ~ The o ~ Lax . Representation made possible the solution of many problems of remarkable physical interest as the ones described, for instance, by ~ o r t e w e g - ~Vries, e s~ne-Gordo~, no~~zne~r Schrodinger equations and l'oda's lattice. All such dynamics are Hamiltonian dynamics on infinite dimensional weakly-symplectic manifolds, on which the classical Liouville criterion of integrability can be extended in terms of mixed tensor fields with vanishing Nijenhuis torsion. Peculiar, in the approach to integrability in terms of invariant mixed tensor fields, i s the direct construction of abelian maximal algebras of symmetries, leaving out the associated groups, so that only the algebraic aspect of the traditional methodology is reproduced, An exemplary case is given by the Kepler dynamics, in which both the integrabi~ityand the degeneration, classical and quantum, are inferred by identifying the correspon~inginvariance groups (SO(3)and SO(4)). On the other hand, it is just by means of the modern theory of Hamiltonian system, based on the analysis of symmetries, that an algebraic group approach arises from the analysis of Lax dynamics. This approach arises from the observation that Hamiltonian dynamics, on the orbits of the coadjoint representation of a Lie group endowed with their natural symplectic structure, are Lax dynamics, provided that an internal product, invariant under the adjoint action, exists in the Lie algebra. The group approach analysis, even if from one side has the merit to be constructive, on the other, is not fit t o investigate, Q priori, the possible integrability of a given dynamics. In these lectures we shafl look at this geometric approach to the study of Hamiltonian dynamical systems, specially in connection with the kinds of problems which arise in completely integrable 2-dimensional field theories. It would have been interesting to include a chapter concerning nonintegrable dynamics, an essential topic for the theory of particles dynamics in accelerators. However, this last one is a vast subject and goes beyond the purposes of this book. We will spend however a few words to delineate the idea of invariant tori in phase space, to define and illustrate the structures for organizing dynamics and the origin of chaotic orbits in nonintegrabb systems. , due to a formal Finally, I also hope to lessen the i m p r ~ i o n sometimes approach, that classical mechanics is a closed subject with no mysteries left to explore.
Contents
Preface Introduction I ~~~~~~1 ~ e c h a n i ~ 1 The L a ~ a n ~ i Coord~ates an 1.1 A Primer for Various Formulations of Dynamics 1.1.1 The Newtonian formulation of dynamics lS.2 A discussion on space and time 1.1.3 Inertial frames revisited 1.1.4 The Lagrangian formulation of dynamics 1.1.5 The Hamiltonian formulation of dynamics 1.2 C o ~ t r a ~ t s 1.3 Degrees of F’reedom and Lagrangian ~oordinates 1.4 The Calculus of Variations and the Lagrange Equations 1.4.1 Historical notes 1.4.2 A digression on the variation methods in problems with fixed boundaries 1.5 Remarks on Lagrange’s Equations 1.5.1 Equivalent Lagrangians 1.5.2 Dynarnical similitude 1.5.3 Electrjcal circuit analysis 2 Hamiltonian Systems 2.1 The Legendre Transformation 2.2 The H ~ i ~ t ~o qnu a ~ o n s 2.2.1 From Lagrange to Hamilton equations ix
V
vii
1 5 5 5
6 11 12 14 16 20 23 24
28 35 35 35 37 39 40
41 41
X
Contents
43 2.2.2 From Hamilton to Lagrange equations 45 2.2.3 Remarks on Hamilton's equations 47 2.3 The Poisson Bracket and the Jacobi-Poisson Theorem 47 2.3.1 The state space 48 2.3.2 The phase space 48 2.3.3 First integrals 50 2.3.4 The Poisson bracket 53 2.3.5 The Jacobi-Poisson theorem 57 2.4 A More Compact Form of The Hamiltonian Dynamics 57 2.4.1 General Hamiltonian dynamics 59 2.4.2 Jacobi-Poisson dynamics 59 2.4.3 More on the Poisson bracket 61 2.4.4 h r t h e r generalizations of the Jacobi-Poisson dynamics 62 2.5 The Variational Principle for the Hamilton Equations 65 3 ans sf or mat ion Theory 3.1 Canonical, Completely Canonical and Symplectic Transformations 65 65 3.1.1 Canonical transformations 69 3.1.2 A general class of canonical transformations 71 3.1.3 Completely canonical transformations 73 3.1.4 Symplectic transformations 73 3.1.5 Area preserving tr~sforInations 75 3.1.6 Volume preserving transformation 75~ 3.2 A New Characterization of Completely Canonical ~ a n s f o r m ~ t i o 80 3.3 New Characterization of Sympletic Transformation 81 4 The Integration Methods 81 4.1 Integrals Invariants of a Differential System 85 4.2 A Primer on the Lie Derivative 89 4.3 The Kepler Dynamics 90 4.3.1 The Laplace-Runge-Lenz vector 92 4.3.2 The hydrogen atom 95 4.4 The Hamilton-Jacobi Integration Method 99 4.4.1 Remarks on the Hamilton-Jacobi equation 101 4.5 The H a m ~ ~ t o n - ~ aEquation co~ for the Kepler Potential 105 4.6 The Liouville Theorem on the Complete Integrability 105 4.6.1 Reduction 107 4.6.2 The Liouville theorem 114 4.6.3 Remarks on the Liouville theorem
Contents
4.6.4 Action-angle coordinates 4.6.5 The action-angle coordinates for the Harmonic Oscillator 4.6.6 The Kepler dynamics in action-angle variables 4.6.7 The perturbations of integrable systems and the KAM theorem 4.6.8 The Poinear6 representation I1 Basic Ideas of Differential Geometry 5 M ~ f o l d and s Tangent Spaces 5.1 Differential Manifolds 5,2 Curves on a Differential Manifold 5.3 Tangent Space a t a Point 5.3.1 Tangent vectors to a curve on a manifold 5.3.2 Tangent vectors to a manifold 5.4 A Digression on Vectors and Covectors 5.4.1 Vector space 5.4.2 Dual vector space 5.5 Cotangent Space at a Point 5.6 Maps Between Manifolds 5.7 Vector Fields 5.7.1 Holonomic and anholonomic basis of vector fields 5.8 The Tangent Bundle 5.9 General Definition of Fiber Bundle 5.9.1 More on the tangent bundle 5.9.2 Analysis of two bundles with 5‘’ as base manifold 5.10 Integral Curves of a Vector Field 5.11 The Lie Derivative 5.12 Submanifolds 5.12.1 The Frobenius theorem 6 Differentiai Forms 6.1 The Tensors 6.1.1 The pcovectors 6A.2 The exterior product 6.1.3 The metric tensor on a vector space 6.2 The Tensor Fields 6.2.1 The Lie derivative of a tensor field 6.2,2 The differential pforms 6.2.3 The exterior derivative
xi
115 116 118
120 121 125 129 129 131 133 133 134 135 135 136 138 139 140 140 143 144 145 146 148 151 154 155 159 159 164 164 166 167 167 170 172
xii
6.2.4 Closed and exact differential forms 6.2.5 The contraction operator ix 6.2.6 A different procedure 6.2.7 A dual characterization of holonomic and anholonomic basis 6.3 The Metric Tensor Field on a Manifold 6.3.1 Killing vector fields 6.3.2 Maximally symmetric manifolds 6.3.3 The Levi-Civita covariant derivative 6.3.4 The Riemann tensor field 6.3.5 The Ricci tensor and the scalar curvature 6.4 Endomorphisms Associated with a Mixed Tensor Field 6.4.1 The Nijenhuis bracket of two mixed tensor fields 7 Integration Theory 7.1 Orientable Manifolds 7.2 Integration on Orientable Manifolds 7.3 p-Vectors and Dual Tensors 7.4 Metric o Volume = Hodge Duality 7.5 Stokes Theorem 7.6 Gradient, Curl and Divergence 7.7 A Primer for Cohomology 7.8 Scalar Product of Differential p-Forms 7.8.1 Exterior codifferential 7.8.2 Hodge theorem 8 Lie Groups and Lie Algebras 8.1 Lie Groups 8.1.1 Local Lie groups 8.2 Building of a Lie Algebra F'rom a Lie Group 8.2.1 Lie algebras 8.2.2 Left invariant vector fields 8.2.3 The adjoint representation of a Lie group 8.2.4 The coadjoint representation of a Lie group 111Geometry and Physics 9 Symplectic Manifolds and ~ a r n ~ ~ tSystems oni~ 9.1 Symplectic Structures on a Manifold 9.2 Locally and Globally Hamiltonian Vector Fields 9.2.1 Integral curves of a Hamiltonian vector field
Contents
173 173 178 180 181 182 183 184 188 189 190 192 195 195 196 197 200 202 205 206 208 208 210 211 211 213 213 213 214 219 225 227 231 231 233 233
Contents
9.3 H ~ i l t o nFlows i~ 9.3.1 Lie algebras of ~ ~ i l t o n i vector a n fields and of Hamilton functions 9.4 The Cotangent Bundle and Its Symplectic Structure 9.5 Revisited Analytical Mechanics 9.6 The Liouville Theorem 9.6.1 The construction of the action-angles coordinates 9.7 A New Characterization of Complete Integrability 9.7.1 From the Liouville integrability to invariant mixed tensor fields 9.8 Applications 9.8.1 A recursion operator for the rigid body dynamics 9.8.2 A recursion operator for the Kepler dynamics 9.9 Pois~n-N~enhuis Structures 9.9.1 Compatible Poisson pairs 10 The Orbits Method 10.1 Wueed Phase Space 10.2 Orbits of a Lie Group in the Coadjoint presentation 10.3 The Rigid Body 10.3.1 The space and the body angular velocities 10.3.2 The space and the body angular momenta 10.4 Rigid Body Equations 11 Classical Electrodynamics 11.1Maxwell’s Equations 11.2 Geometrical Identification of Fields on @ 11.3 Geometrical Ide~tificationof Electroma~eticField in Space-Time 11.3.1 The vector potential and the gauge transformation 11.3.2 Constitutive equations 11.3.3 The wave qua ti on 11.3.4 Plane waves IV Integrable Field Theories 12 KdV Equation 12.1 An Existence and U ~ ~ q u Theorem e n ~ 12.2 Symmetries 12.2.1 Space-time symmetries 12.2.2 Biicklund tra~formation
xiii
235 236 239 241 248 250 252 259 261 261 264 267 267 271 271 280 288 289 291 291 295 295 297 299 300 301 303 304 307 311 311 314 314 315
xiv
12.3 Conservation Laws 12.3.1 Lax represent at ion 12.3.2 The inverse scattering method 12.4 KdV as a Hamiltonian Dynamics 12.5 KdV as a Completely Integrable Hamiltonian Dynamics 13 General Structures 13.1 ~ o t a t i o nand Generaiities 13.1.1 Backward to KdV 13.2 Strongly and Weakly Symplectic Forms 13.3 Invariant Endomorphism 13.3.1 Dynamical invariance 13.3.2 Nijenhuis torsion 13.3.3 Bidimensionality of eigenspaces of T (KdV and sG) 13.4 Invariant Endomorphisms and LiouvilIe’s Integrability 13.5 Recursion Operators in Dissipative Dynamics 13.5.1 The burgers hierarchy 14 Meaning and Existence of Recursion Operators 14.1 Integrable Systems 14.1.1 Alternative Hamiltonian descriptions for integrable systems 14.1.2 Recursion operators for integrable systems 14.1.3 Liouvil~~Arnold integrable systems 15 Miscellanea 15.1 A Tensorial Version of the Lax Representation 15.1.1The LR of the harmonic oscillator as a parallel transport condition 15.1.2 The A-invariant tensor field for the harmonic oscillai;or 15.1.3 The A-invariant tensor field for KdV 15.1.4 The A-covariant tensor field for KdV 15.2 Liouville Integrability of Schrodinger Equation 15.2.1 Comparison with the nonlinear Schrodinger equation 15.3 Integrable Systems on Lie Croup Coadjoint Orbits 15.4 Deformation of a Lie Algebra 15.4.1 Deformation 15.4.2 Lie-Nijenhuis and exterior-Nijenhuis derivatives 16 Integrability of Fermionic Dynamics 16.1 Recursion Operators in the Bosonic Case
Contents
318 320 322 328 330 337 337 340 342 343 346 347 347 347 352 353 359 360 361 363 365 369 369 372 373 374 374 376 380 385 386 386 387 389 389
Contents
16.2 Graded Differential Calculus 16.3 Poisson Supermanifold 16.3.1 Super ~ ~ j e n h utorsion is A Lagrange: A Short Biography B Concerning the Lie Derivative C Concerning the Kepler Action Variables D Concerning the Reduced Phase Space E On the Canonical D~fferenti~~ 1-Form F Concerning Rigid Body Equations G The Gelfand-Levitan-Marchenko Equation Bibliogr~phy Index
xv
390 394 399 401 404 406 409 410 413 414
427 437
Part I
Analytical Mechanics
The aim of this part is a self-contained treatment of Classical Mechanics in an advanced formulation. Many topics relevant to applications will not be treated, since they can be found in excellent t e ~ t b o o k s Our treatment is inspired by two important classical textbooks, Lezioni di Meccanica Razionale by T. Levi-Civita and U. Amaldi, and The Analytical Foundations of Celestial Mechanics by A. Wintner.36*58 Definitions will be given for a particle, i.e. for a body whose space dimensions can be neglected with respect to the dimensions of the space in which it moves, and naturally extended to systems of particles and to continuous systems Cfields). The simplicity of the formal extension from systems of particles to fields, and the difficulties for a rigorous extension, will limit the treatment to system of particles.
3
.
~
~
Chapter 1
The Lagrangian Coordinates
1.1
A Primer for Various Formulations of Dynamics
1.1.1 The Newtonian formulation of dynamic8 The Newton formulation of classical mechanics is based on three principles: The First Principle or Galilei's Principle of Relativity There exist special observers with respect to which a particle not being acted upon by any force moves with a rectilinear motion. Such an observer will be called an inertial observer or an inertial frame. He can define the time in such a way that the motion appears to be also uniform. Any observer moving with rectilinear and uniform motion with respect to an inertial observer is an inertial observer too.
*Galilea Galilei was born in Pisa on February 15, 1564 and died in Arcetri (Florence), Italy on January 8, 1642. The author of Dialog0 dei Massimi Sistemi (Landini ed. Florence, IS%?), and Diswrsi e dimostmzioni matematiche intorno a due nuove scienze attenenti alla m-niccr e i movimenti locali (Leida, 1638), Galilei is considered aa the inventor of the dynamica. 5
The Lagmngian Coordinates
6
The Second Principle or Newton'sf Second Law 0
In an inertial frame, once the time has been chosen as specified before, the motion of a particle is governed by the differential equation: +.
ma'=F, where m is the mass of the particle} a" its acceleration and 3 the force acting on the particle. It is an experimental observation that forces acting on a particle can change with time t or with the position r' and the velocity 17 of the particle. Thus, the force is represented as a vector function of variables ( t ,r', ."), and the second law is more explicitly written in the form
Third Principle 0
The total momentum 2 and the total angular momentum isolated system of particles do not change in time.
L' of
an
dp - -- 0 , d- =z o dt dt In many elementary textbooks a statement can be found, namely: that the first principle is a particular case of the second principle when the force vanishes. So expressed, the statement is wrong. Actually, it suggests that the distinction between kinematics and dynamics is artificial, and that inertial frames can only be defined dynamicalfy, as the following discussion well shows.
1.1.2
A discussion o n space and time
In Newton's principles, at least three concepts are given as natural and absolute, namely: we are able to state that no forces act on a body; tIsaac Newton was born in the castle of Woolsthorpe, a little village to the south of Grantham in Lincolshire, England, on Christmas 1642, eleven months after the death of Galilei. He died in a suburb of London in 1727. The author of the celebrated Philosophiae Natvmlis Principia Mathematzca (London, f681), in which the foundations of mechanics and mathematical physics are exposed, Newton invented, by himself, the main tool of investigation; i.e. the differential calculus. On his grave, in Westminster Abbey, it is written: Sabi gratulentur Mortales tale tantumque extitisse Humani Generis Decus.
A Primer for Various Formulations of Dynamics 0 0
7
we have a notion of an absolute straight line; we have a notion of an absolute time as “flowing uniformly,” to quote Newton.
Concerning the absence of forces, it is evident that Newton’s definition of a free body as lLabody far away from any other body in the Universe,” presumes that all forces decrease with distance. Thus, Newton was only thinking of gravitational forces. It is clear that it was an attempt by Newton to abstractly generalize the definition of an inertial observer given by Galilei, who defined inertial as a frame which is stationary with respect to the “fixed stars”. However, after Mach, we are aware that the inertia is related to the surrounding Universe, so that the more pragmatic definition given by Galilei is much more acceptable. Galilei recognized, as a result of clock measurements, that approximately free bodies move in an approximately inertial frame, along approximately straight lines with approximately constant velocities. His tools were an inclined plane to slow the fall, a water clock to measure its duration, and a pendulum to avoid rolling friction.
“Inoltre, k lecito aspettarsi che, qualunque grado d i velocitd si trovi in un mobile, gli sia per sua natura indelebilmente impresso, purchk siano tolte le cause esterne di accelerazione e di ritardamento; il che accade soltanto nel piano orizzontale; infatti nei piani declivi k d i gid presente una causa di accelerazione, mentre in quelli acclivi d i ritardamento; infatti, se k equabile, n o n scema o diminuisce, ne tanto meno cessa.” (G. Galilei, Discorsi e dimostrazioni matematiche intorno a due m o v e scienae, Terzo giorno) Newton was aware that Galilei’s conclusion might be only approximately true, but he waa very impressed by the existence of numerous coordinate transformations leading to coordinate systems, in which the Galilei description cannot be given. Then he elevated the Galilei approximate empirical discovery to the position of a rigorous principle, the inertia principle, and stated that absolutely free bodies move, in an ideal inertial frame, with absolutely constant velocities along perfectly straight lines.
“Absolute space, in its own nature and with regard to anything external, always remains similar and unmovable. Relative
8
The Lagmngian Cooniinates
space is some movable dimension or measwe of absolute space, which our senses determine by its position with respect to other bodzes, and is commonly taken f o r absolute space.” From that, Newton also defines an absolute time congruence. As far as the notion of a straight line is concerned, we need a structure of vector space, and we know that, on the same space, we can give different vector space structures. Thus, the notion of straight line is observer-dependent. The same can be said about Newton’s allusion to time, for a rate of flow can be recognized as uniform only when measured against some other rate of flow taken as standard. In other words, we need a c o m p a ~ s o n$ynam~cs. Even if, from it theoretical point of view, the law of inertia should allow us to get an accurate determination of congruent intervals, the impossibility to observe freely moving bodies, due to the presence of frictional and gravitational forces, suggested to define a frame to be Galilean if a perfectly rigid sphere rotating without friction about an axis, fixed in the frame, has a uniform or constant rate of rotation. Here, constant is understood as measured in terms of the standards of time congruence, defined by a freely moving body under the ideal conditions required by the principle of inertia. The previous definition is still far from perfect, but at least, is coherent with a definition of time congruence based on the principle of causality which, following Weyl, can be given as follows: “If an absolutely isolated physical system reverts once again to exactly the same state as that it was at some earlier instant, then the same sequence of states will be r e ~ a t ~indtime, and the whole sequence of events will constitute a cycle. I n general, such a system is called a clock. Each period of the cycle lasts equaZly long.”
We now come to Einstein’s definition of Galilean frame, as implicitly given in special relativity: The velocity of a light ray passing through a n inertial frame will be the same regardless of the relative motion of the luminous source and frame, and regardless of the direction of the ray.
Remark 1 Actually, this property of the light defines the conformal group which contains the Lorentz group as a subgroup. The optical definition presents a marked superiority over those of the pre-relativistic physics. While, with earth’s rotation, we had to assume the
A Primer for Various Formulations of Dynamics
9
correctness of Newton’s law to determine the corrections required by earth’s breathing and by the friction of the tides, the new definition is just based on the most highly refined experiments known to physicists. The relevance of Einstein’s definition lies in the consequences which follow from the attempt to correlate space and time measurements between two inertial frames in relative motion. The concepts of space and time congruence lose the classical attributes of universality given to them by the Newtonian physics. It is then found that congruence can only be defined in a universal way (independent from the observer) when we consider the extension to the 4-dimensional space-time.
“And now, in our time, there has been unloosed a cataclysm which has swept away space, time and matter, hitherto regarded as the firmest pillars of natural science, but only to make place for a view of things of wider scope, and entailing a deeper vision.” H. Weyl (Space, Time and Matter). Wonderful as they may appear, Einstein’s previsions have thus far been verified in every detail. After our short discussion on
the Galilean and Newtohian principles of relativity, the Einstein special principles of relativity, it appears useful, after 115 years from the appearance of Mach’s after 83 years from Einstein’s to also discuss
and
The Einstein general principles of relativity. The principle is assumed after the results of the mentioned Galilei’s experiments on free falling bodies, later confirmed by Eiitvos’ (1889) and Dicke’s
(1967) measurements, which suggest that, at any point in space-time, a reference frame can be chosen, henceforth called locally inertial frame, such that, in a sufficiently small neighborhood of the point, the motion of a free falling particle is described by the equation
d2ta
-= o , dr2
where the t ’ s are the coordinates in the locally inertial frame, and r is any parametrization of the curves (principle of equivalence).
The Lagmngian Coordinates
10
Thus, by assuming that a differentiable map exists between the coordinates xa in the laboratory b a m e and the coordinates J in the locally inertial frame, by the above equation we obtain
o=-d
dr
3
at" =-c ( ) d", ) dJ" dr
3
p=o
dxp -(axp dr
3
p=O
so that, multiplying by dzx/a 0, says that the Newtonian motion will be bounded in space only if the total energy E is negative.
The above analysis can be also carried out, of course, by means of Lagrange's equations. The reader is invited to do it himself.
2.3 2.3.1
The Poisson Bracket and the Jacobi-Poisson Theorem The state space
The solution of the Lagrangian equations, describing the motion of a given arbitrary holonomic system S with n degrees of freedom and Lagrangian coordinates q1, q 2 , . . . ,qn, is given by n functions of time qh=qh(t),
V h E {1,2,..*,n},
(2.12)
representing, at each time t , the position of S . Their derivatives ~h
= &(t),
V h E { 1 , 2 , . ..,n},
(2.13)
will give at each time t the corresponding velocities. From Cauchy's theorem view point, the state of S is characterized by the 2n parameters ( q / q ) . In this way, it appears convenient in the analysis of the motion, to use a hyperspace representation, considering the 2n parameters q and q as Cartesian coordinates of a 2n dimensional space E . Since each point of this space represents a state { P , v p } of S, E is called the state space. The motion in E , or to be more precise, the sequence of states in E will be represented by the parametric equations (2.12) and (2.13).
Harniltonian Systems
48
2.3.2
The phase space
An analogue geometrical representation is introduced for the canonical coordinates p , q , by considering them as Cartesian coordinates of a 2n dimensional Euclidean space @ called, after Gibbs, the phase space. In this space any solution,
of the canonical system (2.8) is represented by a (integral) curve often called, regarding t as a measure of time, a trajectory. In this way, we will have ooZn trajectories corresponding to possible choices of the 2n arbitrary constants, from which the general integral of the canonical system depends. 2.3.3
First integrals
Let us consider the following canonical system again:
As for any first order differential system of equations, any function f, such that the relation f ( p / q / t )= constant
(2.14)
is identically satisfied for all solutions of the system is called a f i r s t integral, or shortly, an integral of the canonical system. In other words, if
denotes an arbitrary solution of the given canonical system, representing the parametric equations of an integral curve in the phase space, the function f in the left hand side is such that f(p(t)/q(t)/t= ) constant.
For this reason, the function f is also called an invariant.
The Poisson Bmcket and the Jacobi-Poisson Theorem
49
Of course, the function f may take on different constant values for different trajectories in the phase space. More precisely, denoting with po,qo,to, the corresponding initial values of p , q, t , the constant must be chosen to coincide with fb0,qo, to). Let us recall that, if the Lagrangian does not depend explicitly on time t , the total energy is a first integral:
a.c - L = constant.
E ( q / v / t )= X
V h G
h
As for the equivalence between any Lagrangian system and the corresponding canonical system, it is expected that the Hamiltonian function, if not depending explicitly on time t , is a first integral of the canonical system. It is interesting to prove the above statement directly, since the result follows from a general identity which will turn out useful later. The rate of change of any function f, depending on the 2n coordinates ( q / p ) and the time t , under the evolution described by Eq. (2.8), is given by
(2.15) Thus, for f = 31, we have
- _ -- a31 dt
at
*
Therefore, if the Hamiltonian does not depend on time t , the Hamiltonian function 31 defines, for the canonical system, a first integral which can be also called the generalized integral of the energy. Simple first integrals exist when the characteristic function 3t does not depend on some q's. For instance, if a3c/aqr= 0, from Eq. (2.8) it follows: p , = constant
These types of integrals also are called generalized momenta, as they coincide with the ones given by Lagrangian systems for cyclic coordinates. In this case, in fact, if the Lagrangian function does not depend, for instance, on qrl
Hamiltonian Systems
50
the same is true for
and for their inverse oh = a h ( q / p / t )
, v h E { 192, . . .,n ) .
It follows also that the characteristic function 31 will not depend on qr. Vice versa, if q,. does not appear in the characteristic function 3 1 ( p / q / t ) of a canonical systems, it does not appear in qh =
2.3.4
az
-, 'dh E { 1 , 2 , . . . ,n} , aph
The Poisson Bracket
Equation (2.15) can be written in the following form:
where the bracket of any two functions, f and g , defined by (2.16)
is called the Poisson$ Bracket of f and g. The Poisson Bracket satisfies the following identities:
antisymmetry {f,g) = -{g,fI,
(2.17)
{f,(9,h ) ) + (9,{ h ,f)) + { h ,{f,g)) = 0 >
(2.18)
Jacobi identity
$Sirneon Denis Poisson, author of the %it2 de mdcanique (Paris, 1831), was born in Pithiviers (Loiret) in 1781 and died in Paris in 1840. He was a professor of mechanics at the Sorbonne University.
The Poisson Bracket and the Jacobi-Poisson Theorem a
51
derivation
{f,9 + h ) = if, 9) + if, h ) , If,Sh) = {f,d h + d f ,h ) 9
(2.19)
{h,c} = 0 v c E R . Properties (2.17) and (2.19) follow easily from the definition, and their proof is left to the reader. Property (2.17) simply expresses the antisymmetry of the bracket, while properties (2.19) simply say that the Poisson bracket has a natural compatibility with the usual associative product of functions, on which it acts as a derivative. The Jacobis identity also follows directly from the definition and the reader can check it by “brute force.” An elegant proof can be given as follows: Let us observe that the left hand side of Eq. (2.18) is a sum of terms, each one being a product .of first partial derivatives of two of the three functions f,g , h with a second partial derivative of the remaining function like
Therefore, the Jacobi identity will be proven if we are able to show that the left-hand side of Eq. (2.18) does not contain any second partial derivative. For this purpose, let us introduce, for any function f , the first order differential operator Xp defined by xfg = { f , & J1 )
which will be called the Hamiltonian vector field associated with f. The explicit expression of X f is given by
h
$Karl Gustav Jacobi was born in Postdam in 1804 and died in Berlin in 1851. He is universally known for the investigations on elliptic function, for his papers on determinants and particularly the Jacobian determinant, for the Jacobi identity, which is basic almost everywhere in physics and mathematics, and for the Hamilton-Jacobi theory, which was a starting point for quantum theory. Most of the results of the researches are included in his Vorlesungen uber Dynamik.
Hamiltonion Systems
52
In terms of these operators, the left hand side of Eq. (2.18) can be handled as follows:
Therefore, the following remarkable relation holds:
{f!(9, h ) ) 4-(9, {h,f)) + {h9 if,9)) = [Xf, Xglh - X{f,g}h
1
(2.20)
where the bracket [X,Y ]denotes the commutator of the differential operators X and Y . The final observation is that (a) the last term Xif,,)h does not contain second derivatives of It. (b) the commutator [X,Y ]of two first order differential operators is again a first order differential operator. Indeed, if
a
X =xXi(z)-
axi
i
a
and Y = x Y i ( ( z ) i
denote two first order differential operators, we have
i
a ij
ij
ij
a x j axi
The Pobson Bmcket and the Jacobi-Poisson Theorem
53
As a consequence of properties (a) and (b), the left-hand side of Eq. (2.20) does not contain any partial second order derivatives of the function h. The same is obviously true for the functions f and g. Therefore, the left-hand side of Eq. (2.20) does not contain any second partial derivative. The conclusion is that the s u m of all terms in the left-hand side of Eq. (2.20) is vanishing.
2.3.6
The Jacobi-Poisson theorem
Let us suppose now that, for the canonical system
8%
ZPh
{I
= --
Zqh=
aqh
’
a3c aph ’
V h E {1,2,...,n},
two first integrals are known, namely, f and g. These first integrals satisfy the following relations:
af + {f,3t} = 0 , at
g + {f,3t}
=0.
It is easy to prove that the Poisson bracket { f , g} of f and g is also a first integral. As a matter of fact, let us first observe that
Then, by using the previous relation and the Jacobi identity,
Hami~ton~an Systems
54
we can write
Therefore,
- + {f,31} = 0
[:
+
at (g,%} = 0
*a { f 1 g } + { { f , g } , ? i } = 0. at
Of course, the Jacobi-Poisson theorem will not always give new first integrals, since their number is bounded to be 2n - 1, where n is the number of degrees of freedom. In some cases, in fact, the Poisson bracket of first integrals simply reduces to a function of them or to a numeric constant. Two functions, f and g, with vanishing Poisson bracket { f , g} = 0 , are said to be in involution.
Problems 1. Show that, if one of the functions f and g coincides with a momentum or a coordinate, the Poisson bracket simpIy reduces to a partial derivative
As a particular case, notice that
where 6 h k is the Kronecker delta. 2. Evaluate the Poisson bracket of the Cartesian components of the momentum $and the angular momentum I"= ? A $of a particle.
Solution
The Poasson Bracket and the Jacobi-Poisson Theorem
55
so that
The remaining brackets follow from a cyclic permutation of the indices 5,y, z. Finally, denoting with ( p 1 , p 2 , p 3 ) and ( Z 1 , 1 2 , / 3 ) the components of p' and respectively, we write
17
3 {li,Pj}
= C e i j h ~ ,h v i , j E { 1 , 2 , 3 } , h=l
where Eijh is the Levi-Civitav tensor densaty defined by
if i, j, h is an even permutation of 1 , 2 , 3 if i , j , h is an odd permutation of 1,2,3
in the other cases; i.e., if two indices coincide. 3. Show that
4. Evaluate the Poisson bracket of the components of the angular momen-
tum of a particle between them by using only the algebraic properties
(2.19). Solution
TTuIlio Levi-Civita was born in Padova in 1873 and died in Rome in 1941. He obtained his degree at Padova University in 1894. He was a professor of mathematical physics, at the age of 24 years, at Padova University, where he taught until 1919. In this same year he moved to Flome University. In 1938, withthe introduction of racial fascist laws, he was removed from the chair and his now classical books, including the Lezioni di Meccanica Razionale (1929)and Lezioni di Calcolo Differenziale Assoluto (l923),was interdicted. Fortunately, thanks to Whittaker, the last had been translated in English and published on 1926 by Blakie & Son. He died j u s t in time to avoid to be forced also to hide because of racial persecutions.
Hamiltmian Systems
56
Since the momenta and the coordinates of different particles are independent quantities, it is easy to verify that the resulting formulae of the previous problems still hoid for the total momentum 9 and for the total angular momentum L' of an arbitrary system of particles. 5. Prove that for any scalar11 function p of the coordinates and momenta of a particle, the following relation holds:
Solution
A scalar function depends on the vectors r' and p'only by means of the combinations r2 = r'. r', p 2 = 3 .@ and r'. p. Therefore, we can write
acp p',
acp ap-- -2g+ ar' ar2
a(F.p)
The relation (I,=,p} = 0, then follows by applying the definition of Poisson bracket (2.16). The same relations hold for the remaining compon~ntsof ( so that, for any scalar function cp of the coordinates and momenta of a particle, we can write
{L,cp)
= U y , 'PI =
{LPI = 0
*
6. Prove that for any vector function fof coordinates and momenta of a particle, the following relation holds:
(Iz,
f)= n' A $,
where ii is the unit vector along :he z axis. Analogous formuIae hold for the remaining components of t .
Solution Any vector function f o f ?and @canbe written in the form f = cplr'+ 'pzp'+ cps(r'A $, where cp1, p2, cp3 are scalar functions. Thus, by using the algebraic properties of the Poisson bracket, we finally obtain the soiution to the problem. IIHere, scalar and vector functions are understood with respect to the rotation group SO(3).
57
A More Compact Form of the Hamiltonian Dynamics
2.4
A More Compact Form of the ~
~
i
lDynamics t o ~
~
Let us start by considering a Hamiltonian system with n = 2 degrees of freedom. By ~troducingthe cofumn vectors
and the skew-symmetric matrix
.=(; -;),
(2.21)
Hamilton's equations can be written in the form
or in components, as follows: (2.22)
where the s u m over the index k is understood. The Poisson bracket can then be written as follows:
where the bracket .) denotes the Euclidean scalar product. Of course, the previous nota~ioncan be also used for a H ~ i ~ t o n i system an with n degrees of freedom. In this case, the matrix E: is given by (a,
(2.23)
where 0 and I denote the n x n nu1 and identity matrices. 2.4.1
Geneva1 ~ a r n ~ l ~ ~o ~~ n~ ua rnn ~ ~ s
Let us now consider a generic dynamics described by an equation similar to Eq. (2.22): (2.24)
Hamiltonaan Systems
58
where the matrix A may depend on the point u. The evolution of a generic function f , defined on the phase space, will be given by
In order to have a Jacobi-Poisson theorem for this type of dynamics, we must require 0
slcew-symmetry
e
Jacobi identity
(V(Gf, AVgf, A V ~ ~ + ( V ( VAVh), g, AVf)+(V(Vh, AVf), AVg) = 0 . In this case, the bracket (V f , AVg) will be called the Poisson bracket of f and 9, and will be denoted with { f,g}A, or simply, if no ambiguity arises, with { f,g}. IR terms of the matrix A, the previous requirements are expressed by the following: 0
s k e ~ s y m ~ e t :r A y = -AT,
Of course, these properties are trivially satisfied by the matrix E. Definition 11. We shall call Hamiltonian a dynamical system with n degrees of freedom, and then with a 2n-dimensional phase space, if it is described by the equation
where the bracket, beyond the usual derivation properties, satisfies the prop erties
A More Compact Form of the Hamiltonian Dynamics
2.4.2
59
Jacobi-Poisson dynamics
Let us finally observe that in the previous definition, no role is played by the even dimensionality of the phase space. Thus, it is natural to define more general dynamics according to the following definition.
Deflnition 12
A dynamics, described by the equations df
dt = {f,'tC)P , with the bracket satisfying the properties
is called a Jacobi-Poisson dynamics.
Of course, a Hamiltonian dynamics is also a Jacobi-Poisson dynamics. 2.4.3
More on the Poisson bracket
We notice that properties expressed by Eqs. (2.17) and (2.18) endow the set 7 ,of differentiable functions defined on @, with a Lie algebra structure.
Remark 6 A Lie algebra A is a vector space endowed with a n internal composition law, denoted by [., and called a Lie bracket, satisfying the properties: a]
[ X , Y ]= - [ Y , X ] , V X , Y E A ,
[X, [Y,211 + [Y,[z, XI] + 12, [ X ,Y ] ]= 0 , V X ,Y ,
E
A.
Examples of Lie algebras are
0 the set of vectors in !R3 endowed with the Lie bracket vector product
[a,
.] given by the
Ham~~t~n Systems ~an
60
0 the vector space of rt x n ~ ~ ~ T i~c ne sd o ~~i te fthe ~i Lie bracket f., +]given by the commutator [M,NJ M N - N M . A geometrical definition of Lie algebra will be given in Part II. Since it can be written in the following equivalent alternative forms:
Xi,,g}h = [XfiXgIh,
Xf{S, h ) = {XfS, h ) +
(2.25) (91
Xfh) ?
(2.26)
the Jacobi identity is equivalent to the following alternative s t a t e ~ e n t ssuggested by Eqs, (2.25) and (2.26), respectively: 0
The map
f
-
Xf =
u 1 . 1
{f,9) +-+ X{fd is a Lie algebra morphism
(3, .I) r--) (XFl I-, -1) (‘1
between (3, -}) and the set of ~ a m i ~ t o n ivector an fields XF endowed with the Lie bracket given by the commutator I-, .I, The operator X f = { f,.} is a derivation of the Poisson bracket. {ml
*
Last statement, as it will be shown in the next subsection, suggests the introduction of more general structures named n-Poisson brackets. For instance, a 3-Poisson bracket on 3 is bracket .} satisfying the following properties: {a,
{fl
Sl h ) =
-{f,
s,
h,9) = -b,f,h ) 1
{f191 {u, v , w } l + { v ,w,if, s,
- (B,w,{f,9,v)}
+ {%V,
{f,9, w ) ) = 0 I
{f,9,h i h z ) = {f,g1hi)hz i-hi{ f ,g1hz) { f , g , c }= 0 v c E R . An algebraic fomulation
We observe that properties expressed by Eqs. (2.17), (2.18) and (2.19) are purely algebraic in nature, so that the following abstract formulation can be introduced.
A Mom Compact Form of the Hamiltonian Dynamics
61
Let M be a Poisson manifold and F the ring of functions defined on it. This means that on M a bracket {-,-} is defined such that (1) it yields the structure of a Lie algebra on T; i.e.
{fld = --{91fL {f,(9,h H f (91 {hl f 1) f {h,~ f 1 g I = l0
1
(2) it has a natural compatibility with the usual associative product of functions, which is
Therefore, we can define an abstract Poisson aZgebra as an associative commutative algebra endowed with a Lie bracket satisfying Eqs. (2.17), (2.18) and (2.19). It is natural to generalize the notion of a Poisson manifold by relaxing condition (2) and requiring only that {f,g} be just a local type operation: support
{f,g ) C (support f ) n (support 9).
The bracket {f,g} is then called a Jacobi bracket and the corresponding manifold a Jacobi manifold. 2.4.4
f i r t h e r generalizations of the Jacobd-Poisson dgnamico
The possibility of further generaiizations of Jacobi-Poisson dynamics rely on the possibility to generalize the Poisson bracket. Let us consider a dynamical system described by the equations
d
-f dt
= {flxl32} t
where the ternary bracket in the right-hand side is supposed to be skewsymmetric. This dynamics will be called a ternary Jawbi-Poisson dynamics if the ternary bracket allows for a Jacobi-Poisson theorem on first integrals. In such a case the ternary bracket will be called ternary Jacobi-Poisson bracket. We are thus looking for a property of the ternary bracket such that
H a ~ ~ l t o Systems n~a~
62
For this purpose it is useful to recall the form of Jacobi identity, for bracket, given in Eq. (2.26):
nary
X f b , h ) = { x f g ,h ) + {!7>Xfh). This form can be immediately generalized to skewsymmetric brackets with an arbitrary number of entries. Indeed, given the ternary bracket { f , g , h } , we require that the operator Xf, (vector field), defined by Xfgh := {f,Q, h) f
be a derivation of the bracket; that is
Xfg{hl,h2,h3)
= {Xf&l,h2,h3) $-
(hl,Xfgh2,h3} f (hl,h2,Xfgh3}(2.28)
The above formula can be explicitly written as follows:
(f,9,{hl, h2, h3))
= { {f, hl), h2, h3)
+ {hl, {f,9 , h2), h3)
+ {hl, h2,if,% h3))
I
which would be difficult to invent without a deep understanding of the significance of the usual Jacobi identity. It is not diEcult to prove that Eq. (2.27) is equivalent to Eq. (2.28). We will not go on further on this subject. Much more details can be found in Ref. 152 and references therein, where examples of n-ary Jacobi-Poisson dynamics are explicitly given, and the following important property, here reported just for the case n = 3, is proven:
If { f , g, h } is a ternary Jacobi-Poisson bracket, the binary bracket {f,g}h = { f,g, h } , obtained by fixing one of the functions, is a binary Jacobi-Poisson bracket. h r t h e r m o r e , a linear combination of two of them c l { f , g } h l
+
cz{ f ,g)ha is agaan a b z n a r ~J a c o b ~ - ~ ~ i s sbracke~. on 2.5
The Variational Principle for the Hamilton Equations
It has been shown that Lagrange’s equations (2.1) are differential equations for the unknown functions qh(t), which are required to be an extremwn of the action
The Variational Principle for the Hamilton Equations
S[qI =
/””
63
G?/Q/t)dt.
ta
On the other hand, from Eq. (2.6), we have ( L ) * d t = x p h d q h - ‘fldt , h
where C, is the Lagrangian L in which the velocities qh have been expressed in terms of momenta and coordinates by means of Eq. (2.3). It is then natural to argue that Hamilton’s equations can be obtained as the equations for the extrema of S*[q/p/tI = f
b ta
L*(q/p/t)dt,
the transformed functional of S(qJ:
This is easily verified, since
As before, by imposing 6qh(tA) = dqh(tB) = 0 , we obtain
In this way, Hamilton’s equations of the motion follow from the vanishing of 6s’ for any choice of 6Ph’S and 6Qh’S.
Chapter 3
Transformation Theory
3.1
3.1.1
Canonical, Completely Canonical and Symplectic Transformations Canonical transformations
The differential equations of motion have been brought into a particularly desirable form, the canonical form:
However, no direct integration method of the canonical system is known. There exist indirect methods which allow to highly simplify the integration problem. One ofthem is the method of coordinate transformations, whose goal is to find new coordinates, namely ( r ,x), in which the characteristic function %! of the canonical system is “more simple.” For a generic coordinate transformation, the canonical system is not form invariant; i.e., its form is not preserved. Therefore, the interesting preliminary problem is to characterize the set C of invertible differentiable transformations which preserve the canonical form of 65
hnsformation Theory
66
equations of motion. Any transformation satisfying this requirement, will be called a canonical transformation.' It was already clear from the Lagrangian form of dynamics that a proper choice of coordinates can greatly faci~itatethe search for the s o l u t i o ~of the differential equations of motion. For instance, since a first integral of the Lagrange equation is known whenever one of the Lagrangian coordinates is cyclic, it is of great interest to produce cyclic coordinates by transforming the original ones. Let
be an invertible differentiable tr~sformationfrom the coordinates (~/p)to (x,n),which may depend on time t. It was proven by Sophus Lie? that
A suficient condition for Eq. (3.1) t o define a canonical transformation is that there exist two functions 310 and f of ( q / p / x / ~ / t such ) that the relation
h
h
identical@ holds. The new characteristic function is k; = (31 - '?fo),* where the symbol * indicates that all coordinates ( p , q ) have been expressed in terms of ( T / X ) . *The theory of canonical transformations is essentially due to Jacobi whose efforts were too much bent on the integration problem to which Hamilton was only incidentally interested. The resulting integration theory played an important part in the modern development of atomic physics. tMarius Sophus Lie was born at Nordfjordeide (Norway) on 1842 and died in Oslo on 1899. He was a professor at Oslo University from 1872 to 1885, at Lipsia University from 1866 to 1987 and again at Oslo University from 1898 to 1889. It is difficult t o illustrate, in a short note, its enormous contribution t o mathematics. He invented, in particular, the theory of contact transformations, the theory of (finite and infinite) Lie groups, the theory of minimal surfaces, the theory of translation surfaces, the theory of surfaces with geodesical groups, the theory of surfaces with constant curvature. Many results of M.S.Lie have been recovered, independently, by excellent modern mathematicians, after almost 100 years. We just mention here the Konstant-Kirillov-Souriau symplectic structure. *Here identicalty means that once the transformation (3.1) has been pedormed, the relation (3.2) reduces to an identity.
Canonical, Completely Canonical and Symplectic hnsjonnations
67
It is a trivial exercise to verify that the transformation
is a canonical one; the new canonical system being
with iC = a/3%!*, but it does not satisfy the condition (3.2). We can argue that the set C of canonical transformations is larger than the one characterized by the Lie condition (3.2). It was later proven by Lee H ~ a - C h u n ghow ~ ~ the condition (3.2) can be generalized, in order to express a necessary and sufficient condition for Eq. (3.1) to be a canonical transformation. A heuristic way to find a necessary and sufficient condition for a differentiable invertible transformation to be canonical is the following. By using the variational principle, the Hamilton equations can be written as 6S=O,
where
In this way, associated with the transformation
we have the following picture:
‘ l h n s f o n a t i o n Theoiy
68
$as*= o
$as=o
Therefore, the necessary and sufficient condition for a differentiable invertible transformation to be canonical is that
where S*[x/r/t]is the transformed of S [ q / p / t ] . The above equivalence will be certainly true if differential forms, up to a multiplicative constant c, differ by an exact differential form dF:
h
/
\ h
It was shown by Lee Hua-Chung that the condition is also necessary. We can conclude that
A necessary and s u ~ c ~ e condit~on nt for a d i ~ e ~ e ~ t i ainvert~b~e b~e trunsf o ~ a t i o n(3.1) to be canon~calis the existence of a wnstant c and of two functions, 3 t o and F , of ( q / p / x / T / t ) , such that the relation phdqh = c h
nhdXh
+ Xodt + d f
(3.3)
h
i d e n t ~ c a ~hold^. l ~ The new charucte~stzcfunctzon is X: = l/c(% - ?lo)*, where the symbol * indicates that all coordinates ( p , q ) have been expressed in t e r n s of (r/x). A simple example of canonical transformation, with %O = 0 and F = pq, is given by
X = arctan(Aq/p).
(3.4)
Canonical, Completely Canonical and Symplectic lhnsformations
Indeed, pdq - ndX
3.1.2
== pdq
- 2x 1 (p2 -tX2q2)d(arctan
69
($))
A general ctase of canonical ~ ~ n s ~ ~ ~ u t ~ o n ~
A particular class of canonical transformations is generated by an arbitrary function V which depends on “one half” of original coordinates and on “one half” of the new ones, for instance on q’s and 71%. The fmction is only required to satisfy the condition that the mixed hnctional determinant
does not identically vanish 3 # 0. It is easy to see that the relations
implicitly define an invertible differentiable coordinate transformation between the p , q’s and r ,x’s. In fact, since 9 # 0,by the implicit function theorem, the second of relations (3.5) can be solved in the form qh
qh(n/X/t)
and associated to the first one to give an explicit one-to-one transformation between the coordinates ( p ,q ) and ( x , x ) . As for the canonical character, it is enough to observe that Lie’s condition is satisfied:
Then, the transformation defined by relations (3.5) with with
3’#
0, is canonical
and leads to the new characteristic function
aV at
K=%+-,
expressed, of course, in terms of Ip/q) coordinates. The function V is called the generating function of the canonical transformation. A different choice could be to choose a function V’ depending on q’s and on x’s, and satisfying the condition that the mixed functional determinant
does not identically vanish. The relations (3.6)
implicitly define an invertible differentiable coordinate transformation between the p , q’s and x , x’s. In fact, since J’ # 0, by the implicit function theorem, the second of relations (3.6) can be solved in the form qh = q h ( ~ / X / ~ } 2
and associated to the first one to give an explicit one-to-one transformation between the coordinates (p,q> and ( T , x). As for the canonical character, it is enough to observe that Lie’s condition is satisfied:
71
Canonical, Completely Canonical and Symplectic 'Pransformations
from which we obtain
Then, the transformation defined by relations (3.6) with with
3'# 0, is canonical
and leads to the new characteristic function dV'
K : = x + -at
'
expressed, of course, in terms of ( p / q ) coordinates. The function V' is also called the generating function of the canonical transformation. 3.1.3
Completely canonical transformations
It has been shown that a canonical transformation of a given canonical system with characteristic function 'fl leads to a canonical system having Ic = (l/c)(% - '?LO)* as characteristic function. When the function K = (l/c)(% ?LO)* reduces to K: = (l/c)(%)*the transformation is called a completely canonical transformation. Then, a necessary and sufficient condition for a canonical transformation to be a completely canonical one is that 'tlo = 0. It is easy to see that a canonical transformation
which does not explicitly depend on time t , is a completely canonical one. The Lie condition (3.3) for the transformation (3.7) gives k
h
Runsfomation Theory
72
or equivalently,
[
(mk f
~)
dpk
where the functions @ k and @k(P/Q)
f (@k -b ikk
apk
x Pi-
(.. $)
dql] f
4-
dt = 0,
are defined by
a$JiI
=c
2)
@ k ( P / q ) = c c ( o i ~ - P k .
i
i
It follows that F has to satisfy the following relations: df
Moreover, as @ k and
ikk
aF No + at = 0.
do not explicitly depend on time t , we &so have
or equivalently,
which implies that d F / & does not depend on the p’s and the q’s but only on t . From
BF dt = f ( t >, it follows that
and
Therefore, Lie’s condition (3.3) becomes
so that the new characteristic function is simply Ic = (l/c)(N)*.
Canonicul, Completely Canonical and Symplectic Thnsformations
3.1.4
73
Sgmplectic transformatione
A canonical transformation, which leads to a new characteristic function K of the form K = (%)*, is called a symplectic transformation. Therefore, a symplectic transformation is a completely canonical transformation with c = 1. The following picture summarizes all the cases well:
K
1
= -(N - No)* (canonical), C
K: = -(N)* 1
(completely canonical) ,
(3.8)
C
K
= (N)*
(symplectic) .
The transformation (3.4) of the previous example is a symplectic transformation with a generating function given by
3.1.5
Area presenting transformations
An invertible differentiable map from R2 to itself,
will transform a given Lebesgue measurable region S C R2 in a measurable region C 2 R2,
The map will be said an area-preserving transformation, or simply, an equivalent transformation, if the measures of S and C coincide: mz(S) = m2(C).
Theorem 13 A necessary and suficient condition for an invertible transformation on R2,
to be symplectic is to be an area preserving transformation.
Tlvlnsformation Theom
74
Pro0f Let the trunsformution (3.9) be ~ymplectic. Then there exists a f u n c t ~ o nG, such that, ~ ~ e n t ~ c a l ~ ~ , pdq = ndX
+ dG ,
i. e.
Since the RHS of the above equation is an exact differential, the equality of crossed derivatives
a dX (
P ~ = )
(Pg
-
.>
?
gives
~ h i c hcan be, e q u i v a ~ e n t lexpressed ~~ by
I t follows that
, Con~ersely,zf the transfoTmataon (3.9)is area p r e s e ~ n g then
The arbitrariness of C and the continuity of the Jacobian determinant imply !I) -=& l. a(n,x )
A New Chamcterization of Completely Canonical lhnsformations
75
If the Jacobian is 1, the transformation (3.9) isf symplectic. The same is true i f the Jacobian is -1; it is suficient t o interchange the names of the variables 7~'s and x's to go back to the first case. 3.1.6
-
Volume p r e s e r v i n g transformation
An invertible differentiable map from 52'
(P,q) E !RZn
to itself,
(n,x>E %",
(3.10)
will transform a given Lebesgue measurable region S C 8 '' in a measurable region C C R",
S-Z. The map will be said a volume preserving transformation, or simply, an equivalent transformation, if the measures of S and C coincide: m,(C) = mn (C). It will be shown, in the next section, that a symplectic transformation on !R" is also equivalent. I t is worth mentioning that the converse is true only in the case n = 2.
3.2 A New Characterization of Completely Canonical Tkansformations The Lie condition is not easily handled for checking the canonical nature of a differentiable invertible transformation. Thus, we are looking for conditions to be directly required to the functions v h , $h, in the invertible differentiable transformation Th
= (Ph(Q/P) 1
Xh
= $h(q/P) >
to define a completely canonical transformations. The condition from which we start is the usual one, namely
$With the usual aseumption that the C and S be linearly simple-connected.
(3.11)
~ n s f o ~ ~ Theory t ~ o n
76
in which the transformation (3.11) must be performed. In this way, we have -dG(p/q) = x ( @ h d 4 h
+ @h&h),
(3.12)
h
where
(3.13)
If the right-hand side of Eq, (3.12) has to be an exact d ~ f f e r ~ n t ithe a ~ ,f o ~ ~ o w ~ n g conditions d@k - -- a @ h aqh
&k
8$k -
- -a @ h
dph
apk
a@k d@h - -&h
aqk
' ' '
must be satisfied. Replacing Eq. (3.13) in the above conditions, we find =
----aVk
' i + (a$k
aqi
&j
1
-saj
,
h!'k a ' p k ) = 0 ,
aqj aqi
The above relations can be written in a very compact form by introducing the Lugrange bracket, which, for given 2n functions P h , +h of variables w,w, are defined by3
qMmy authors define as Lagrange bracket the analogue, but associated with the inverse transformation.
A New Characteritation of Completely Canonical hnsfonnations
77
By using this bracket, the necessary and sufficient conditions for a transformation to be completely canonical can be expressed as follows:
It has been already shown that, in the plane R2,the symplectic transformations coincide with area preserving transformations. Thus, it appears interesting to analyze, from this point of view, the properties of a completely canonical transformation more closely. Let then
{
rh
= [Ph( q / p )
Xh
= ‘$‘h(q/p)
be a completely canonical transformation and let
be its Jacobian determinant, which, more explicitly, can be written in a block f o m as
By performing, on the Jacobian matrix
the following interchanges: 0 0 0
0
+
row number j with row number n j, V j E (1,. . . ,n}, column number j with column number n j , V j E ( 1 , . . . ,n}, inversion of the sign in first n rows, inversion of the sign in first n columns,
+
lhnsfonnation Theory
78
we obtain the matrix
M'
, i , h E { l , ...,n } .
= a'Ph --
a'Ph -
Of course, since an even number (4n) of interchanges, each one introducing a factor -1, has been performed, it turns out that
J = det M = detM'
It follows that the product of MT, the transposed of M , with M', has the following determinant det(MT . M I ) = (det M)(det MI) = J 2 . More directly, the matrix M' can be obtained from M , by using the matrix
introduced in Eq. (2.23) of the previous chapter. Accordingly,
M' = E
M . ET ,
with ET being the transposed of E. Trivially, ET = -E = E-l. The elements cij of the product MT M' can be divided into the following four groups: (i 5 n , j 5 n), (i 5 n , j > n), (i > n , j 5 n), (i > n,j > n). In other words, similarly to M and M ' , also the product MT M' can be divided into four sectors, which separately evaluated, give
-
A New Chamcterixation of Completely Canonical Transformations
79
Therefore, we have
the last equality following from Eq. (3.15). Thus, the Jacobian determinant J of a completely canonical transformation in satisfies the relation J2 = c
- ~ ~ .
(3.16)
From the expression
we have
so that
M . (M’)T
1
= ;((M’)-l)T
.(M’)T
=
1
-(MI
. (M’)-’)T
C
By performing the product M
(M‘)T,
1
= -1. C
we finally obtain
Therefore, it follows that
The necessary and suficient conditions for a transformation to be completely canonical can be expressed as: {qt,qj}=Oi
{pi,pj}=O,
1 {qijpj}=;6ij>
V(iij)c{1121**.1n}.
(3.17)
lhnsfomation Theory
80
3.3
New Characterization of Symplectic Transformations
From relations (3.15), it turns out that The necessary and suficient conditions for a transformation to be symplectic is given by [Qi, Qjl = 0
,
[Pi,Pjl= 0
1
[Qi,Pjl= 6ij
, v
.
( i , j ) E { 1 , 2 , . . ,n},
or
{ qi,q j } = 0
{Pi,~
j
=} 0 7
{ ~ i ~,
j
=} 6ij ,
V (ii j ) E { 1,2, .
*
,n}
*
Moreover, from Eq. (3.16),it follows that
A symplectic transformation preserves the volume of any given region of the phase space.
Chapter 4
The Integration Methods
4.1
Integrals Invariants of a Differential System
Let us consider the first order differential system dXi
= X"Z/t), dt
vi E (1,2,. . .n),
(4.1)
and denote with
xi = z y t , 20) the solution, which at time to, takes the value xo : X; = si(to,20). The above relations can be equivalently expressed by
Q = Q(t,Qo)
3
(4.2)
where Q and Qo denote the points whose coordinates are (d, . .. ,zn) and (xi,.. ,eg), respectively. Given any submanifold UOC Rn, whose points will be denoted by Qo, let U be the submanifold, depending on t, of points Q given by Eq. (4.2). In other words, U represents the evolution at time t, according to Eg. (4.1), of 270. Equation (4.2) thus define a map between Uo and U. The map is one-toone by the existence and uniqueness theorem, which is taken to hold for the system (4.1).
.
81
The Integration Methods
82
Let us consider an arbitrary function p of x and t and the integral
which, of course, will generally depend on time t. In the case I does not depend on time, no matter how U is chosen, we shall say that I is an integral invariant
for the system (4.1). In order for I to be an integral invariant, p is required to satisfy suitable conditions. Let us start from the natural characterization of a n integral invariant, which is given by
In the above expression, the transfer of time derivative under the integral sign is not permitted, since the integration region depends on time. The difficulty is easily overcome by using the change of variables given by Eq. (4.2). In this way, we obtain
where i j = P O & is the composed function between p and Eq. (4.2);i.e. ~ ( z ot ), = p ( x ( z o ,t ) ,t ) and J is the Jacobian determinant
(4.4) of the transformation Eq. (4.2).
Remark 7 The theorem o n the change of variables in the integrals requires, really, the absolute value of the determinant. I n our case, however, the Jacobian matrix at initial time t o coincides with the unit matrixZ. Then, by the continuity, it exasts a neighborhood of t o ( a n interval of time) an which the Jacobian determinant is always positive. It follows that
where we have used the property that the Jacobian of the inverse transformation is the inverse of the Jacobian.
Integmls Invariants of a Diflerential System
Therefore,
83
(2
+ p J - l - )d J dt
=
dt
dU.
In order to explicitly calculate the derivative d J / d t , let us observe that J depends on t via the elements gij = axi/&$ of Jacobian matrix. Then dJ _ By wing the Laplace* expansion, the Jacobian determinant can be written as follows:
It is important to note that in the above expression the algebraic complement Gij of the element gij does not contain the elements g i l , . . . ,gin. In this way
dJ w
= C G i j -d X i ij
ax;
jj
axi
=EGij ij
=CGijC-axi a x k
aSI,gkj
k
k
axk
ax$
aXi
=X G i j G g k j ijk
'Pierre Simon Laplace was born in 1749, in a little village of Calvados, a region of fiance, and died in Paris in 1827. He covered public chargea and was a member of the Science Academy of the Institute de h n c e and a professor at the kcole Normale. He was appointed Earl, Marquis and Peer of France by Napoleon. His results on celestial mechanics, acoustics and electromagnetism are very important; the treatises on the celestial mechanics (five volumes) and on the calculus of probability as well as the divulgation works Exposition du Systdme du monde (two volumes) and Essai philosophique sur le probabilitb are now considered aa classical works. His complete production fill up 14 volumes.
The Integmtion Methods
84
and
It thus follows that
dl dt =
(2
+ p div
2)d U .
(4.5)
Since the function in the integral is continuous and the region U is arbitrary, we can conclude that
The necessary and suficient condition f o r I = invariant is that
s,
p ( x ,t)dU t o be a n integral
2 + p div r7 = 0 , dt
(4.6)
A function p satisfying the previous equation is called a Jacobi multiplier. Finally, let us observe that, by using the identity
Eq. (4.6) can also be written in the more familiar form
2 +div(pz) = 0 , dt which the reader has already met, for instance, in electrodynamics, where p has to be identified with the electric charge density and p.? with the current density. Finally, let us address that in the case of divergenceless vector field d , any constant is a Jacobi multiplier. For these types of dynamics we get the important result that the measure p ( U ) of any region U does not change in
A Primer on the Lie Derivative
85
time. This is certainly the case for Hamiltonian systems
631 631 for which we have
It has already been remarked that the Jacobian determinant of a completely canonical transformation is 1, and that this type of transformation are volume preserving. It has also already been remarked that the Hamiltonian evolution is itself a completely canonical transformation between a submanifold UOof the phase space and the submanifold U of the same space whose points are the evolutes, at time t , of points of Vo, having the Hamiltonian function as a generator. Thus, for canonical systems, a double conservation of the measure holds (Liouville remark). 4.2
A Primer on the Lie Derivative
Let us consider the differential system dxi = X"X), dt
v i E (1,2, * . . , n ) ,
(4.7)
where the X ' s do not depend explicitly on time, and evaluate the rate of change of any function of the coordinates x along the solutions of the system, shortly the "time" derivative. We have
Therefore, one can naturally associate, with a differential system, the first order differential operator
i=l
The Integration Methods
86
whose action on an arbitrary function f gives its "time" derivative. Since with each point x of the space we can associate a vector having the real numbers X i ( x ) as components, the previous operator X is also called a vector field. Vice versa, with any vector field we can associate a system of differential equations dx'
dt = X i ( x ) , V i E
(1,2,.
..
defining the curves xi = x i ( t ) ,such that the tangent vector v' = (dx'/dt,. . . , dx"/dt), in a generic point x , is just given by the components of Xi at point x . Such curves are called the integral curves of the vector field X . By denoting, as before, with Q and QO the points whose coordinates are (x1,x2,.. . ,z") and (xh,xi,. . . ,x;), respectively, the equations
represent the solutions of the differential system which, at time to, takes the value Qo. They locally define a one-to-one global map, which depends on a parameter t ,
called the flow generated b y the vector field ( X l 1X 2 , .. . ,X " ) , satisfying the group properties cpo = identity map,
( V 4 - l = 9-t 'Ptl
O
,
9 t z = (Ptl+t2
*
If the differential system is canonical with characteristic function R, the map pt is also called the Hamiltonian flow generated by R. The derivative of a function f along the solutions of the differential system (4.7) is denoted by
and is known as the Lie derivative off with respect to X .
A Primer on the Lie Derivative
87
From equations xi = xi(t, xo), it follows dxa =
axi C -dd. 8x3,
j=1
It is then easy to evaluate the “time” derivative of the differentials dx’. We obtain
d(dxi) dt
a
d
n
.
j=1
j=1
=
dxi
‘ a
xdx3,-Xi j=l 823,
n
It
= j=1
(k=l
axk a --Xi) ax; dxk
axi = C -dxk. axi dXk dXk
(4.9)
k=l
The above “time” derivative is known as the Lie derivative of dxi with respect to X and is denoted with L x d x i , so that the above equation can be written in the following form:
L x d d = dXi , without any reference to the parameter t. By now using
which also follows from xi = xi(t,xO), it is possible to evaluate the “time” derivative of partial derivatives d / d x i . We thus get
Lx- d
axi
c
d a =d n a2 x ia = = --
dt a x z
dt j=l axi 6x3,
axk j=l
k=l
a
axi a x k
(4.10) The last step in the above formula is explicitly performed in the Appendix B.
88
By observing that
axk a
(4.11)
k= 1
Eq. (4.10) can also be written in the following form: (4.12)
The Lie derivative with respect to a vector field X has been defined on functions f, on differentials dxi and on vector fields a/axi, with the transparent physical significance to be a "time" derivative; i.e. a derivative along the solutions of the evolutive first order differential system. Since, by definition, Lx satisfies the Leibnitz rule, the Lie derivative, with respect to a vector field X, is defined for generic differentia1forms Q = ai(x)dxi and vector fields Y = Y ~ ( ~ ) aas /a follows: ~~,
(4.13)
= (X,Y]
(4.15)
Remark 8 Alternatively, once Lie's derivative has been defined on functions, the Leibnitz rule suggests to define
( L x & ) f = L x ( g ) - s ( dL x f ) . The interested reader will prove that the two definitions are equivalent.
The Kepler Dynamics
4.3
89
The Kepler Dynamics
The gravitational potential energy of two bodies with mass ml and m2 located, with respect to a chosen frame, at 3 1 and i5 is given by:
where G = 6.6 Nm2/kg2 is the gravitational universal constant. The Lagrangian function C,obtained subtracting U from the kinetic energy T , is
The coordinates F1, 772 can be expressed in terms of center of mass coordinate 2 and relative coordinate F defined by
We have -+ rl =
mzr' ml+ m2
+R,
mlr'
$2=-
+ 8.
ml +mz The velocities v'l and $2 can also be expressed in terms of the center mass velocity and relative velocity v' as follows: * v'1=v+
mzv'
ml
-
, &=V-
+m2
mlv' ml f m 2
The Lagrangian L expressed in terms F, 8,v' and 1
1
becomes
k
~=-(ml+m2)V~+-pw~+-, 2 2 r where P=
mlmz ml
+mz
is the reduced mass. Thus, the Lagrangian L is the sum of a free Lagrangian C R = [(ml m2)V2)/2and a Lagrangian L, = (1/2)pv2 - k / r of a system with 1 degree of freedom. The first Lagrangian describes the motion of the center mass which turns out, of course, to be uniform. The second describes the motion of a
+
The Integration Methods
95
particle with the reduced mass p in the gravitational field force located at center of mass coordinate. We may notice that if m2 denotes the dimension of the (i,j)-irreducible representation of
su (2) €3su (2)
N
so (4): D(i,j) = (2i + 1)(2j
+ 1).
4.4 The Hamilton-Jacobi Integration Method Important concepts, as first integral and integral invariant, concerning canonical systems have been discussed in the previous sections. It is now time to briefly discuss problems concerning the effective integration of canonical systems. Let us start with the classical Hamilton-Jacobi integr~tionmethod. This method brings the integration of any canonical systems of rank 2n to the determination of a so-called complete integral for a partial differential equation in n 1 independent variables. Given then the canonical systems
+
SNiels Henrik Bohr was born in Copenhagen in 1885,and died there in 1962. Soon after his degree, he moved to Cambridge and then to Rutherford’s laboratory in Manchester. He solved the contradiction between the Rutherford’s atomic model and the electrodynamics classical laws. Indeed he was able to agree on four physical theories: the classical electrodynamics, the quantum black-body radiation by M. Planck, the Rutherford atomic model and the atomic spectra observed by J. J. Balmer. He was appointed t o the Nobel Prize in 1922, and was an associated founder of the CERN in Geneva.
The Integration Methods
96
let us try to find, if any, new canonical coordinates ( n , ~ such ) , that the new Hamiltonian function K: is the simplest one; that is K: = 0. Then the integration of the transformed canonical system
becomes trivial 7fh = constant, Xh
V h e {1,..., n } .
= constant
Let us take advantage of the general method, previously introduced, consisting in generating a canonical transformation
av
ph=->
av
Xh=-,
aqh
anh
by using an arbitrary function V depending on the q’s and the T’S and satisfying the condition
,.7 = det
(-------) a2V # 0. aqhark
The new characteristic function will thus be
where the * indicates that the transformation has to be completed expressing the q’s in terms of the (nl x ) ’ s by using the relations
Our goal will be achieved if V is such that K: = 0. Therefore, V has to be a solution of the celebrated partial differential equation (4.19)
known as the Hamilton-Jacobi equation.
The Hamilton-Jawbi
Integmtion Method
97
The Hamilton-Jacobi integration method can be summarized as follows: Once given the canonical systems d &tph =
{~
-qh=
--aa31 qh
t l h {~I , ...,n } ,
a31 aph
'
replace the momenta p's in ~ ~ p / qwith / t the ~ symbol
av
ph=-t &h
where V is an u n ~ o w nfunction. Write down the Hamilton-Jacobi equation
Find a complete integral V(q/T/t)of the Hamilton-Jacobi equation; that is, any solution V of the equation
depending, besides the q's and the time t , also on n arbitrary integration constants, namely ( ~ Q , R Z , . .,zn) and s a ~ i s f y ~the g condition 3#0. Write clown the canonical trans€ormation dV +
i
Ph=-r aqh
av Xh=-,
V h e {1, ...,n ) ,
(4.20)
anh
leading to the trivial solutions T h = constant, X h = constant of the new canonical system. Explicitly write down the above transformation in the form
representing the general integral of the canonical system.
The Integration Methods
98
Fix up the values of constants 7 ~ 'and s x's according to initial data:
0
Compose the two mappings (4.21) and (4.22) to obtain
representing, finaily, the integral of the canonical system in terms of the initial conditions.
Example 14 Let us consider the harmonic oscillator with 1 degree of freedom whose ~ a ~ i ~ ~ iso given n ~ abyn H = %1( d + m
w
2 2 Q2 >
I
so that the c o ~ ~ s p o n d i nNumil~on-Jacobi g e q ~ a t ~ ocan n be itt ten as f o ~ ~ o
Let us try to find a solution of the form
V =: -Et
+- W ,
where E is an a r b i ~ ~ constant. ry Then, the Namilton-Jacobi equation simplifies to
f o r which the solution is easily found in the f o r m 4 2
+E
mu4
W = - d 2 m E - m2w2q2 - arcsin W
&GiZ'
so that
V = -Et
E arcsin mwq + Q-2 d2mE - m2w2q2 + W &z'
(4.24)
The Hamilton-Jucobi Integration Method
99
The function V generates now the canonical map dV
1
2 m E - m2w2q2, = -t
x=
w q + -W1 arcsin &GE' ,
with
-
d2V m dqdE - J(2mE - m2w2q2)*
J=--
The explicit canonical transformation turns out to be
I
mwq , x = -t + -1 arctan W
P
and its inverse
( p = &cos(w(x
+t))
represents the general integral of the canonical system. 4.4.1
Remarks on the Hamilton-Jacobi equation
The Hamilton-Jacobi equation can be considered the most elegant form of dynamics and gives an important physical example of the deep connection between first-order partial differential equations and first-order ordinary differential systems. It was first introduced in 1834 by W. R. Hamilton,'12 in his investigation on analytical dynamics, and it has been the starting point for Schrodinger to state the wave equation in quantum mechanics, of which is the approximate version in the cases in which the Planck constant can be neglected. The proof that once a complete integral is found, then the dynamical problem can be completely solved by using Eq. (4.20), is instead due to J a ~ o b i , " ~ hence the name "Hamilton-Jacobi" for the equation and the ensuing method of solution.
100
The ~ n t e ~Methods t ~ o ~
Each complete integral of the H ~ i l t o n - J ~ o bequation i gives rise to a family of solutions of Hamilton’s equations, and according to D i r a ~ v “while ,~~ the famzly does not have any importance from the point of view of Newtonian mechanics, . . . i t . . . corresponds to one state of motion in the quantum theory, so presumably the family has some deep significance in nature, not yet properly understood.” Once the full dynamical problem has already been solved, an explicit solution of the Hamilton-Jacobi equation is given by
where t and f a r e two time-instants, q’ = d q / d r , Ifl and L the Hamiltonian and the Lagrangian functions, and the integral has to be taken along the actual trajectory of the dynamical system. The right-hand side of the above equation does indeed satisfy the Hamilton-Jacobi equation and also the additional equation32
Remark 9 For conservative systems, S depends actually only on the dzf’erence t - f, so that
qPau1 Adrien Maurice Dirac was born in Bristol in 1902, and died in 1984. After his degree, obtained at Bristof University in 1921,he moved to Cambridge University. In this university, he was Lucasian professor, a chair already covered by Newton, from the year 1932. Dirac has been one of the most important physicist of our age and can be considered the father of modern physics. We just need to mention the Dirac e ~ predicting ~ the ~ existence of the positron and more generally of antiparticles, the Femi-Dirac statistics and the constraints method, which is an essential tool for the Hamiltonian formulation of Einstein’s equation, considered then as a step towards a quantum theory of gravity. The constraints method has been also a fundamental step for the quantization of gauge theories. His books are now considered as classical works. Together with Schrodinger, Dirac was appointed to the Nobel Prize in 1933.
~
The Hamilton-Jacobi Equation for the Kepler Potential
4.5
101
The Hamilton-Jacobi Equation for the Kepler Potential
In terms of spherical-polar coordinates ( T , 19, p), the Cartesian coordinates (2, y, z) of a point are expressed as follows:
x = rsinGcosp, y = rsingsincp, z = rcose.
+
+
The line length ds = (da2 dg2 d z 2 )4 , representing the infinitesimal distance between two points of coordinates (x,y, z ) and ( x + d x , y+dy, z+dz), is given by ds2 = dr2
+ r2d@ + r2sin229dp2.
The kinetic energy 7of a massive particle will be w r i t t e ~as
= 5n(+2
2
+ r262 + r2 sin2 t+2) ,
so that the Lagrangian of a particle in the potential U ( 3 can be written as follows: 1 t = -rn(t2 + ~~9~+ r2sin2.~(t+~)- ~
2
(3.
By introducing the conjugate momenta ( p r , p ~ , p , )of ( r , d ,p), pr = m+, pe = mr9 , p , = mT2sin2#+,
the corresponding Hamiltonian will be given by
Since the Hamiltonian does not depend explicitly on the time, the Hamilton-Jacobi equation
The Integration Methods
102
can be reduced, with V = W - Et , to the form
For a central potential U ( r ) , it is possible to find a complete integral of previous equation by using the method of separation of variables, which consists of searching for a solution W ( r ,6, cp) of the form W ( r ,6 , cp) = Wr(r>+ WG(4+ W,(cp>; that is, for a solution that is the sum of three different functions W,, W,q and W,, each one depends only on one of the variables r, 6 and cp. In this way, the Hamilton-Jacobi equation for W becomes
and with a simple manipulation, it can be written in the form
(2) 2
=r2sin26 2m[E-U(r)]-
($)2-
1 7
(w,”). dW8
The left-hand side of the above equation depends only on cp, while the righthand side depends only on r and 19. Since the variables r,6 and cp are independent, each side must be equal to a constant, namely 7r;. Thus, we obtain
Once again we observe that the left-hand side depends only on 19, while the right-hand side depends only on r. Since r and 6 are independent variables, both sides must be equal to a constant, namely T:.
Remark 10 The constant T , has a clear physical meaning: it simply corresponds to the component p, of the angular momentum along the z axis. Thus, it expresses the uniformity of the spanning, by the projected vector radius 8 = F - t.& of the areas in the (x,y ) plane.
103
The Hamilton-Jacobi Equation for the Kepler Potential
The constant
~ f corresponds i
to the modulus of the angular m o m e n t u m
so that, if a denotes the angle between the orbit plane and the (x,y) plane, we have p , = $1 cosa ,
and also T,
= 7r~COSa!.
(4.25)
Therefore, the Hamilton-Jacobi equation for W is, for this solution, equivalent to
In the case of the Kepler dynamics we choose U ( r ) = - k / r , so that the above equations can be written as follows:
Thus, we have
104
Problems 1. Find a complete integral of the Hamilton-Jacobi equation for the harmonic oscillator with 3 degrees of freedom,
by separating the variables in spherical-polar coordinates. 2. Find a complete integral of the ~ a m ~ l t o n - J a ~ oequation bi for the Kepler dynamics,
by using parabolic coordinates
(t,v,p), defined by
} be a covering of M . The intersection I of the intervals I,, corresponding to the open sets U(p0) can be empty, but if the manifold N is compact, the covering ( U ( p 0 ) ) contains a finite subcovering and the intersection I of the corresponding intervals is certainly not empty. In such a case the can be extended to the entire manifold M , with T E I d i f f ~ m o r p h i s(5.12) ~ and the vector field X said to be complete,29 F'rom Eq. (5.11), we have
r(7+ g,PI
= Y'+"(P)
7
$7,
r(0,PI> = YT(Y(O,P ) ) = r r 0 Y"(P) >
The Lie Derivative
151
and then
pff = 77
0
yo.
Moreover, every rT is supplied by an inverse; that is by the diffeomorphism y M T ,so that we can define yT for every T E
8.
What has been said above can be summarized by the following theorem.2g
Theorem 21 With e u e q Ch d~fferentia~ze vector field X ( p ) , on a Ck ( h I k - I) compact, differential manifold M , a one parameter group is associated
y‘:M-+M.
(5.13)
This grou~of ~ ~ ~ e o ~ oof ~Mh ini itse2f s ~ is s sack t ~ a ~
The group rT is also called the Bow of the vector field X ( p ) , and it is also denoted by 7:. The group (5.13)is then well defined if the manifold M is compact. In the general case, the y7 are defined just like in Eq. (5.12)only in neighborhoods of a point po E M and for small T . 5.11
The Lie Derivative
Let X ( p ) and Y ( p ) be two vector fields on an n-dimensional differential manifold M , and y;; be the flow of the vector field X . The Lie derivative LxY of the vector field Y is the vector field defined by the relation
(5.14) where (rj-T)*~T(,~
:~
T ( ~+ ) T,M M
Let us calculate the Lie derivative of the basis vectors {afax:”}. In order to simplify the notation, let us indicate with z E 8%the coordinates of the points of M and let us set f(s) = y. If # ( T , Z) are the coordinates of rjT(x), and ~ ( T , zthose ) of 4-T(y), then @(O,p) = zi and #(O,y) = yz.
~ a ~ a f and ~ ~ Tangent d s Spaces
152
From Eq. (5.14),we have
The components of since
a/azi, in the natural coordinates system (5.5), are $,
(&)p=Ji($)
' P
while the components of { ~ - ~ ) * are ~ given ( ~ by ) ~
Then, we can write
By using Eq. (5.161,Eq. (5.15) can be rewritten in the following fonn:
Since
we have
and
The Lie Derivative
153
so that
and
a')
r=O
&Ip
=
-% (&) .
(5.17)
P
The Lie derivative is an additive operator; i.e.
Lx(U
+ V ) = LXU + L X V ,
where X, U,and V are vector fields on the manifold M . Moreover, it satisfies the Leibnitz rule
L x ( U @ V )= (LXU) @ V + U @ L X V , where the symbol @ denotes the tensor product defined in the next chapter. By using the Eq. (5.17) and the relations (dx', a / a x j ) = dj, we can calculate the Lie derivative Lxdxi of the basis differential 1-forms {dx'}. Indeed
(
L x dx'&) so that
= Lxd; = 0 ,
(Lxdxi, &) = - (d z i ,L x &) (dx', 64 axk gap ) =
=
(-dxk, axi -) a axj ask
Therefore, we obtain
Lxdx' = d X i .
= (dX',
s).
~ a ~ ~ and f ~~ al n ~g esn Spaces ~
154
The Lie derivative of a differentiable function f on the manifold M has the following expression:
where f = f o @-l is the function representing f in the chart p is represented.
5.12
(U,$) in which
Submanifolds
Examples of submanifoIds are given by a sphere S2or a curve y in the space g3. In some neighborhood U C %I3 of any point p E S2,a coordinate system (2,y, z ) can be introduced for such that the points of S2 n U are characterized by z = 0. Similarly, in some neighborhood U C g3,of any point p E y,a coordinate system ( x , y , z ) can be introduced for IR3, such that the points of y nU are characterized by y = z = 0. A sphere S2 (or a curve y) is said to be 2-dimensional (l-dimensional) submanifold S of the manifold M = %I3. Thus, it is natural to say that an dimensional s ~ ~ ~ u S, ~ of~ an f o ~ d n-dimensional manifold M , is a set of points of M such that, in some neighborhood U C M of any point p E S, a coordinate system ( X I , . , .,xn) can be introduced for M in which the points of S n U are characterized by xm+l = p + 2 = . . , = x" = 0. More formally, a one-to-one map f : Q -+ M is said to be an embedding of the m-dimensional manifold Q in an n-dimensional manifold M , ( m 5 n),if at every q E Q there is a neighborhood V 5 Q of q and a chart (U,p) of M at p = f (g), such that ( Y ,p o flv) is a chart of &; that is, p o flv : Q -+ Rm are coordinates on Y for Q. The manifold Q is said to be embedded in the manifold M . The image S = f(Q)is called a s ~ ~ m u ~ of z ~the o~ manifold d M, provided with the manifold structure for which f : Q --+ S 5 M is a diffeomorphism. If f is not one-to-one, we shall speak of immersion. In other words, a map f : Q -+ M is said to be an immersion of the manifold Q in a manifold M , if at every p E Q, there is a neighborhood V G Q of p and a chart (U, p) of M at f ( p ) ,such that ( V ,po f) is achart of Q; that is, p o f : Q -+ 8" are coordinates on V for Q. The manifold Q is said to be immersed in the manifold M .
Submanifolds
155
By recalling what has been said in Sec, 5.6, concerning maps between manifolds, a vector field defined on a submanifold S is also a vector field on M , and a wvector field on M is also a covector field on S. A suggested reading on the subject and its applicatio~is given by the Marmo, Sabtan, Simoni, Vitale book. 41
5.12.1
The fiobeni2~stheorem
It has been shown that, given a smooth vector field X on an n-dimensional manifold M ,one can find a curve (integral curve) that, at every point p E M , the value X , of the vector field X coincides with the tangent vector to the curve at the same point. In other words, since a vector field X is an assignment at every point p E M of a vector X , in the tangent space 7 , M , we can paraphrase the previous statement saying: Given, at every point p E M , a 1-dimensional subspace D, of the tangent space 7 , M , one can find a 1-dimensional submanifold N such that Dp = T ~ ,E M ~ .p
It is interesting to have an answer to the analogous problem: Given, at every point p E M , a 2-dimensional subspace D, of the tangent space 7,M (i.e. a pdrsne), does a 2 - ~ ~ m e n s i o n as l~ b m a n ~N ~ ,l such d that LIP = 7pN, exist V p E M?)
The answer is generally: No. In order to discuss the general case, it is advisable to introduce the f ~ l ~ ~ w i n g useful definitions: a
0
An assignment D at every point p G M , of a h-dimensional subspace LIP of the tangent space 7 , M , that is, a hyperplane, is called a h - d ~ ~ e n s ~ # nd ai sl~ T ~ b u t ~oonnM , or also, a d i ~ e ~ n t i systems al of h-planes on M . A h-dimensional distribution D is said to be C" if, at every point p 6 M , there exists a neighborhood U of p and h Co3-vector fields, namely XI,. . . ,x h , defined in U and defining, at every point q E U,a basis X , ( q ) , . . . , X h ( q ) for D,.The vector fields X I , ,, . ,Xh are then called a local basis for D. A vector field X is said to belong to D if X, E LIP at every point
PEM.
M a n ~ ~ and o l ~Tangene §paces
156
A CM d~stributionD is called invol~tiveif
x E D , Y E D * [ X , Y ]E D . The above relation is equivalent to say that a local basis { X I ,. . . ,Xh} of a involutive distribution has the following property:
[Xi, X j ] = CfjXk , since the Lie bracket of any two vector fields X and Y, which are their f -linear (i.e. the coefficients are f ~ c t i o n s combinations )
x = fi(P)Xi,
Y =g i ( p ) x ~ f
will be linear combinations of X i : [ X ,Y ]= [f”xi,gjxj]
+
= figqxi,xj] f i y i ( g k ) X , - giE(f”xk = (figjcFj -t- f i X i ( g k )- g i X i ( f k ) ) X k= d $ X k .
A connected submanifold N of M is called an integral manifotd of the d ~ s t ~ ~ uDt ifi f,(T&) o~ = Dq for all p E N , where f is the embedding of N into M . The subma~foidN is called a m ~ i m ail n t e ~ ~a ~u ~ i f o $ ~ of D,when no other integral manifold of D,c o n t ~ n i n gN,exists.
It can be proven (see for instance Refs. 11, 29 and 50) that
Theorem 22 (Frobeniw) If D is an involutive distribution on a diflerentiat manifold M , through every point p E M , there passes a unique maximal integral manifold N(p) of D. Any integral manifold through p is an open subrnanafold of N ( p ) . In other words, if X I , . . . ,Xh are h(< n ) vector fields defined on a region U of an ~-dimensionalmanifold M such that
[Xi,Xjj = C t X k , the integral curves of vector fields mesh to form a family of submanifolds. Each submanifold has dimension equal to the dimension of the vector space these fields define at any point, which is at most h. Each point of U belongs to one and only one submanifold, provided that the dimension of the vector space
Submanifolds
157
defined by the fields is the same everywhere in U.This family of submanifolds is called a ~ 0 of U,~ and each ~ submanifold ~ ~ a leaf of~ U. 0 ~ The central idea underlying the F’robenius theorem is that, if the integral curves, of the vector fields X I , .. . ,X h defining a distribution, are to define a submanifold to which the vector fields must be tangent, they have to mesh one another as cotton threads in a web. In other words the flows &, of the vector field Xd have to transform an integral curve of a vector field X j , in the integral curve (of the image) of a vector field constructed as linear combination (with functions) of X I , .. . ,Xh. This will be guaranteed if all their X j ] are themselves tangent; that is, belong to the distribution Lie brackets [Xi, [Xi,Xj]= ctjXk. This just means that the distribution has to be involutive.
Chapter 6
Differential Forms
6.1 The Tensors In previous sections it has been shown how to construct the dual space E* of a given vector space E. The elements of such spaces constitute the simplest examples of tensors. A more interesting example is given by the area A(U, V ) of a given parallelogram constructed by two vectors U , V . Its most important property is expressed as follows:
+
A ( X + Y,2 ) = A ( X ,2 ) A(Y, Z ) Thus, the area of a parallelogram is a rule
A : (U,V) E E x E -+
A ( U , V ) E %,
which associates a real number with two vectors linearly in the entries U,V . Any bilinear map T , from the Cartesian product E x E to R, is called a tensor of (0,P)-type. The space of all such tensors is denoted with
C(E)E Lin(E x E , R) , 159
Dafferential F o n s
160
and can be endowed naturally with a vector space structure, defined by
(Tl f T 2 ) ( X ,Y ) = Tl(X,Y )f T 2 ( X ,Y ),
( ~ T ) ~ X= , Y~ ) ~ ~ ( X , VYk ~E 32. ) , A basis of such vector space can be easily constructed by using a basis { e i } of E and its dual basis {di}. In the given basis {ei}, the vectors X and Y can be written as
x = Xiei,
Y = Yjej ,
and we have
T ( X ,Y ) = T ( X i e i ,Y j e j ) = XiYjT(e,, e j ) = T ! j X i Y j , with Tij _= T ( e + , e j )E R. Since, by definition, for all X E E , S i ( X ) = X i , the previous relation can also be written in the form
T ( X , Y )= ~
~
~
~
~
{
~
~
~(6.1)(
Thus, by introducing the tensor product @ of two covectors, a! E E*, p E
E*, by Y f := a!(X)P(Y)>
(a! Qd P)(X,
the reIation (6.1)becomes
T ( X ,Y ) = (Ti$@Qd &j){X,Y ), or for the arbitrariness of X, Y ,
T = Tij.tsiQd . t s j , Since the tensor T is an arbitrary element of the vector space q ( E ) , the last relation shows that a basis for this space is given by the n2 elements (6*rip @}.Thus, a basis in E will fix a basis in its dual space, E*, and dso a basis in the vector space of (0,2)tensors. For this reason the n2 elements of R are called the components of the tensor T in the given basis. Similarly, any ~ ~ l ~ nmap e u rR, from the Cartesian product E* x E* to 92,
R : (cr,/3) E E' x E* -+
R(a,P)E R ,
Y
The Tensors
161
is called a tensor of (2,O)-tgpe.The space of all such tensors is denoted with
c ( E ) = Lin(E* x E*,3), and can be endowed naturally with a vector space structure defined by
(R1+ Rz)(X,Y)= &(X,Y)+ M X ,Y ) 1
V k E R. ( k R ) ( X , Y )= k(R(X,Y)),
By defining the tensor product X CED Y ,of two vectors X,Y of El to be the (2,O) tensor given by
(X@ Y)(a,P)= a(x)p~Y)V a E E*, p E E* f
1
a basis {q (&, e j } of q ( E ) is fixed in terms of a chosen basis (ei} of E. Once more, any ~~~~n~~~ map S,from the Cartesian product E* x E to 92,
S : ( a , X ) E E* x E --+S(a,X) E W, is called a tensor of ( l , l ) - t y p e . The space of all such tensors is denoted with
7,'IEf) = Lin(E* x E , 3), and c&n be endowed naturally with a vector space structure defined by
(4+ S z ) ( a ,X ) = $1
X)+ Sz(a,X)
(a1
1
( k S ) ( a ,X ) = k ( S ( a ,X)), V k E R *
Exercise 6.1.1 Show that a basis of T1(E)can be giuen bg (8' an obvious ~ e ~ n ~for t ~this o ntensor product.
@ ej},
with
Previous examples exhaust the concepts of tensor of rank 2. More generallyl any ~ ~ l tm a~p ~ ~ ~ e ~ r
T : E x E: x x E x E* x E* x - * * x E* + R l -v e . 1
p times
q times
is called a tensor of ( p , +type. The tensor T is also said to be of rank p The space of all such tensors is denoted by
T ( E )= Lin(E x Ex,...x 5 x q times
,E* x E* ; ptimes
x dT13),
c q.
and c m be endowed naturally with a vector space structure defined by
+
(TI T2)(X,Y,. . . ,Z,a, P,
= Ti(X, Y,. Z,a,P, * . ,Y)
CT2(X,Y,..
a,P,
-12,
f
*
.,TI ,
( k T ) ( X ,Y, . *.,2,CX, 8,. . . ,y)= k(T(X,Y,. . . , 2,a , P I ., . ,y)), V k E 9 , A basis of e, 1 *
*
> ekt
*
*rrT
a', PpI * *
- piq~:k:. y i y 3. * . p(-@,.. . TT -~ =
(
~ ~ : ~ $ I ~* ..d'(Z)e,(a)e,(p) ( x ) ~ ~ ( Y *)..e,(r> ~
1~7
-
~ @ e pFQD
~ @ q @ ~* * @ ~
eq~ r i ~* ~* *
e,)
x (X,Y,. * * , Z,a,P,. . . $ 7 )*
Since X I , Y,. . . 2,a,p, .. . ,y are arbitrary vectors and covectors, we can write
T = T ~ q ~ @-Sj : ~ -@S ~ @-Sk
@ e p 09
eq @
- QD e, ,
(64
is given by
which shows that a basis for the vector space
P ~ ~ P Q ~ .r p. e,p Q ~ e gDQ ~ D -~. . @ e r . "
g times
'-
ptimes
Remark 13 According t o the previous definition we can say that e A covector is a tensor of (o,l)-tgpe. The c o ~ e s p o n d ~ nvector g spuce E* i s also denoted, besides v ( E ) ,with A(&), or simply A. So A(E) = ~(~~ = E'. A vector is a tensor of (l,O)-type. The c o ~ e s p o n ~ 2 nvector g spuce E could be also denoted with $'(E). The elements of 8 are called tensors of (0,O)-type.
The Tensors
163
A tensor T of (0, 2)-type is said to be 0
symmetric if T ( X , Y ) = T ( Y , X ) ~ n ~ ~ a if~T (mX ,Y~ ) = e -T(Y, ~ ~ Xc )
The same definition can be given for tensors of (2,O)-type and, more generally, for tensors of (0,p) or (q,O)-type. As for a tensor of (1,l)-type, no meaning can be given to the i n t e r c h ~ g eof a vector with a covector. The set of ail antisymmetric tensor of (0,2) type is, of course, a vector subspace A2(E) of the vector space Q ( E ) . A basis can be easily found by considering a generic element A of A2(E). In a given basis { e i ) of E, the antisymmetric (0,2)tensor A can be written as
where the 4n(n - 1) distinct numbers Aij = A(ei,ej) are antisymmetric for the interchange i +) j. Thus
Then, by introducing the exterior (or wedge) product di A 03, of the basis elements Gi and dj by
the a n ~ ~ s y ~ e(0,2) t r ~tensor c A can also be written in the form 1 2
A = -Aijd‘ Ad’. Thus, a basis for A2(E)is given by the an(n - 1) elements {tJi A @}. More generally, a tensor T of (0, q)-type is said to be
s y ~ ~ if eT (~X t a~, Xcb r..Xc) , = T ( X I , X 2 , . . .X,) for ail permutations ( a ,b , . . ,c) of (1,2,. . . ,q) antisymmetric if T ( X a lXb, . . . x,) = - T ( X l , x2,.. . x,) for all odd permutations ( a , b , ,. .,c) of (l,2,. .., g )
.
A similar definition can be given for tensors of (q, 0)-type.
Differential Forms
164
The p-covectors
6.1.1
Antisymmetric (0,p ) tensors are calfed pcovectors or pfomzs and u s u d y denoted with small Greek letters. So a pfomn w is a function w : (Xi,..., X p ) F: E X
x
E -+ w(X1,...,Xp)E R ,
which is ptimes linear; that is, for all i = 1 , 2 , . .., p
w ( X .~. .,,X i - l , ~ Y i + b Z i , X i + l..., , Xp)
x,Xi+l, .,X,)
= aw(X1,. . . ,xi-1, f 0
I
.
bWk(X1,. .,xi-1,zi,Xi+f, . ,xp> ; I
f
I
(6.3)
completely antisymmetric
where la\= 0 or 1 according to the parity (even or odd, respectively) of the permutation CJ = (il,. . .,ip) of ( l , Z , . . ,,p). The set of pcovectors is a subspace AP(E) of the vector space q ( E ) . A basis of A p ( E ) can be found by applying the usual procedures which require, however, the notion of wedge or exterior ~d~~~ o f p covectors. 6.1.2
The exterior product
Let al,a2, , . . ,ap be pcovectors on a vector space E . Their exterior product a1 A a2 A A aP is the pform on E defined by
--
that is, by the determinant of the matrix ( a i ( X j ) ) . The properties of the determinant show that the exterior product defined by (6.5) is a pform.
The Tensors
165
By using the procedure already used for 2-forms, it is easy to check that any pform w can be written, in a given basis (ei} of E, &s follows
with wil...i,= w(ei,, . . . ,ei,) and (@}the dual basis of (ei}. Thus, the )(; = n!/p!(n- p ) ! distinct elements
make a basis in the vector space, A p ( E ) , of the p-forms on E ; that is, any p-form w can be expressed in terms of them, and then dimAp(E) =
(;)
.
The exterior product between a pform a E A p ( E ) and a q-form p E AQ(E) is the (p q)-form a A p E Ap+q(E) defined as
+
(QA
€ ) ( X I , ... ,x k + Z ) = C ( - l ) ' " ' a ( x i. l ,Xi,)p(xj,, . .. ?
xjl)
,
0
where the sum is over all permutation (T = ( i l , . . . ,i k , j l , . . . ,jl) of (1,.. . ,k + l ) and la1 = 0 or 1, according to the parity (even or odd, respectively) of the permutation. It is easy to verify that 0 0
+
a A /? is truly a ( p q)-form, that the product is
0 anticommutative: a A p = (-1)P q p A a, 0 distributive with respect to the sum: (aa+ba)AP = a a ~ p + b a ~ p , 0 associative: (aA p) A 7 = a A (p A r), 0 coincides with the product defined by Eq. (6.5) if a and p are monomials; that is, if a is the exterior product of p covectors a1,.. . ,ap and p is the exterior product of q covectors P I , . . . ,p,, respectively:
Differential Forms
166
The pair
of all pcovectors with arbitrary p , endowed with the exterior product A, is called a Grassmann algebra.
The metric tensor o n a vector space
6.1.3
A metric tensor on an n-dimensional vector space E is a (0,Z) tensor g satisfying the following requirement: 0
0
symmetry: g ( X , Y ) = g(Y,X),X , Y E E not degenerate: g ( X , Y ) = 0 , V X E E t3 Y = 0.
n t s equivalent, in a given Exercise 6.1.2 Show that previous ~ e q u ~ ~ e m eare basis {e,}, to
*
symmetry: gij = gji not degenerate: det(gdj) # 0 ,
where gcj = g(ei, e j ) are the components o f g in the basis {ei}. Since the matrix = (gij) is symmetric, there exists a basis {ci = U!ej}, with U = (U,”) an orthogonal matrix, such that g(Ei,Ej)
= Mij >
where the eigenvalues Xi, i = 1,.. . ,n are not vanishing by the hypothesis that g is not degenerate. Thus, if g is a metric tensor, a basis
exists such that gLj z g (ei3e i ) = ;tSij.
A metric tensor provides an isomorphism between vectors and covectors. In fact, with any vector X E E , we can associate the covector x = i x g defined by
The Tensor Fields
167
whose components in a given basis are (iX9)j = x 4 g * j .
xj E
Since det(gij) f 0,the previous map is invertible, so that xi = g .y*y j , where (9'j-i)is the inverse of (gij) defined by ih
9
ghj
= 6ij
6.2 The Tensor Fields
A (m,n)-tgpe tensor field S on a manifold M is a rule that associates, with every point p E M , a (rn,n)-type tensor S, E TT(7,M);i.e. a map S :p E U C M
+ S(p) = S,
Er(T,M).
By applying the same algebraic procedure used to obtain the Eq. (6.2), it is easy to see that, in a local coordinate system, a tensor field S can be written in the form
cp). The tensor field where si = cp'(p) are the coordinate of p in the chart {U, S(p) is said to be Ch ~ z ~ e r e ~ ton ~ ~abCk l emanifold M , with h 5 k 1, if the b c t i o n s S$::i(p) are Ch differentiable on the manifold M .
-
6.2.1 The Lie derivative of a tensor field Since the Lie derivative has been defined on functions, differentia^ 1-forms and vector field, it is also defined, by the Leibnitz rule, on a general tensor field as the one given in Eq. (6.7). One of the most important use of the Lie derivative in physics is to check if a tensor field is invariant under some transformation. If the transformation is generated by some vector field A, then the invariance of the tensor field S is expressed by
LdS===O.
Difleerentid F o m s
16%
The invariance condition preserves its elegance, also locally. For instance, let the mixed tensor field
s : ( X ,a)+ sjx,a) be locally represented by
Its Lie derivative, with respect to the vector field A , is given by
Thus, the invariance of S is expressed by
In terms of the matrices
it can be written as follows:
dr For the interested reader, an intrinsic ~ e ~ n i t i can o n be given as in the case of a vector field. The Lie derivative, with respect to X of a tensor field S, is defined by
169
The Tensor Fields
where +r
is the flow of the vector field X
bT: 0
p M ~+br'(P) E M ,
(@)* its derivative
to the whole tensor algebra
m,n=O
Thus, in our case ( 4 - T ) * b q p ): ~ ( f ( P ) + ) 7 ( P )*
This definition can appear formally complicated. In reality it is very simple from a geometrical point of view. As a matter of fact, a map 4 between two manifolds M and N , transforms a curve trough p E M to a curve trough the point &(p) E N; then, it also transforms a tangent vector to M at p in a tangent vector to JV at +(p). So 4 induces a map, &, between the corresponding tangent spaces 7,M and T&,)N at corresponding points. Of course, it also induces maps between the tensor spaces z m ( 7 , M ) and zm(%T(,>M) at corresponding points and, finally, between 7 ( p ) = C&,=ax m ( 7 , M ) and 7(dT7(p))= CZ,n=o%m(%~(p)M). If S is a (1,~)-tensorfield, the relation (a) S(Y1,Y2,.. , ,Y')) E S ( a ,Yl)Y2,.. . ,Y')
defines a vector field s ( Y 1 , Y 2 ., .. , Y r ) . It can be easily proven that, for any vector field X,we have
(LXS)(Y')Y2,.. .,Y') = [X,S(Y',Y2,.. * ,Y')] T
i= 1
s ( Y 1 , . . ,[ X ,Y i ] ., . . ,Y')
.
(6.10)
Daflerential Forms
170
The Leibnitz rule gives the following general properties of the Lie derivative:
s + R 63 (LxS ) , = (LxT)(a1,.. . ,ffp, x1,. . . , X,)
L x ( R@ S) = (LxR)€?J
Lx(T(ff1, * . . , a p , x 1 , . . , X,)) *
+
c
(6.11)
t l
T(a1,. . . , L X W , .. * , a p , x1,. . * ,X,)
i=l
Equation (6.10) is just a particular case of Eq. (6.12).
Exercise 6.2.1
Show that f o r any vector fields X and Y
:
The differential p-forms
6.2.2
A diflerential 1-form a on the manifold M is a regular map
a:TM+R of the tangent bundle of the manifold
(6.13)
M in R,linear in every tangent space
7,M: a p ( a X + b Y ) = a a p ( X ) + b a p ( Y ) , V a , b E X , VX,Y E T M . In this way, a differential 1-form on M is a covector on 7,M differentiable in p . Let us suppose that the functions d ,. . . , zn are a system of local coordinates in a given domain U of the manifold M , zi: po E u
-+ 2 ( p o ) = z;
E 8 vi = 1,. . . , n *
These functions are differentiable, and their differentials dzk, at the point PO, dxko : X E T O M+ dzk,, (X)E R ! are covectors on
TOM.
V i = 1,. , . ,n ,
The Tensor Fields
171
The values of the differentials dxbo,.. . ,dzgo on the vector X are the components X', . . ,X" of the vector. If am is any covector on &,N, because of the linearity of ape, we have
.
= (ai(Po)dz;o,X ) , with
The covector apocan thus be expressed locally in the form Qpo
+ + a,(po)dzjf,
= al(po)dz;o
* ' *
Therefore, every differential 1-form a (Eq. (6.13)) in the domain U can be locally expressed as Q
=q(p)dx'
+ * . . + a,(p)dz"
.
A Ic-covector wp at the point p E M is a k-times linear (Eq. (6.3)) and antisymmetric (Eq. (6.4)) function, up: ( X I , . . , X k ) E G M
X
X
&M + w p ( X 1 , . .. ,Xk)E 8 .
(6.14)
A di8erentira.lk-form w is defined on the manifold M if the form (Eq. (6.14)) is given at every point p in M and, moreover, if it is differentiable. Every differential k-form w can be uniquely expressed in a domain with local coordinates I' , , . . ,z" as 1
w = -w'z l . . . i k(d, . . . , zn)dZil A
. * A dxik , (6.15) k! A dxik are the exterior products of the basis 1-forms
where dxil A d x l , . . ,dzn. Operations such as the sum of k-forms, the product with real numbers, the exterior product between forms are always point-wise possible; that is, at every
.
point p E M the corresponding exterior forms on the tangent spaces T,N can be summed and multiplied with numbers or exteriorly.
Lie derivative of a differential k-form F'rom Eq. (6.9),defining the Lie derivative of a tensor field, we obtain, for a differential k-form w , the useful formula
(L*w)(Y',Y2,... , Y k )= (X,W(Y',Y2,* . ., Y k ) ]
which is similar to the one given, for a (1,k)-tensor field, by the Eq. (6.10).
6.2.3
The exterior derivative
On the space of differential k-forms we can define an operator d , called exterior derivative, having the following properties:
If a E A k ( M ) 1,3!
E
ifk(&),
y E A~(M),
+
(1) d(.. 4-P } = da d p ; (2) d ( a A yf = d a A r + ( - l ) k ~A d r ; (3) d 2 a = 0; (4) On the differential 0-forms; that is, on functions, the operator d coincides with the differential defined in Sec. 5.5. The operator d, as it easily follows from its properties, transforms differential k-forms in differential (k t 1)-forms. By using the properties (l), (2), (3) and (4), we can easily calculate the exterior derivative of a k-form in a coordinate basis. For w given by Eq. (6.15), we obtain
because ddxi = 0.
The Tensor Fields
6.2.4
173
closed and exact diflerential forms
A differential k-form is said to be closed if
dw = O , and to be exact if there exists a E A k - l ( M ) ,such that
w=da.
Since dl = 0, an exact pform is also closed. The converse is not true and the following is a classical example in R2,
Example 23
Consider the differential 1-form W =
xdy - ydx X2fY2
which it is easy t o see to be closed, h = O .
In polar coordinates x=rcosd, y = rsind,
it becomes
Thus, one i s tempted to say that w i s an exact differential form also. But the angles do not exist really! The misunderstanding is solved by obseruing that w is not defined at the point (0,0 ) , as well as the transformation from Cartesian to polar coordinates. 6.2.5
The contraction operator ix
-
Let E be an n-dimensional vector space and h T ( E )be the vector space of r-covectors defined on it. If w E A'(E) is an antisymmetric multilinear map from E x E x . x E to R, and X I , X 2 , . . .X , are vectors of El then
-
4x1,x 2
* *
.X r )
(6.17)
Dzflerentiol Forms
174
is a real number antisymmetric under the interchange of any two vectors. Therefore, an r-covector is defined once the number (Eq. (6.17)) is given V X i , i = 1 , . . ,r. It is natural, starting from any r-covector w E A'(E) and a vector X E El to define a (r - 1)-covector,namely i x w E A'-'(E) by the following equality:
.
( i x w ) ( X , ,x2,... X?.-I) := W ( X , X l , xz,. . .Xr-1). In this way, ( i x w ) is the (r - 1)-covector, built from w f A'(E) and X E El which evaluated on (T - 1)vectors X I ,X Z , . X,--l, gives the same real number given by w on the r vectors X , X i , X 2 , . . X r - l . The operator ix is called the contraction operator with respect to X. We already met it in the case in which w is a simple covector. In fact, by denoting with CY an element of A'(E) = h ( E ) E*, the previous definition simply reduces to
.
..
=
ixCY = a ( X ) 5 (a,X ) In order to illustrate the given definition, let us represent the r-covector w in a basis ( ~ 9 2 )as ~ follows: 1
w = -wij...kdi A 6' A * * * A 8"
(6.18)
r!
Thus,
1
(ix,w)(X2,. . . ,XT)=: 2 w i l i2...irdet
1
(6.19)
and then, by using the Laplace expansion (first column) of the deter~inant, i x w is represented by
The Tenaor Fields
175
(6.21)
What are the properties of the operator ix? 0
It is easy to see that ixiy=-iyix
(6.22)
This easily follows by observing that, 'dw E A'(E),
w ( X , Y,XI, x2, . . Xr-2)= (ixw)(Y,x1,x2,. * . xr-2) =
W(Y,X,Xl,XZ,. .x 1
(iyixw)(Xl,x2, ...xr-2)
4= (iYW)(X,Xl,X2,. . .Xr-2) =
(ixiyw)(X1, x2,. . . Xr-2).
As a particular case, it follows that
ia 0
If a E A r ( E ) ,and p
E
=o.
(6.23)
h 5 ( E ) ,with r + s I n, then
This easily follows from the following formula:
+
where the sum is over all permutation (jl,$2,. . . ,j r + s ) of (1,2, . . . ,r s) and CT = 0 or 1, according to its parity (even or odd, respectively).
Properties of Eqs. (6.23) and (6.24) extend in a natural way, with the only additional requirement that on 0-forms f E .F(M), ixf = 0, to r-forms on a differential manifold M . Let X be a vector field on M and a E h k ( M ) .
Da;tferentiafForms
176
The operator i x which, acting on the differential k-form a, transforms it in a differential (k - 1)-form i x a (also called interior product between X and a),is defined po~nt-wiseas (ixa)p(xl,.
xk-1)
== a p ( x ( P ) ,X I , *
* *
,xk-1)
1
\JP E M ,
(6.25)
where X I , .. . ,X k - 1 are tangent vectors to M at p . The i x operator fulfills the following properties: (1) ix(a1+ ( 2 2 ) = i x a 1 f ixff2; (2) i x ( a A P ) = ixa A p ( - I ) ~ c xA i x p ; (3) if Q: E A ' ( M ) , ( i x a ) ( p )= ( X ( P ) a*) , = ap(X(II))); (4) i f f is a 0-form, then i x f = 0 .
+
Thus, the properties of the interior product i x on a differential manifold
M are a~gebraicallysimilar to the ones of the exterior derivative d, namely d2 = 0 , d(a A @ ) = ( d a ) A P
+ (-1)'a A d P .
Of course,
ix : A'(M) -+ Ar-'(M) d : h'(M)
+ Ars'(M),
and ixd : A'(M)
-+ A'(M)
dix : A'(M)
-+ A r ( M ) .
The operators ixd and d i x do not coincide, as it is easy to verify on the simple example of a = dxi . Indeed, denoting with X ithe components in the basis fa/8xi} of the vector field X , we get ( i x d ) d ~= ' 0,
(6.26)
8X' ( d i x ) d z i = d(ixdzi)= d X i = - d d . Moreover, the operators ixd and dix are not derivations, since ixd(a A p) = ix[(daaf A p
+ (-l)raA dp]
177
The Tensor Fields
+ (-l)'+'(da)A ixp + (--l)'(iXa) A dp + (-1)'+'o A i x d p ,
= ( i x d a ) Ap
and
d i x ( c u A p ) = d [ ( i x d a ) A P +(-1)'aAixp]
+ (-1)'d[a A ixp] = (dixa)A p + ( - l ) ' + ' ( i x ~A) d p + ( - l ) r ( d ~A )ixp + (-1)"'a A dixP.
= &[(ixa) A p]
However, by adding ixd(a:A p) and dix(a A p) from the above relations, we obtain the result that the operator i x d dix is a derivation, since
+
(ixd
+ dix)(aA P ) = [(ixd+ dix)a]A p + a A ( i x d + d i x ) P .
(6.27)
Finally, let us remark that the three operators L x , i x and d are not independent on A'(M). It is easy to see that, on r-forms w E A'(M), they satisfy a very useful relation, the so-called homotopic or Cartan identity:
Lxw = ixdu + dixw ,
(6.28)
Lx=ixod+doix.
(6.29)
or in operator terms,
Pro0f, 0
I f f E F ( M )= A'(M), since i x f = 0, we have ixdf
0
+ dix f
= ixdf = ( d f ) ( X )= X f = L x f
,
For a generic 1-form cr = fdg E A(M):
ixda = ix(df A dg) = (Xf)dg- (df)Xg,
+ fd(Xg).
dixa = d(fXg) = (df)Xg
(6.30)
178
Thus,
ixda
+ d i x a = ( X f f d g + fd(Xg) = ( L x f ) d g + f L x d g = LXQ, (6.31)
where the Cartan identity on functions has been used:
+
f d ( X g ) = f d ( i x d g ) = f(dix)dg = f(&x i x d ) d g = fLxdg. The proof proceeds now by induction.
A more elegant proof can be found in the Kobayashi-Nornizu book, and it consists in observing that (1) i x d + dix is a derivation of degree 0; (2) every derivation of degree 0 commuting with d is the Lie derivative with respect to some vector field; ( 3 ) the derivations L x and ixd dix give the same result on f E F ( M ) .
+
From Eq. (6.29) directly follows the useful formulae
6.2.6
A ~
~pr0cedup.e ~
e
~
~
t
The fact that the three operators d , L x and ix are not ~nde~endent on differential forms, suggests the following different procedure to define the exterior derivative in terms of the interior product and the Lie derivative. Let us observe that, by using the Cartan identity, we have
0 for a function f: ixdf == ( d f , X ) E L x f
,
0 for a differential 1-form a E h ( M ) : ( d f f ) ( XY , ) = iyixda = iy(Lxa - d i x f f )
= (Lxa, Y }- i y ( d i x a ) = (Lxa,Y )- iyd(ixa) = ( L x a ,Y ) - LY (a,X )
The Tensor Fields
179
where the property that f zi ixa! = (a, X ) is a function, to which the previous formula can be applied, and the Leibnitz rule has been used,
0 for a differential 2-form w E A2(M):
0
for a function f, as
0
for a differential 1-form a E A ( M ) ,as
+
( d a ) ( X Y) , -= (Lxa,Y ) - (LY%X)
(0,
[X, Y]),
Daflerentaol Forms
180 0
for a differential 2-form w E A2(M),as
& ( X , y, 2 ) = L X W ( Y , 2 )- LYW(X, 2 )+ (LZW)(X,Y )+ W ( [ X , Y], 2) 0
1
for a differential pform w E hP(M),as
&(XI,.
. . ,X,,,)
E
~ ( - l ) l " l L x i w ( x a ., ., .Xi,) -
C ( - l ) ' " ' W ( [ X iX , i l ] ,. . . , X i , ) , (6.32) U
where the sum is over all permutation o = (i, i l , . . . ,i p )of (1,.. . , p + l ) and lo(= 0 or 1, according to the parity (even or odd, respectively) of the permutation.
Exercise 6.2.2 Prove, b y using as definition the one given in the Eq. (6.32), all the properties of the exterior derivative. 6.2.7
A dual characterization of holonomic and anholonomic basis
Let us return to the discussion in Sec. 5.7.1 and consider a generic basis { e i } of vector fields on an n-dimensional manifold M : [ei,ejl = c$eh.
The dual basis {gi}has the point-wise property ( S k , e j )= hjk
.
By taking the Lie derivative, with respect to the vector field ei of the previous expression, we obtain ( L e i 8 ' , e j ) = -(6',[[e,e.1) = -c$(6',eh) = - c ikj . z,
3
Then, by using the Cartan identity, we have ( d d k ) ( e ie, j ) = -cFj
.
The exterior derivatives ddk are differential 2-forms and the above formula allows us to evaluate their coefficients dfj in the given basis, in which ddk = dF,d' A 6".
The Metric Tensor Field on a Manifold
181
We obtain
d,k,(Sr A P ) ( e , , ej) =
-~tj,
or 2dFj = -cajk
I
Therefore, the elements of the dual basis {Sa} have the following property: dt?k f --cijS l k 2
A @'.
i
(6.33)
We c a n summarize the previous results as follows: If {ei} is a basis of vector fields and { d i } its dual basis on an n-dimensional manifold M , then 1
[e,, ej] = ckeh H dSk = ---cfj29' A @ . 2 Therefore, for a holonomic basis, given c& = 0, the dual basis consists of closed differential 1-forms Sk,dSk = 0,and then, locally, coordinates functions { z i } exist such that
dk = dxk As
B
.
consequence,
a
ei = axi *
Thus, besides the one given in Sec. 5.7.1, a new characterization of a holonomic basis {ei} is given by the closure property of the differential 1-form which composes its dual basis. 6.3 The Metric Tensor Field on a Manifold
A metric tensor field g on a manifold M is a rule that associates with every point p E M a symmetric and not degenerate (0,2)-tensor g ( p ) . Thus, at every point p E M , g ( p ) is a metric tensor for the tangent space T,M, and the considerations, already done for a metric tensor on a vector space, can be repeated. In particular, in every tangent space 7 , M , a basis can be chosen such that gij ( p ) = rtdij.
Dafferential Forms
182
Since a metric tensor field is required to be at least continuous and integers do not change continuously, the canonical form of g has to be constant everywhere and we speak of signature of the field g . The collection of the bases in which g takes on the canonical form, defines a globally orthonormal basis on M , but this global basis is not generally a coordinate basis. In this sense the space Xn,considered as a manifold endowed with the Euclidean metric tensor field (6ij at every point), constitutes just an exceptional case. Even in that case only the Cartesian coordinates generate an orthonormal basis. 6.3.1. Killing vector fields The Killing vector fields play a relevant role in the study of the isometries of a metric tensor field; this is why they are usually used in general relativity. They are defined to be the vector fields A preserving a metric tensor field g ; that is, by the invariance condition
The above equation, given g , admits very few solutions for A . Let the metric tensor field 9 : (X,Y) -+
g(X,Y)
be locally represented by g = gijdxi 8 d x j . Its Lie derivative, with respect to the vector field A, is given by LAg = LA(gijdxi @ d x j ) = (LAgij)dXi 8 d d
+ gij(LAdXi) 8 dxj + gijdxi 8 (Lad.')
(6.34) Thus, the invariance of g is expressed by
(6.35)
The Metric Tensor Field on a Manifold
183
In terms of the matrices
Eq, (6.35) can be written
ELS
foIIows:
where the symbol T denotes matrix transposition. 6.3.2
~
~
i symmetric m a manifolds ~ ~ ~
We may now ask the following question: how many vector fields, leaving a metric tensor field 9 invariant, exist on an n-dimensional manifold M ? By introducing the differential l-form t by
(4, x>= s ( 4 XI
8
Eq. (6.35) can also be written in the following form: (6.36)
where (6.37)
are called the ~ h r ~s ~s~ b ~o Z s~. ~ ~ l The number of independent differential equations, in the partial differential system (6.36), is (1/2)n~n+ l), while the number of unknown functions is n, so that the system (6.36) is overdetermined for n > I, and the number of Killing vectors will be upper-bounded, By taking the derivative of Eq. (6.361, we obtain
; (b) f(0, ). = f(., 0) = z; (c) there exists a differentiable map
I(., 4.))
E
:A
+ W, such that
= f(E(ZC), 2) = 0 *
Thus, given a Lie group G it is always possible to build a local Lie group, with the identification A f yQ') c an. Two local Lie groups, (A1,f l ) and (A2,f 2 ) , are isomorphic if there exists any two neighborhoods, A{ c A1 and A', c A2, of the origin of R" and a diffeomorphism 2c, : A', Ah such that the diagram I
1
is commutative, as to say
+(fl(., 9)) = f 2 ( ( 2 c , x 1cl)(z,Y))
= f 2 ( l L ( Z ) , Ilr(Y))
1
v (2,Y> E A', x
A:
Of course, all local Lie groups obtained from the same Lie group G, with the previous procedure, are isomorphic among themselves.
8.2 Building of a Lie Algebra from a Lie Group 8.2.1
Lie algebras
The algebraic definition of Lie algebra has been already given in Part I. Here we are going to give just a mention about three types of algebras with great
hie Groups and Lie Algebras
214
relevance; they are Abelian Lie algebras, simple Lie algebras, and s e ~ ~ - $ i m p ~ e Lie a l g e ~ ~ a s . Vector spaces endowed with an identically vanishing c o ~ m u t a t o cr o ~ t i t u t e the so-called commutative or Abelian Lie algebras. The definitions of simple and semi-simple Lie algebra need the introduction of a further concept, that of ideal in a Lie algebra. A subspace Zof a Lie algebra A is said to be an ideal if
that is z E Z, if [y,x]E Z for every y E A. Of course, since [y,4 = -[z, 3) and Z is a vector spacet if [z, yf E Z for every 3 E A, then x E Z.Notice that this implies Z is a subalgebra. ~ ~ ideals ~ in zA are a (0) ~and A. A Lie atgebra. coIitaining just trivial ideals is called simple. A Lie algebra containing nontrivial ideals, but none of them Abelian, is called a semi-simple Lie algebra. There exist different methods to build a Lie algebra from a Lie group. Here we are going to give account of the two most significant methods. The first of them is based on the use of differential operators on the group. 8.2.2
&eft inwariant vector Pelds
Let G be a finite dimensional Lie group. For every g E G, the left translation
L, : h E G -+ L,(h) = g h E G is a diffeomorphism from G to itself. Any neighborhood of e is mapped by left translation along a particular g onto a neighborhood of g , so that the map carries curves through e into curves through g, and curves through h into curves through gh. Then, the derivative of the map at point h, namely (A@)&, is a linear map from the tangent space ThG to the tangent space q h G ,
If by
v is a vector field, its value V ( h )at point h belongs to ThG. Its image
(&)+h,
which belongs to 7&G, will be denoted by ~ g V ) ( g ~i.e. );
215
Building of a Lie Algebra from a Lie Group
so that we have
(gV)(h) =(
(8.5)
~ g ) * g - d W W'
A vector field V on a Lie group G is said to be left i n v a ~ a n tif
(SV)(h)= V(h) v g ,h E G7 )
or equivalently, if
V(gh)= (Lg)*hV(h) * The addition of vector fields on G and their product with real numbers can
be naturally defined as follows:
(V + W ( g ) = V(g>+
wl) t/g 9
G,
(.lV>(g) = .lV(g) 7
so that, by the linearity of the operator (Lg)*gr it follows that
The set of left invariant vector fields o n G is a vector space over !I?. Moreover, e
A left invariant vector field on G is uniquely determined by its value at the identity element e of the group G .
Indeed, if V ( h )and W ( h )are left invariant vector fields on G
The vector space of the left invariant vector fields is isomorphic to ZG,
the t ~ ~space ~ to e G~ at te, Indeed, with every vector V, E 7,G, we can associate a vector field V ( h ) on G by means of the operator (Lh)*e
V ( h )= ( L h ) * e ( V e ) 1 V h E G ,
(8.6)
Lie Groups and Lie Algebras
216
the left invariance of V ( g ) following from
Let us consider now the set of differential l-forms a on G that constitutes a vector space on !R if the sum and the product with real number T are defined as
A useful notation for the operator (Lg)*is given by the symbol dL,. Then, relations (8.5)and (8.6) can be rewritten as follows: ( 9 W h )= d W % J - l h ) ),
V ( h ) = dLh(V,)
(8.7) (8.8)
1
Let us introduce the transposed operator dLi of dL,, which acts on the differential l-forms a, by
v,dL;(a))= (dLg(V)
I
4 1
(8.9)
where V and a are a vector field and a differential 1-form on G, respectiveIy, and the brackets (., denotes, as it is usual, the interior product. Thus, dL, and dL: are the following operators: a}
dLg : G G -+GhG, dLz :Tj,G
+7 z G .
(8.10)
A differential l-form a on G, is transformed by means of the translation dL: in a differential l-formga on G,according to the relation (9ff)(h)= d ~ ~ ( ~ ~ g h ) ) .
A differential l-form a! is said to be left invariant if (9~)(h) = ff(W
f
v9,h E G .
Since the linearity of dL, implies the linearity of dL:, we have that
217
Building of a Lie Algebm from a Lie Group
The set of left invariant d ~ ~ e ~ eI-forms n t ~ ~on l G is a vector space on 92. Moreover e
A left invariant diflerential 1-form is uniquely determined by its value in e.
Indeed, if a and a‘ are two left invariant differential 1-forms, such that a(e>= d ( e ) , we have
a ( g ) = (9-’a)(g) = dL;-l(a(g-lg)) = d L i - l ( a ( e ) )
= d L ~ - ~ ( a ’ ( e=) )d L ; - ~ ( ~ ’ ~ g - ’ g )=) a’(g) . As the vector space of the left invariant vector fields is isomorphic to T,G, so the vector space of left invariant differential 1-forms is isomorphic to T G . If a, denotes a covector on 7,Gl the differential 1-form a ( g ) , defined by a ( g ) = dLrf-i ( ~ l e )
Vg E G,
is a left invariant differential 1-form. Indeed, since
dL& = dLS; o d L i , we have
(ga)(h)= dLS;(a(gh))=
* dL&h)-l)(@e)
= (dL: o dL;-l9-1)(ae) = dL;-l(ae) = a(h)
VghE G.
Thus, with every covector on 7,G we can associate, in a unique way, a left invariant differential form on G. An interesting and useful result is the following: I)
The contraction (a,V),between a left invariant differential 1-form a and a Zeft i n v a r i a ~vector ~ field V, is constant on G.
Indeed, (a1V ) ( g )=
(ff(d1 Vb))
= (d-q-l(a(e)), dLg(V(e)))
Lie Groups and Lie Algebras
218
=W =
g - 1 0
d L g ) ( ( f f ( e )V, ( ~ ) ) )
(44,V ( e ) )
7
vg E
G.
There is a converse to this, namely a
A vector field V on G, for which (a,V ) i s constant o n G for every left invariant differential 1-form, is a left invariant vector field.
Indeed, { ~ ( h9V(h)) ), = ( 4 h ) ,~ ~ ~ ( V ( g = - l{ h~) ~~ ~ ( v(g-'h)) ~ ( ~ ) ) , = ( 4 9 - lh),V(9-'h)) = {ff(h~, V ( h ) }*
Since the left invariant form a is arbitrary, then
(gV)(h)= V ( h ),
Qg,h E
G.
These two properties, together with the useful relation*
d 4 X , Y ) = L x ( ( f fy>> , - LY((&,X ) ) + ( a ,[ X ,'YI) ,
(8.11)
allow us to prove the following statement:
* If X
and Y are two left i n v u r i a ~ ut e c ~ o ~ ~ eolnd as Lie group G, their Lie ~ r a ~ ~[ eX t,Yj s is a left i n v a r i a ~vector ~ field.
To this purpose, we just have to prove that {a,IX, Y]) is constant on G for every left invariant differential 1-form a(g). Indeed, if a is left invariant, then (a,X ) and (a,Y ) are constant on G, because X ( g ) and Y ( g ) are, by hypothesis, left invariant vector fields, so that the Lie derivatives Lx((a,Y ) ) and L y ( ( a ,X ) ) vanish identically. Therefore, we have
W X ,Y ) = (a,[ X ,yi>*
(8.12)
Since d a is an exact %form, for which d d a = 0, and the right hand of Eq. (8.12) is a O-form; that is, a function on G, then
(a,[ X ,Y ] )= constant. *In this chapter the Lie derivative, with respect to a vector field X , has been denoted with the symbol L x , instead of L x , to avoid confusion with the left translation Lx.
Building of a Lie Algebnz from a Lie Group
219
Thus, by using the isomorphism between 7,G and the vector space of the left invariant vector fields, it is possible to introduce, in the tangent space Z G , a commutation relation which, being bilinear, antisymmetric, and satisfying the Jacobi identity, endows it with a Lie algebra structure. To be specific, if X e and Ye denote two vectors belonging to 7,G, the Lie brackets of the two left invariant vector field on G corresponding to them, still is a left invariant vector field which also is uniquely determined by its value at the identity element of the group. Thus, given Xe and Ye belonging to 7,G, we define the Lie commutator of X and Y as the value, at the identity element e of the group G, of the Lie bracket of the corresponding left invariant vector fields
This Lie algebra is called the Lie algebra of the Lie group G.
8.2.3
The adjoint representation of a Lie group
There exists a second method, &s well, which allows us to introduce a Lie algebra structure in the tangent space 7,G. Let us observe that the map
A, : h E G -+ A,(h) = ghg-' E G , composed of the left translation by g and the right translation by 9-'
A, = Rg-iL, : h E G + (R,-i L,)(h) = ghg-' E G , is a onsto-one, differentiable map. Actually, since A;' = A,-I, the map is a diffeomorphism of G into itself. Since
A, is a homomorphism of G into itself. Actually, A, is an isomorphism of G into itself, as to say an inner automorphism of G, since A,-I = A;'. Notice that each A, maps the identity element e into itself, so that every curve through e is mapped into a, possibly different, curve through e.
Lie Groups and Lie Algebras
220
Therefore, the derivative
at the unit e , usually denoted with Ad,,
Adg : Z G
-+7,G,
is an invertible linear map of any tangent vector of 7,G to another one in 7,G. For the automorphism A,, we have
and for derivatives
so that
A d f g = Ad$ o Ad,
.
The set of all invertible linear maps of Z G into itself is a group whose internal composition law is the usual composition of maps. This group is denoted by Aut 7,G. Thus, the map
Ad : g E G 3 Ad(g) = Adg E Aut Z G
(8.13)
is a homomorphism of G into the group Aut7,G of the invertibIe linear maps of the vector space 7,G. Once a basis in 7,G is chosen, the map Ad becomes a homomorphism of G into the group GL(n,S), where n is the dimension of 7,G. The group GL(n,8) is the group of nonsingular real matrices n x n and can be endowed with a differential manifold structure. The compatibility of group and differential manifold structures promotes !I? ~is), the group GL(n,92) to a Lie group. Obviously, the dimension of G ~ ( n2.
Thus, the map Ad is a representation of G on 7,G and is called the adjoint representation of the Lie group G. The tangent space to GL(n,R)at the identity I (the unit matrix) is the of the ~ not 8 necessarily ) , invertible real matrixes space, denoted with ~ ~ t ~ n x n. The map Ad is differentiable and its derivative (Ad)*, at the unit e is a linear map of Z G in EndKG, the vector space of the (not necessarily
Building of a Lie Algebra from a Lie Group
221
invertible) linear maps of 7,G into itself, that are endomorph~smsof 7,G. In other terms, End7,G
= T(Aut 7,G) .
The derivative (Ad),, is denoted with the symbol ad,
ad : V E 7,G 4 ad(V) = adv f End7,G. A one parameter subgroup of a Lie group G is a representation of R in G, as to say a homeomorphism of !I? in G; that is, a differentiable map
R -+ p(t) E G ,
p :t E
such that p(0) = e , p(t i-t') = p(t)p(t'),
Let
V t , t' E R.
Ve be an element in 7,G and let p v , :t E R
+ pv,(t) = etve E G
(8.14)
be the integral curve of the left invariant vector field ( L g ) * e ( Kon ) G. Let us also fix s E R and define the map X l :t f
92 -k xft) = PV. (S)PV* ( t )= L,ve (o)Pv, ( t )E G
9
where Lpve(dl is the left translation by p v , (9). Since the vector field (La)*e(V)on G is left invariant, we have
SO
that XI($)is an integral curve of V = (LQ)*e(Ve)through p v , ( ~at ) t = 0. On the other hand, the map
xz : t E R! -+~
2 ( t= ) pv.
( s - t t )E G
is also an integral curve of V = (Lg)*e(Ve)through p v , (3) at t = 0. , the integral curve of V = (LQ)*e(&) through Thus, x l ( t ) = ~ 2 ( t ) since pv,(s) at t = 0 is unique. As a consequence, we have
Pv, (8 4- t ) = PV. (S)PV, (t)
(8.15)
222
Lie Gmups and
Lie Algebras
and
&om Eq. (8.15), it follows that the map (8.14) is a homeomorphism of R in G, and then a one parameter subgroup of G. This subgroup is unique, Indeed, if
u : t E 92 + 6(t) E G is another one parameter subgroup of G such that
then
a(t f s) = a(t)a(s)= L V ( t ) 6 ( S ) . Thus ,
that is, G ( t ) is an integral curve of V = ( L ~ ) * e through (~) e a t t = 0. Since, Eq. (8.14) shows that pv, is an integral curve of V = ( L g ) * e ( V e ) through e, then pv = 0. We can conclude that, with every vector V, E 7,G, there is associated a unique one-parameter subgroup pv, (t ) of G. By using the notation pv, ( t ) = etv- , we can write
The explicit expression of the operator a d v can be easily found. Indeed,
so that the value of the operator adv, on a vector We E T,G will be given by
Building of a Lie Algebm from a Lie Group
223
On the other hand, etVe is just the value at e of the flow of the vector field V ( g )= ( L g ) * e ( K ) , as it is easily followed by
&$)
= getv= = Retveg,
so that
etv = & ( e ) .
Thus,we have (8.16)
By Eq. (8.16) and by the well-known properties of the Lie derivative, we can define the following bracket in 7,G:
[. 1 *]
=L:
(Ve, We) E 'ZG x T",G-+fVe, We]= adv,(We) E 7,G
(8.17)
which can be easily seen to be bilinear because
a~c,xl+caXa(y)= cladx, ( Y )+ czadx,(Y), adx(c1YI + c2Yz) = c1adxfY1) 4- a a d x ( Y 2 ) , e
whatever X I ,XZ, V,Y I ,Y2,Y E 7,G and antisymmetric because
adx(Yf = - u d y ( X ) ,
C I , c2
E 8 are
chosen;
V X , Y E 7 , G ; and
satisfying the Jacobi identity
U d x ( u d y ( 2 ) )f d Y ( a d z ( X ) )-t- ~ d z ( u ~ ~=~0.Y ) )
224
Lie Groups and Lie Algebras
The composition law, defined by Eq. (8.17), provides the vector space 7,G with a Lie algebra structure. In conclusion, given a Lie group G, it is always possible to build from it, a Lie algebra. Now we can ask whether, given a Lie algebra A, there exists a Lie group of which, vice versa, A is the algebra. The answer to this question is given by the following theorem:
Theorem 26 (Cartan) Every Lie algebra is the Lie algebra of some Lie group.
In the previous section we spoke about local Lie groups. They are also related to the Lie algebras because every Lie algebra is a Lie algebra of some local Lie group. Moreover, the local Lie groups are isomorphic if and only if the corresponding algebras are isomorphic as well. Let A1 and A2 be two Lie algebras, and GI and G2 the corresponding Lie groups. By the last we can build two local Lie groups Gi and G;, which will be isomorphic if A1 and A2 are isomorphic. However, the fact that G\ and GL are isomorphic does not imply that G I and G2 are so. In this case, we speak of local isomorphism between G I and G2. For a simply connected Lie group G , the following theorem holds: Theorem 27 (Monodromy) If G is a simply connected Lie group and F any Lie group, every local homomorphismt of G in F is uniquely prolonged in a global homomorphism of G in F . Let us denote with GL(n,R) the Lie group of n x n invertible real matrices and with GL(n,R) the corresponding Lie algebra, which is given by the vector space of n x n real matrices with the commutator as Lie bracket. A very important result is the following:
Theorem 28 (Ado) Every Lie algebra of a Lie group is a subalgebra of GL(n,R) for some vaEue of n. For Lie groups, the analogous statement holds only locally; i.e.
Every Lie group is locally isomorphic to a subgroup of G L ( n , R ) for some value of n. By this theorem the local isomorphism between GI and G2 is prolonged to a global isomorphism. *A local homeomorphism is a homeomorphism of the correspondent local groups.
225
Budding of a Lie Algebmfim a Lie Group
Thus, we can conclude that just one simply connected Lie group G corresponds to a Lie algebra A. 8.2.4
The ~
~ ~ ~ ~ ~a e n of j~ aa Lie ~ o~g ~ o un F ~
~
We can introduce translation operators also in the dual space T G of 7,G. As for the left translation, we can use relations (8.9) and (8.10); the right t r ~ s l a t i are o ~ defined in a perfectly analogous way.
(V,dR;;(4)= (dR,( V )I 4 dR, : XG
+x,G,
dfi; : T g G -+ T G .
It is also possible to define the operator Adz, dual of the operator Ad,, as follows:
(Vt Ad~(a!)} = (Adg(V),af
-
(8.18)
B y Eq. (8.18) and by the properties of Ad,, we argue that
Ad: : T G -+ r G is an invertible linear map of 7,G into itself. The map
A d * : g E G - + A d * ( g j = A d 3 ;~ A u t r G ,
(8.19)
as the one in Eq,(8.13), is also a representation of the Lie group G. It is called c o ~ $ j o i ~ t p ~ ~ e ~of tthea Lie t ~grogp o ~G The map (8.19) is differentiable. Its derivative (Ad*)*,at the unit, denoted by ad*, is the map ad* : V E 7,G -+ ad*@') = ad; f End T G .
The operator ad? is the conjugate of a&, and
( W , a d ; ( ~ )=) ( a d v ( W ) , a ! ) , Va! E r G , VW E 722. The vector spaces 7,G and T G ,endowed with a bracket giving them a Lie algebra structure, are usually denoted by the symbols G and G*. The coadjoint representation of a Lie group has an important role in classical mechanics. As we will see, the orbits of the group under the coadjoint representation are symplectic manifolds,
~
Lie Groups and hie Algebms
226
This will be shown in Part 111, in the chapter Orbits method$ after the in~roductionof some preliminary concepts, An exhaustive discussion an this subject can be found in Ref. 41. The second part of this book is in fact completely devoted to Reduction, Actions of Group and Algebras.
Further Readings
R, Abraham, J. E. Marsden, and T. Ratiu, ~ a n g f o ~Tensor ~ , A n ~ y s and ~, A ~ ~ Addison-~esley, ~ ~ c 1983). ~ ~ ~ o ~ e B. Dubrovin, S, Novikov, A. Fomenko, Gkomktrie C o ~ ~ e .&&ions ~ ~ ~Mir~ n e (Moscow 1979, 1982). * C.J. Isharn, Modem D~~erential Geometry for Physicasts (World Scientific, 1989). * J. L.KOSZUI,Lectzlres on Fibre 3undles and D ~ ~ eGeometry ~ t z (Tata ~ Institute of ~ n d ~ eResearch, n t ~ Bombay, 1960). A. "kautrnan, ~ a f f e r e ~Geometry ~ ~ a ~ for Physicists (Bibliopolis, Naples, 1984). ~
Part I11 Geometry and Physics
Part 111is devoted to a revisiting of analytical mechanics in terms of geometrical structures. Chapter 9 is devoted to the intrinsic formulation of Maxwell’s differential equations in terms of differential forms,so that it can be considered as an introduction for Gauge Theories.
229
Chapter 9
Symplectic Manifolds and Harniltonian Systems
9.1 Symplectic Structures on a Manifold If M is a 2n-dimensional differentiable manifold, a symplectic structure on M is a differential 2-form w , required to be a
closed &=O,
and not degenerate
( w P ( XY , ) = 0 VY E 7 , M ) + ( X = 0 ) V p E M
.
(9.1)
A pair ( M , w ) ,with M a 2n-dimensional differentiable manifold and w a symplectic structure, is called a syrnplectic manafold In a given basis { e i } for vector fields on M, we may write X = X i e i , Y = Y'ei ,
= wp(eirej), the relation (9.1) becomes ( X i Y Y i w i j ( p= ) O V Y i ) + (Xi =O) V p E M ,
so that, with w i j ( p )
or equivalently,
( X i w g = 0)
* ( X i = 0).
231
Symplectic Manifolds and Hamiltonian Systems
232
Thus, a differential 2-form is not degenerate iff
A generic differential 2-form w on a manifold defines a homomorphism w :7,M
+T M
of the vector space 7 , M , of tangent vectors at the point p E M , into G M , the vector space of differential 1-forms to the manifold M at the point p E M , since with the vector X p G M , w associates the differential 1-form c y p , defined as a p
= iXpW(P).
As a consequence, with the vector field X , w associates the differential 1-form a,defined point-wise as
When the differential 2-form is not degenerate, the above relation can be point-wise solved with respect to the vector field X . Then, a not degenerate differential 2-form w defines an isomorphism, between the vector spaces 7 , M and T d M , given by
X = A ( a , *) , where the 2-vector field
A:T M
+ 7,M ,
is the inverse of w ,
The above relation in a given coordinate basis, in which 1 w = -wijdxi A dx' 2
is simply written as follows:
1 ..a , A = -Az3-
2
ax*
A
a -
dxj '
(9.3)
Locally and Globally Hamiltonian Vector Faelds
9.2
233
Locally and Globally Hamiltonian Vector Fields
A vector field X on symplectic manifold ( M ,w ) is called a (locally) Hamiltonian vector field if
Lxw=O, that is, if the symplectic structure is invariant under the flow generated by X . Since a symplectic form is closed, the above relation can also be written, by using the Cartan identity, in the following form:
Thus, a locally Hamiltonian vector field on M is a vector field satisfying the requirement that the differential l-form CY, defined by
(Y=ixw, is closed. If the differential 1-form CY = ixw is also exact; that is, a function H on M exists such that
ixw = -dH ,
(9.4)
the vector field is called a globally Hamiltonian vector field, or simply a Hamiltonian vector field, and the function H is called the Hamiltonian function corresponding to X . The minus sign in Eq. (9.4) is introduced just for historical reasons. Vice versa, any differentiable function on a symplectic manifold M ,
defines a Hamiltonian vector field Xf by the relation
ix,w 9.2.1
= df
.
Integral curves of a Hamiltonian vector field
In a coordinate basis, we may write
Symplectic Manifolds and Hamiltonian Systems
234
so that
and Eq. (9.4) becomes
or
dH w..X% = --. 31 I
dXj
Since det(wij(x)) # 0, the last relation gives
Thus, the first order differential equations for the integral curves of the Hamiltonian vector fields X have the following form: dxi - AaJ..dH _
dxj '
dt
(9.5)
and they are very similar to the Eqs. (2.24) of Sec. 2.4.1. Equations (2.24) and (9.5) coincide, provided that the antisymmetric matrix, whose elements are A i j , satisfies the Jacobi identity
Actually, the Jacobi identity is satisfied because of the closure of the symplectic form w. Indeed, in a coordinate basis, we may write
so that
The reader can easily check that, if
Aihwhj = dj,
to dwij
dwjk
-+ -+ B X ~
ax%
dwki
-= o . ax3
Eqs. (9.6) are equivalent
235
Hamiltonian Flowa
9.3 Hamiltonian Flows What has been said in the previous section can be repeated, more geometrically, as follows. Let us consider a function f defined on the differentiable symplectic manifold (M,w). Its differential df, at the point p E M belongs to T M
dfp:7,M+!R, V p E M . The bi-vector field A associates to df, a tangent vector to M at the point p E M as follows:
X f ( P )= A(df (PI1 -1* With the vector field X,(p), a one-parameter group of diffeomorphisms is associated (Eq. (5.13)) as
ut:M+M, such that
The group ot, which is called Hamiltonian flow with Hamilton function H , preserves the symplectic structure, that is Ut*W
=w
(9.7)
where at*is the derivative of at. More explicitly, Eq. (9.7) can be written in the following form:
( a t * 4 p ( X ,y > = w u t ( , ) ( f f ~ , ( X ) , ~ l , ( = Y )%) J ( Xyl >9 where X,Y E 7,M and ofp: 7,M
+ 7,t(,lM
is the derivative of at at the point p . The Lie derivative of the %form w along X f is given by
where the relation (9.8) has been used.
(9.8)
Symplectic Manifolds and Hamiltonian Systems
236
Since w is closed, we may write
LX,W=O
H
dix,w=O.
(9.9)
Of course, ix,w is an exact differential 1-form, since
9.3.1 Lie algebras of Hamiltonian vector fields and of Hamilton functions It is worth recalling that a Lie algebra is a vector space A supplied with a bracket
which is 0
bilinear
0
antisymmetric
(9.10) [X,Y1= 0
-[Y,ZI V G Y E A ;
(9.11)
and satisfying the Jacobi identity “Z,YI,Zl
+ “Y,Z1,4+ “Z>ZI,Yl= 0 ,
‘dX,Y,ZE
A.
(9.12)
The Lie bracket
[ X ,YI = LXY ,
(9.13)
which satisfies the relations (9.10), (9.11) and (9.12), provides the infinitedimensional vector space of differentiable vector fields, on a manifold M , with a Lie algebra structure. Let X and Y be two vector fields and ox and u+ be the corresponding flows, respectively. As already said, such flows are diffeomorphisms defined over all M , if the manifold is compact. Otherwise, ok and a+ are defined only in open sets in M and for small values of the parameters t and s. However, this suffices for our purposes.
Hamiltonian Flows
237
An important property of the Lie bracket, of two vector fields, is that its vanishing is a necessary and sufficient condition for the corresponding flows to commute2: [ X ,Y ]= 0
*
. : a +
= a+ax. t
(9.14)
Let ( M , u )be a symplectic manifold, and f and g two differentiable functions on M . The bracket { f ,g } , defined by (9.15) where a; denotes the Hamiltonian flow corresponding to the Hamiltonian vector field X f defined by ix,w = df, is called the Poisson bracket of the functions f and g. From definition (9.15), we have that the vanishing of the Poisson bracket { f , g } is a necessary and sufficient condition for the function g to be a first integral of the flow c; with Hamilton function f . Became of the isomorphism (9.2) between vector fields and differential forms, the Poisson bracket (9.15) can be written in the following form: { f , d ( P ) = Ap(df,dg) = ql(Xf,XB) *
(9.16)
Indeed, V p E M
Exercise 9.3.1. Prove, by using (9.16), that the Poisson Bracket is bilinear, antisymmetric, and satisfies the Jacobi identity { { f , g } ,h} + ((91 h l l f 1 + { { h l f 1191 = 0 *
(9.17)
Thus, the Poisson bracket provides the set T ( M ) ,of differentiable functions on M , with a Lie algebra structure. This Lie algebra, as it has already been shown in Part I, modulo the constants, is isomorphic to the Lie algebra of differentiable vector fields on M .
Symplectic Manifolds and Hamiltonian Systems
238
Exercise 9.3.2. Let w be a closed differential %form and X and Y be any two vector fields o n a manifold M , locally represented by
W e have L x i y w - i y L x w = d i x i y w +i x d i y w - i y d i x w = d(w(Y,X ) )
+ixdiyw - iydixw
= d(-wijXiY3)
+ ixd(wijYidd - wjjYjdxi)
-iyd(wijXidxj
-wijXjdxi)
= w i j [ X ,Y]adxj - w i j [ X ,Y ] j d x i = i[X,Y]W.
Prove the relation i [ x , y ] w= L x i y w - i y L x w ,
(9.18)
without use of the coordinates. Let X f and X, be the Hamiltonian vector fields associated with the functions f and g , respectively; i.e.
ix, w
= df
, i x , =~dg .
By using Eq. (9.18) for the Hamiltonian vector fields X j and
i[x,,x,lw = Lx,ix,w = dix,ix,w
+ ix,dix,w
Yj,we have
= dix,ix,w = d { f , g } ,
(9.19) so that L[X,,X,]W = 0 .
(9.20)
Therefore, [X,, X,] is a globally Hamiltonian vector field with Hamilton function given by
H ( P ) = WP(Xf,X,) = {f,S H P ) * Thus, the set of globally Hamiltonian vector fields on a symplectic manifold close on a Lie subalgebra of all vector fields.
The Cotangent Bundle and Its Symplectic Structure
239
Exercise 9.3.3. Prove that the set of first integrals of a Hamiltonian flow constitute a subalgebra of the Lie algebra of all differentiable functions. Exercise 9.3.4. Prove, by using Eq. (9.18), that the Lie bracket of two locally Hamiltonian vector fields, X and Y, is a globally Hamiltonian vector field, with Hamiltonian function given by H ( p ) = wp(Y,X ) . It follows that the set of locally Hamiltonian vector fields constitute a subalgebra of the Lie algebra of all vector fields too. The considerations developed in Sec. 2.4.4 (Further generalizations of the Jacobi-Poisson dynamics), can be repeated, of course, also in this new context. A useful reading on the theory of ordinary Jacobi-Poisson manifolds is given by Vaisman’s book.54 9.4
The Cotangent Bundle and Its Symplectic Structure
An example of symplectic manifold is given by the cotangent bundle ‘T*Q of an n-dimensional manifold Q. An element 29 of T*Qis a differential 1form on 7,Q, the tangent space to Q at a point p . In a coordinates basis (q’,. . . ,q n ) , a differential 1-form 6 has components P I , . . . ,pn and the 2 n numbers (PI,. . . , p n , q’, . . .,qn) can be taken as local coordinates of a point in T’Q. Thus, the cotangent bundle M = T*Qhas a natural structure of a 2ndimensional differential manifold. Moreover, it can be proven (see Appendix E) that T Qhas a natural symplectic structure w, which, in local coordinates, can be written as follows: 211
or
wC = d6,, with
The differential forms 6 , and w, are called the canonical differential f-form and the canonical symplectic structure, respectively.
240
Symplectic Manifolds and Hamiltonion Systems
But there is much more, in the sense that any symplectic manifold can be locally considered as a cotangent bundle. This is guaranteed by the Darboux+ t h e ~ r e m , ~ according *'>~ to which:
Theorem 29 (Darboux) At every point po of a 2n-dimensional symplectic manifold M , there exists a chart (U,po)in which the symplectic structure w assumes the form w=dxiAdxi+n,
i = l , ..., n .
Such a chart (U, po) is called a Darboux chart. In a Darboux chart, setting (PI
= x 1 ,.
* .
,pn
= x",ql
f xn+l,. .
. ,q" = P ) ,
the symplectic structure w and the bivector field A, given by Eqs. (9.3), assume the canonical forms
wc = dpi A dq' and
respectively. Moreover, the Eq. (9.3),
dxi ..aH - = A"-, dt 6x3 become the familiar Hamilton equations
(9.22)
An atlas for M , composed by Darboux charts, is called a Darboux atlas or a symplectic atlas. 'Gaston Darboux, born in Nimes in 1842 and died in Paris in 1917,has been a professor at the Sorbonne University for about 40 years. His work in four volumes on Thdorie des Surfaces is considered a classic. Besides giving new and remarkable contributions t o differential geometry, he deeply influenced the development of the theory of differential equations and, thanks to a deep geometrical insight and a sagacious use of algorithms, gave solutions to relevant problems in calculus and mechanics.
Revisited Analytical Mechanics
241
At this point it is clear that the Hamiltonian formulation of the dynamics, described in Part I (Analytical Mechanics) is, at least for systems which do not depend explicitly on time, the local version (i.e. in a Darboux chart) of the theory of Hamiltonian vector fields on a symplectic manifold M . 9.5
Revisited Analytical Mechanics
The reader can discover by himself the global version of many results obtained in Part I. Indeed,
0
A system of particles has n-degrees of freedom if its configurations define an n-dimensional differential manifold Q. The state space of the system is the tangent bundle TQ,while the phase space is the cotangent bundle T"Q. A Lagrangian function L is a differentiable map
0
from TQinto 8. Lagrange's equations
can be written in the intrinsic form,
Lad= = dL ,
(9.23)
iAdfiL: = -dEr:,
(9.24)
or
where
- A is the vector field given by A = vha/aqh+ LhkFk(q/v)a/avh; - Lhk are the elements of the matrix L-', with L = (a2L/avhvk); - 19cthe differential l-form on TQdefined by d r = (aL/avh)dqh;
Symplectic Manafolds and Hamiltonian Systems
242
-
EL is the energy EL = iAdL - C.
We notice that, if the Hessian determinant of the Lagrangian is not vanishing, then W L = d d L is a symplectic structure on ‘TQ. The intrinsic form of Lagrange’s equation allows us to introduce the Nother theorem as follows. Consider a complete vector field X on ‘TQ; i.e. the generator of a oneparameter group cp7 of diffeomorphisms on TQ.Let us calculate the infinitesimal transformation, 6.C = L x C , which X induces on the Lagrangian function C. From Eq. (9.23), we have
612 E L x C = i x d C = ~ X L A I=~(LL A I ~XL ), =L
A(~L X ,) - ( G r , L A X )
= i [ x , A ] d L-k L A i X d L .
It follows that
L x L = 0 , and [ X ,A] = 0 =+ L ~ i x f i=~0: , i.e.
Theorem 30 (Nother) A symmetry X of both the Lagrangian L: and the dynamics A gives rise to a first integral given by L A ( i x d L ) . The translation of the previous geometrical formulation in coordinate language gives back the original formulation by Emmy Nother.
Remark 17 I n order for i x d L to be a first integral, it sufices that i ( x , ~ ]-dL~x C vanishes, which is less stringent than the separate vanishing of each term. 0
The Legendre transformation defines a vector bundle isomorphism between ‘TQ and PQ. Indeed, the map
f
: ( q , v )E
‘TQ+(Q,P)E 7*Ql
with p h = (a.C/awh)(q,v), induces the derivative map
f*
X E
q q , w ) (TQ)I--+
X* = f*X E q q , p ) (TQ) *
Revisited Analytical Mechanics
243
The Legendre transformation is then defined by (q, v/Q,V ) -+(q, P / Q , P>,
where Q,V denote the sets of “q” and the “v” components of the vector field X, respectively, and Q, P the ones of the “q” and the “p” components of the vector field X,. h a
X = Q -+Vh%lh
a bvh ’
a + Ph-.a x,= Qh-aqh aph In matrix notation, setting f*=(
M I
L0 ) ’
with I the n x n identity matrix, and
we have
where the tilde indicates that the velocities v’s must be expressed in v). terms of the q’s and p’s by inverting the relations ph = (aC/bvh)(q, If the Lagrangian is degenerate; i.e. the Hessian determinant vanishes, the Legendre map defines only a vector bundle homomorphism from TQinto T Q . The theory of constraints by Dirac and just starts from this observation. A geometrical analysis can be found in Refs. 41, 108,177,146 and 136. An algebraic formulation of Lagrangian dynamics, suitable to be used in a general context, including situations with no global Lagrangian and/or fermionic variables, can be found in Ref. 76. N
0
Symplectic Manifolds and Hamiltonian Systems
244 0
A symplectic transformation, as defined in Part I, is just a map between two Darboux chart (U, cp 3 ( p / q ) ) and ( V , f (r/x)); that is, it is a $J
map such that dpi A dqi = dri A dXi
-
0
The above relation is the exterior derivative of
0
which is just the Lie condition for a transformation to be symplectic. A completely canonical transformation is a map between two almost Darbow charts, in the sense that
0
The Lagrange bracket % , q ,3
dxh axh - -arb axh - --
b I-
api aqj
aqj api
I
in which the inversion of the position of covariant and contravariant indices is just caused by the old notations, can be then obtained much more easily by expanding the previous equality to the form
which gives the familiar conditions for a transformation to be completely canonical, c[Pa,qj]= 6;. 0
The Poisson bracket { r h , x k } as defined in Part 1 is, vice versa, obtained expanding the inverse equality a d d A-=c-A8.h
aXh
in the inverted direction, to obtain
api
a aqi
Revisited Analytical Mechanics
245
which gives the ofd conditions for a transformation to be completely canonical, C(rhThr Xk)
0
=@ 3
this time with the right covariance of indices! The operator
that we called Harniltonian vector field in Part I, is just the local exi ~ field pression of the H ~ i l t o nvector
ix,= ~ df , 0
here introduced. A compfete andogy exists between the intrinsic Lagrange equations and the Hamilton equations,
i
0
~ =w- d E & ~,
.
i x w = -dH
The main difference between them, consists in the fact that, in the Hamilton equations, the “interaction” is present only in the Hamiltonian function, while in the Lagrange equations, the “interaction,” via the Lagrangian function, is also present in the symplectic structure WE. In other words, the symplectic structure w, in the Hamilton equations, is universal, in the sense that it does not depend on the considered dynamical system. This is not true for Lagrange’s equations. This feature is a consequence of the fact that the cotangent bundle T*Q,of a manifold Q, carries a natural symplectic structure, while the tangent bundle ‘T& has not such a structure. A Nother-type theorem, connecting a symmetry to a first integral, c m be stated in the Hamiltonian formalism M well as in the Lagrangian, even more easily. Indeed, let A and Xj be globally Hamiltonian vector fields, with ~ a m i I t o ~ i functions an given by H and f , respectively; i.e.
iAw = - d H ,
i x , w = -df
.
We thus have
Lx,H
= 0 es ix,d H = 0 ~3 ix,iAw = 0 H w ( X f ,A) = 0 es
{HJ}
= 0,
Symplectic Manifolds and Hamiltonian Systems
246
so that
Lx,H = 0 H L A f
= 0,
that is, to any symmetry of the Hamiltonian corresponds a constant of the motion and, vice versa, any constant of the motion is the infinitesimal generator of a symmetry transformation. In other words, A first znteg~ul,for a ~ u ~ z l t o n dynamics, iu~ generates a one-parameter group of symplectomorphisms, which leaves the Hamiltonian function H invariant and, vice versa, with any one-parameter group of symplect o - m o ~ ~ z s m~s , e ~ H~ i n ng ~ ~we~ can a associate ~ ~ , a first ~ n t e The Sec. (4.1), on the Integral invariants, can be revisited as follows. Let us observe that the Lie derivative, with respect to the vector field X , of the differential n-form CY
= p ( ~ ) d ~Adz2 ' A***AdXn,
on an n-dimensional manifold M, is given by
Lxa = d i x a = d i v ( ~ ~ ) d X A1 dx2 A - . A dxn , so that the relation (4.5),for a function p not depending explicitly on time, simply says that
It follows that a necessary and sufficient condition for fu a to be invariant is Lxa = 0. What has been said can be generalized as follows. A differential k-form a E hk((M),on an n-dimensional manifold M, is said to be an u b s o ~ i~ ~ et e 2~ ~a~ ~u ~ofa the n t complete vector field X , if Lxa=O. The latter is equivalent to c p : ( 4 C p T ( P ) ) ) = 4 P )t
where (p7 denote the flow of the vector field X,
(9.25)
~
Revisited Analytical Mechanics
247
If U is a k-dimens~o~al submanifold of M and i the immersion map
i:Ut,M, (p7o i )(U)is a new k-dimensional submanifold of M , and
J
a = ~(~~ o i)*a= ~ ( io V;)Ly. *
(Vp+Oi)(W
It follows, from Eq. (9.25), that if a is invariant, then
Vice versa, if the relation
holds, for any choice of U,i and T , then (i*o &)a = i*a or, equivalently, cp:a = a. We can conclude that a necessary and sufficient condition for a differential Ic-form to be an absolute integral invariant is that
for any choice of U ,i and r . A differential (k-1)-form j3 E A"-'(M), on an n-dimensional manifold M , is said to be a ~ ~ ~i tn ~t i vn~ ve ~u ~~of~the ~ tcomplete vector field X,if dj3 is an absolute integral invariant; that is, if
0
A revisiting of the ~ ~ i ~ t o n - ~ theory a c o ~can i be found in Ref, 149.
Another difficult task is to globalize the Liouville theorem. Undertaking this task would also be useless, since as we shall see in the next section, it has already been acc~mplished,~~ lS5, t 3 .
248
Symplectic Manafolds and Hamiltonian Systems
9.6
The Liouville Theorem
Let ( M ,w ) be a 2n-dimensional symplectic manifold on which n differentiable functions are defined
fi:M+FR,
Vi=l,
..., n ,
Let us suppose that the functions f l , .. . ,fR are in involut~on;i.e. {fi,fj}=O,
Vi,j=l,
..., n ,
(9.26)
and that the n differential 1-forms dfl,. . . ,dfm are linearly independent a t every point p of the level set M f ( % defined j by
Mf(,,)= { p ~ M : f a ( p ) = n i i, = l , ...,n}. &om the implicit functions theorem, the level set M f ( = )is an ndimensional submanifold of M , which is called the level ~ a n i f o ~ ~ Because of the isomorphism (9.2), with each differential l-form dfd, we can associate a vector field Xf, on M such that
ix,,w = dfi . These vector fields X f ,, which are supposed to be complete, are linearly independent at every point of M f ( z )since the differentials dfi, . . ,dfR are linearly independent and the symplectic form w is not degenerate. In addition, by Eq. (9.26), the vector fields Xfi commute each other,
.
fXfi,Xfj]= 0 , v i , j = 1,..* , n. Moreover, since ( L x , , ( p ) f i ) ( ~=) ( i x f j d f i i ( p )= dfilp(Xfg(p))=
{fi,fj)(P)
0,
the fields X f , ,. . . ,X f , , are tangent to M f ( + Thus, there exist n commuting tangent vector fields on M f ( T )that are linearly independent at every point. These vector fields form a local basis of an involutive distribution which, by F'robenius' theorem, is completely integrable. Moreover, M f ( T )is invariant with respect to each one of the n commuting flows ct associated with the functions fi. It can be proven that the differential manifold M f ( = )if, compact and connected, is diffeomorphic to an n-dimensional torus T", which admits the angles
The Liouville Theorem
249
' p l , . . . ,cpn, as local coordinates, being Tn the product of n circles. Indeed, let us observe that, by hypothesis, on M j ( m )there exist n functions fi, which define an n-dimensional Abelian Lie algebra with the Poisson bracket as a Lie bracket. They generate, at each point, n independent flows under which Mf(?,) is invariant. It follows that, a priori, M f ( T )N x P , but if M j ( m )is compact, we can only have k = n. Under the action of the Hamiltonian flow, generated by H = f l , the angular coordinates 'pa will change according to
dpi dt
--=wi, where w i= wa(f1,.
Vi=1,
...,n ,
. .,fn), so that the motion on M f ( m ) p*(t>= 'pi(0)+ w i t ,
V i=I,.
. . ,n
(9.27)
is almost periodic. Let ua consider a neighborhood U G M of M f ( = )If . we use the functions f l , . . . ,fn as coordinates in U,we can find a neighborhood U' C U c M of M f ( * ) ,which is diffeomorphic to the direct product T" x S", where S" is a sphere of an n-dimensional Euclidean space; i.e. a neighborhood of T in 8". The Hamiltonian flow, generated by H = f l , expressed in terms of coordinates (cp', .. .,'p", f1,. . . ,f n ) becomes (9.28)
The system (9.28) can be directly integrated to f i ( t ) = f,(o),
(pi(t)= 'pi(0) +- w i ( f l ( o ) ,. . . ,fn(0))t, V i = 1 , . . . , n .
The integration of the original canonical system is, then, equivalent to finding the angular variables 'p', . . . ,cpn. This can be done by only using quadratures. What has been previously said concerning the compact case, can be summarized by the following theorem.2
Theorem 31 (Liouville) If o n the Bn-dimensional symplectic manifold M are defined n functions f1,. . . ,fn in involution {fi,fj}=O,
Vi,j=l,
...,n ,
Sympleetic Manifolds and Hamiltonian Systems
250
and the n differential l-forms d f l , . . . ,df,& are linearly independent at every point i n the level manifold
Mf(?,)= {p E. M
: fifp) = TIT^,
.
i = 1 , .. , n } ,
then (a) M f ( n )is an n-dimensional submanifold of M , invariant with respect to the ~ a ~ ~ l t o nflow i a ng~neratedby H = fl; (b) i f compact and connected, M f ( , ) is d ~ f f e o m o ~ hto i c the n - d i ~ ~ n s ~ o ~ a torus T”, with angular coordinates (pl, . . . ,p”); ( c ) the motion on Mf(,l, determined by the Hamiltonian flow generated by H , is almost-periodic
(d) the canonical equations with Hamilton function pure quadratures.
H
can be integrated by
..
Let us now observe that, in general, the coordinates ( f i t . .. ,fa, cpl,, ,p”) do not form a symplectic coordinates system. However, there exist functions
Jh = Jh(f1,. , . ,f n ) ,
\di = 1,.. . , n ,
(9.29)
such that the coordinates (J1, .. . ,.In, PI,. . .,p”) are symplectic; that is, such that the original symplectic form w can be expressed as
w=dJhAdph. The variables (9.29), which conjugate with the angles, are called action variables; they are first integrals of the Hamiitonian flow generated by H . In terms of these coordinates, the system (9.28) takes the form
d Ji -=O, dt 9.6.1
dpi -=d(J1,...,Jn), dt
Vi=1,
...,n .
(9.30)
The construction of the action-angle coordinates
An analysis for the construction of global action coordinates can be found in Refs. 19 and 13 and a general analysis on the possibility to introduce “actionangle type” coordinates can be found in Refs. 158 and 41.
The Lioudle Theorem
251
Let us consider the case in which the manifold M is a cotangent bundle, so that w = d$, = d(phdqh). Let us then consider the immersion
i : MI(,) + M of the level manifold Mf(,) into M and the pull-back i*w to M f ( , ) of the symplectic structure, Since di' = i*d, we have di'w = i'du = 0 . Thus, i'w is a closed differential 2-form on the torus. It is not an exact differential form since the torus is not simply connected; that is, there exist curves on the torus which cannot be contracted to a point. We have i'w
= i'dd, = di'19,.
On the other hand, the vector fields e f i = X f i are a basis for vector field, which are tangent to M f ( , ) , so that, for any two such fields X and Y , we may write X = Xiefi, Y = Yiefi
I
It thus follows that ( i * w ) ( XY , ) = X i Y j ( i * w ) ( e f , , e f i= ) x i y j { j i , fj} = 0. Therefore, over any bidimensional region C on the torus, we have
Since two homotopic curves y1 and 7 2 on the torus will be the boundary of C,we obtain
a two dimensional region
o = p =J
71U{-72)
by Stokes' theorem. As a consequence, we have
i*?J,,
Symplectic Manz-foldsand Hamiltonian Systems
252
The curves on the torus T" can be divided in n equivalence classes ['yh], each class containing noncontractible homotopic curves. The action variables are then defined as follows:
where h = 1 , 2 , . . . ,n. The construction can be finally completed as in Sec. 4.6.4. 9.7
A N e w Characterization of Complete Integrability
In general, the peculiarities of a given dynamics A can be characterized by the invariance of some geometric structure. For instance, the symplectic character of a dynamics is characterized by the invariance of a symplectic structure. This is the case of both Lagrangian and Hamiltonian dynamics. It is then interesting to note the question whether the integrability properties of a dynamics can be characterized from this point of view. As we shall see, this can be done. Let us start with the following considerations.
Meaning of the vanishing Nijenhuis torsion of a mixed tensor jield.
A consequence of the vanishing Nijenhuis torsion h / ~ of , a mixed tensor field T , is that, given a vector field A l , the vector fields of the sequence
close on an Abelian Lie algebra
and that the transposed endomorphism T generates sequences of exact differential l - f ~ r m sin , ~the ~ sense that
(da=O,
dTa=O,
N;.=.O)+d(Fna)=O,
n>l,
a~7,*M.
Moreover, the invariance of T , under the flow generated by a vector field A , implies the invariance of the vector fields TAn and of the differential 1-forms Pa.
253
A New Chamcterimtiora of Complete Integrability
Propertiee of eigenvectors. It is also interesting to analyze the properties of vector fields which are e~genvectorsof a torsionless diagonal~zablemixed tensor field. mixed tensor Let M be a differen~iableani if old and T a d~a~onalizable field; i.e.
,
rfgk
pek = & e k ,
where { e k ) is fi generic basis of 7 , M , k
[ei,e j ] = cijek
and
3
its dual basis of T M ,
{t9*}
&9& = --crs6 l k
T
A$"
*
2
W e recall that the N~jen~uis torsion of T is defined by
NT(%xi Y ) = (a, %T(X,Y ) > t
with
- rf.(&XT)"Y = [PX,W ]+ !P[X, Y ]- P[PX,Y ] - qx, PY]
?t!T(X,Y)= (LrffxT)"Y
t
Let us evaluate %T on the basis { e k } : 'Hr(ei,e j ) = [Pq,f'ej]
+ f"[ei, e,] -
?[f'ei, ej]
- p[ei,Pej] .
(9.31)
Since, for any two differentiable functions f and g, and any two vector fields X and Y on M, we may write
[fX, 9Y1 = f$[X, I"]+ S ( L Y f ) X - f(Lxg>Y
1
and have [ P e i , f e j ]= [ ~ i e i , ~ j = e j~]
II;[Pei,ejl = P(Xd[ei,ejl
~ , [ e a , ei-jXj(L,Xi)ei ] - Ai(L,,Xj)ej,
+ (LegX,)ei) = XiP[ei,e j ] i- Xi(L,,Xi)ei, - ( ~ , * X ~ ) e=~ XjQeiiejl)
II;[ei,fejl = P ( A j [ e i , e j ]
~j(L~,Xj~e~.
Thus, the relation (9.31) becomes N T ( e i , e j ) = (P
- ~i)(f - ~ j ) [ e i , e+j ]( X j - X i ) [ ( L e , X j j e j i-( ~ ~ A i ) e i ]
and the vanishing of Nijenhuis torsion ~
~e j ) = 0~ implies e the ~following: ,
(F - A,)(? - Xj)[ei,
(9.32)
ejl = 0 ,
(A, - Aj)C,,Aj
(9.33)
= 0*
It follows that, if the eigenvalues X k of T are supposed to have nowhere vanishing differentials (dAj)p#O,
VPEM,
and to be doubly degenerate, then the two vector fields ei and to the same eigenvalue X i = X j , satisfy the relation lei, e j ]
= aei
+ bej .
ej,
belonging
(9.34)
Therefore, the vector fields ei, ej are a local basis of a 2-dimensional involutive distributio~and, by Frobenius' theorem, define a 2-dime~sionalsub~anifold of M . A dual point of view is that, by contracting the relation (9.32) with the elements of the dual basis, we also find
=C
1 : ~ - ( X ~
2
= C,.,-((Xk k 1
2
= C:j(Xk
- X % ) ( X ~- X j ) ( e j , S [ S '
- Xi)(&
-S~V)
- A j ) ( S ~ S-~6;di)
- &)(Xk - X j ) ,
where the relation lei, ej] = ckjeh has been used. In this way, the relation (9.35) simply says that c ki j = O , Q k # i a n d k # j ,
so that we discover again Eq. (9.34). Moreover, we also have dt?k = cEs@' A 19k
(no sum over k) .
(9.35)
(9.36)
A New Chamcterization of Complete Integrability
255
The last relation implies that
dk Addk = 0 , which again, by F'robenius' theorem (in the dual form), ensures the holonomicity of the basis. In conclusion, the relations (9.32) or (9.35), which directly follows from the Nijenhuis condition, ensure the holonomicity of the basis { e k } , in which the tensor field T is diagonal,
Invariance of the eigenvalues of an invariant mixed tensor field. It is easy to check that the invariance of T , under the flow generated by a vector field A, implies the invariance of its eigenvalues A. Indeed, let V E 7,M and a E T M be eigenvectors of T and Tl respect ively,
PV=XV,
Tff=Xff,
belonging to the same eigenvalue A, such that iva # 0. If T is supposed to be A-invariant, we have
La(TV) = (LaT)*V + T(LAV)= T(LaV)
V
E
7,M,
so that
!fv= Xv
P ( L A v ) = LA(PV) = (LAX)V + X(LAv),
and
Then, from
we finally have
(LaX)(V,a)= 0 + LAX = 0 ; that is, we obtain the invariance of X under the flow generated by A.
(9.38)
Symplectic Manijotds and Hamiltoprian Systems
256
If a tensor field T is invariant under the flow generated by a vector field A, the vector field A is said to be an autornorphism of the tensor fieid T.
P ~ ~ l ~ofa~ ~ tt o~ r en ~ of~ ah~ ~o ~s s~~ sa mixed ~ e s steraeor field. The A-invariance implies Eq. (9.38), so that (Xi -
X.j)Lei(A,d’)= X i L e , ( A , d )- XjLei(A18j) = Xi(LeiA’dj) - Xj(LqA+t9’)
?!q+ (LAei,Xjt9’> = -Ai(Laei, I 9 j ) + (LAei,TIP) = -Xi(Laei, 8’) + (PLAei,d ) = -Xi(LAei,
+ (LaPeg,d’} = -&(LAe$, 8’) + ,&(LAe$,83) = -Xi(LAei’ @’}
= 0.
At this point, it is worth recalling that a dynamical vector field A is said e ’ dynamics with smaller dimensions, in an open set 0 I:M , to be s e ~ T u ~ l in if a frame {ei} exists such that
Lei ( A ,19’) # 0 =t i = j , where {&} is the dual basis of (ei). If 0 coincides with M , we’ll say that A is separable. s has been shown, Since, ~ 1 it
LAT = 0 + ( X i - Xj)L,,(A,8’) = 0 , the A-invariance of T implies the separability of the dynamics.
Remark 18 This not~onof s e p a T u is ~ ~ ~ i~ ~~~ from e the ~ one e (see ~ Ref. ~ 65) in the Hamilton-Jacobi theory. Equation (9.33) can also be written in the form
XILe,Xj = XjL,,Xj, so that
TdXj = PdiLeiXj = XiSiLeiXj = XjdiLeiXj = X j d X j ,
(9.39)
A New Chamcterization of Complete Integmbility
257
Since the eigenvalues of T are doubly degenerate, the decomposition (9.37) can also be written in the form n
T C A j ( e j Q @ + e,+j
Q &+”).
j=1
By meam of Eq. (9,39), which implies the functional independence of the Aj’s, and, as a consequence, the Iinear independence of dAj’s, it is now possible to choose the basis in such a way as T has the following expression: n
T=
C
~j
(ej QP 4
+ ea+j t ~ d3 ~ j, )
(9.40)
j=1
that is, as if dAj dAj were part of such basis. expIicit~y shows the ~ t e g r ~ b i ~ of i t ythe projected Equation ~9.40~ dynamics, The equation & = A(s) can be decomposed in the foIiow~ngd ~ ~ u ~ I e d systems: (9.41)
Equation (9.40)can be rewritten in terms of the coordinates ((pi, A j ) in the form
where the A’s are defined globally on M, while the p’s, such that dp = @, can be defined only locally on M; in this way, all fields satisfying the equation LAT = O can be expressed as follows:
a and the systems (9.41)become (9.42)
Sgmplectac M u n ~ ~ o ~and d s H ~ m ~ ~ ~ Systems on~an
258
It is easy to check that the separable and integrable vector field A is also a H ~ i l t o n i a nvector field. In fact, given A, we can build many invariant symplectic structures w fk(Ak,~
w=
k ) dA~dXk k ,
(9.43)
k
where fk are arbitrary functions required to ensure the invariance of the differential 2-form (Eq. (9.43)). If we suppose that the field A has not got singular points, the generic symplectic structure w will have the form
Choosing, as a basis, the one associated to the action-angle variables (Jk, q k ) the , tensor field T becomes
and w takes the following form: k
What has been said in the present section can be summarized as
follow^.^^^^^
Theorem 32 (DMSV) Let A be a dynamical vector field o n a manifold M which admits a diagonalizable mixed tensor field T which is inuariant
LaT=0, has a vanishing Nijenhuis torsion
N-=0, a
has doubly degenerate eigenvalues X j with nowhere vanishing diflerentials deg X3 = 2
~
(dXj), f 0 ,
V pE M
.
Then, the vector field A is separable, completely antegrable and Hamiltmian.
A New Chamcterization of Complete Integmbihty
259
Remark 19 The conditions LAT = 0 and NT = 0 and the bidimensiona ~ i t yof the eigenspaces of T was extracted j k m the e ~ t e n c eof dynamics with infinitely many degrees of freedom, admitting a Lax representation (see Part IV). The fact that nonlinear f i ~ l dtheories, i n t e ~ b l e~ t the h inverse scattering method show an endomorphism, invariant under the dynamics, with v a n ~ h ~~ ~~ g e ~ t ohr ~u~ oand ~ n s b ~ d i m e ~ i o ni na ~v ~ ~ aeni gt e n s p a ~ ss~ggested , that the analysis of the integrability of dynamical systems could be realized, instead that in terms of a mixed tensor field T,rather than s ~ p ~ e c tstrucic ture w. The integrabi$i~yconditions in terms of symplectic structures w strictly depend on the finite dimensionality of the space and cannot easily be extended to the infinite-dimensional case. O n the contrary, the integrability in terms of T is expressed by conditions which do not depend on the finite number of degrees of fieedom of the dynamical system A.
Remark 20 It is worth remarking that the vector field A is not taken t o be a priori a ~ ~ m i ~ t o n vector i a n field. A s we shall see in Part I V , i n t e g ~ b i ~ i t y of dissipative dynamics can be put in the same setting by assuming diflerent spectral h y ~ t h e s i sfor the tensor field T , 9.7.1 EFom the L ~ o ~ v~~ ~ l ~e
t to
~ ~ n ~~ mixed ~r ~n ~ ~ n
tensor fields
Let us now study the problem of constructing invariant mixed tensor fields, with the appropriate properties (also called a recursion operator), for a given Liouville’s integrable Hamiltonian dynamics A. If H is the Hamiltonian func.} is the Poisson bracket, we have tion and {a,
Let u8 introduce in some neighborhood of a Liouville’s torus Tn actionangle variables (J1,. .,Jn, p’, . ,cp”). We have
.
..
t~
Symplectic Manifolds and Hamiltonian Systems
260
h
A = - aH .-
a
aJh 8vh'
Let us distinguish the two following cases: The ~ ~ i l t o n i H a nis a separabIe one k
In this case a class of recursion tensor fields can be easily defined as
0
with the A's arbitrary functions required to have nowhere vanishing differentials. Indeed, the tensor field IT is invariant and hap; vanishing Nijenhuis torsion and doubly degenerate eigenvalues. The H a ~ i ~ t o n i ahas n a nonvanishing Hessian
In this case new coordinates
which satisfy the condition
du' A du2 A
+
Aduh # 0 ,
can be introduced. A new sympbctic structure in this neighborhood can be then defined as
with respect to which the Hamiltonian becomes a separable one
Applicntiona
261
The class of recursion tensor fields is then given by
By means of this construction, it is possible to find the second symplectic structure for a completely integrable Hamiltonian system.
9.8
9.8.1
Applications
A Recursion operator for the rigid body dynamics
An invariant mixed tensor field, with vanishing Nijenhuis tensor and doubly degenerate eigenvalues, can be easily constructed,86 for the LagrangePoisson gyroscope dynamics, without gravity for the sake of simplicity, by using the constants of the motion found by Mishenko, Dikii, Manakov, and &tiu. 154,84,141,165 The Hamiltonian function for the rigid body is locally given by
1
H = -2 (
(pficos cp
+ c sin ( P ) ~+ (pa sin cp - c cos cp)'
A
l3
+%) ,
where 29, cp and II, are the Euler angles (of the body principal axes frame Oxyz with respect to a generic fixed frame Otqr), pol p , and p+ their conjugate variables, and A, l3 and C the components of the inertial tensor with respect to Oxcya, and U =
p$ - p , sin 29 sin 29
When A = B the Hamiltonian H reduces to
and the rigid body is said to possess a gyroscopic structure (Lagrange-Poisson gyroscope). Its complete integrability is obviously granted by the Liouville theorem in the open submanifold where H , p , and p+ are independent.
Symplectic Manifolds and Hamiltonian Systems
262
The tensor field defined by
with u
(d,y?,$,p8,p,,p*)
L
0
and the matrix
f'=
0
0
(Tj)given by LT
L +P,
7-
L+P$
-N
T= 0
0
0
0
0
0
PV
0
0
0
0
0
P@
,
fulfills the following properties:
b
deg (eigenvalues of
T)= 2,
where
denotes the Hamiltonian vector field corresponding to H s ~ A = W -d ~Hs,
by means of the canonical symplectic structure wC. The above properties can be easily verified by using the action-angle coordinates ZI = ('p','p2,'p3, J1, Jz,.Js) linked to u = (d,yr,$,pe,p,,p+) by the
Applications
263
following symplectic map: /
19 = arccos
q
+ cos 'PI E
'
cp = p2 - arctan
+ = p3 - arctan Pe =
+ J2) tan:),
J + Q - ~
J( J3
J 1 sin cp'
[p- (7)+ COSyJ1)2]* '
P v = J27 $11 = J3
where
{:1
J?
[(J: - J?)(Ji - J,")]i' J2J3
[(J: - J?)(Ji - J?)]i
In these coordinates the tensor field T has the form
with = diag
(JI, 52,J3, J1, J2, J3).
On the other hand, the complete integrability can be explained in terms of coadjoint orbits of Lie groups62 so that the previous invariant tensor field can be useful to establish a connection with completely integrable systems on coadjoint orbits of a Lie g r o ~ p . ~ ~ 1 ~ ~ ~ ~ ~ ~
Symplectic Manifolds and Hamiltonian Systems
264
9.8.2
A Recursion operator for the Kepler ~
~
n
~
The vector field for the Kepler problem, in spherical-polar coordinates, for !R3 - (01, is given by
A=
m
(pT-
ar
PS a PP a + -+ -r 2 ad r2sin2ddp
-1 r2 [mk+ It is globally Hamiltonian with respect to the following symplectic farm: i=r,d,q,
w=xdpiAdqi,
(9.44)
i
with the H ~ i l t o n i a nI? given by
In action-angle coordinates ( J , p), the Kepler Hamiltonian H , the symplectic form w and the vector field A become
H=-
mk2
(4+ J8 + &)2 ' h
2mk2
A=
(J,+JG+J,)3
a
a
(a,l+@+@)
a '
UnfortunateIy, the Hessian of the Hamiltonian identically vanishes and we cannot apply the previously described methods for the construction of the recursion operator. Nevertheless, starting from the observation that the Hamiltonian depends only upon the sum of action variables, it is possible to define a new coordinates system in which the Hamiltonian appears to be separated. In these new coordinates we can easily apply the previous methods and then, using the covariance of our formulation, construct a recursion operator in the original coordinates. The results can be summarized as follows:'5o
Applications
265
The vector field A is globally Hamiltonian also with respect to the symplectic form w1 where =
~1
s,hdJh A d v k ,
(9.45)
hk
with the Hamiltonian H I given by
HI = -
2mk2
J,
+ JG + Jq ’
or equivalently,
where
and the matrix S is defined by J1
&A(
2
J2
- J3 J3 - J2
51
J2
+
J2
J3
J1
2+
). J2
Remark 21 The matrix S cannot be identified as a transformation Jacobian as at b clear from the fact that the Sdcph’s are not closed 1-forms. In the original coordinates ( p ,q), the symplectic form w1 is simply written as follows: ~1
=C d K i A d d , i
where the functions Ki(p,q ) and ai(p, q), defined by
(9.46)
Symplectic Manafolds and Hamiltonian Systems
266
are considered as functions of p , q by means of the map Ji = & ( p , q), pi = Pi (P,4. As a consequence, a mixed invariant tensor field T, defined for nondegenerate w by
W ( F X ,Y )= w1 ( X ,Y ), can be constructed. The vanishing of the Nijenhuis torsion and the double degeneracy of the eigenvalues of T is more easily checked, however, in the angle-action coordinates, where the tensor field T is simply written as
Moreover, we have
T d H = lc ( - z ) ' d H . Thus, the iterated application of T does not produce new functionally independent constants of the motion. It has been shown that this particular situation prevails for periodic systems when the period P is a smooth function of the initial ~ondition.'~' It is now clear that all various alternative Hamiltonian descriptions that we may build, via a recursion operator T , will satisfy
d P A (T)'dH = 0 , i.e.
d P # 0 + (T')'dH A (T)'"+'dH = 0 .
9,
However, in this finite dimensional setting, {TI@)k,Tl(p)h}= and Trp,Tr(?)2,Tr(p)3are functionally independent. On the other hand, in the infinite dimensional case, it is not easy to give a meaning to the trace of an endomorphism.
The I' scheme Let us observe that the symplectic form w1, given by Eq. (9.46), can be considered'81 as the Lie derivative of the symplectic form w , given by Eq. (9.44),
267
Poisson-Najenhuaa Strvctures
with respect to the vector field
so that
The vector field r generates a sequence of finitely many (Abelian) symmetries according to the following scheme: Ah+l
r]
= [Ah,
1
where A0 = A and where the bracket [,, -1 denotes the usual commutator between differential operators. Such vector fields turn out to be Hamiltonian with respect to both the symplectic structures, so that
and commute between them
9.9
Poisson-Nijenhuis Structures
We may also mention a somewhat different approach to the same problem when the manifold M is supposed, from the very beginning, to be equipped with a Poisson structure A, so that ( M ,A) is a Poisson manifold.
9.9.1
Compatible Poisson pairs
Following Ref. 139, we shall say that a Poisson-Nijenhuis structure is defined on the manifold M ; if on M are defined, simultaneously a Poisson tensor field A and a Nijenhuis torsionless tensor field T that satisfy the following coupling conditions: (a) FA =AT (b) kLpxa - A L ~ ( T & +)(Lj&I')"(X) = 0 ,
(9.47)
for arbitrary choices of the vector field X and the differential 1-form a.
Symplectic Manifolds and Hamiltonian Systems
268
As a matter of fact, we shall see that, on the same manifold, there are infinitely Poisson-Nijenhuis structures, because it turns out that all the tensors T k A , for k = 1 , 2 , . . . , are Poisson tensors too and satisfy the coupling condition. The structure we have introduced seems very specific, but it is interesting to note that it is very natural for soliton dynamics. In fact, almost in every approach to the theory of completely integrable systems, one can notice that a crucial role is played by the so-called compatible Poisson t e n ~ ~ or as they are also called, Hamiltonian pairs.lo3 Two Poisson tensors P and Q are said to be compatible, if the tensor P + Q is a Poisson tensor too. We shall quote now the following theorem139:
Theorem 33 (Magri I) Let P and Q be Poisson tensors o n M . Assume that Q-l exists and is a smooth field of continuous linear mappings p E M + Q;'. Then, the tensor fields T = P o Q-' and Q endow the manifold with Poisson-Nijenhuis structure. Conversely, i f T is Nijenhuis torsionless tensor field, satisfying the coupling conditions with the Poisson tensor Q, then Q and T o Q T Q are compatible Poisson tensors on M . A construction, similar to the one used in the above theorem, can be also applied in the following situation. Suppose we have, on the manifold M , simultaneously a Poisson tensor A and a closed 2-form w (not necessarily nondegenerate), or as it is often referred, a presymplectic form. Then, the following theorem'39 holds:
Theorem 34 (Magri 11) If the form w o A o w is closed, then the tensor fields A and T = A o w define a Poisson-Nijenhuis structure o n the manifold M . It is worth noting that here we consider the 2-form w as a field of mappings p~ M
+ w p : 7,M -+ T M .
(9.48)
An interesting situation arises on a symplectic manifold ( M ,w ) , if in addition, there is a nondegenerate Nijenhuis tensor T for which the following condition is satisfied:
WOT=TOW.
(9.49)
This condition is obviously an analogous of the coupling condition (a) for the Poisson-Nijenhuis structure. In this case, it can be shown that, if the eigenvalues of T are smooth functions on M , they generate a system of integrable
r
Poisson-Nijenhuis Structures
269
vector fields, without the additional requirements which are usually imposed on w and T (see for example Ref. 139). More precisely, we have the following (see Refs. 100 and 139):
Theorem 35 (Florko-Magri-Yanovski) Let ( M ,w ) be a Bn-dimensional symplectic manifold o n which there exists a Nijenhuis torsionless tensor field T , such that T* o w = w o T. Let, for every point p E M , 7p be a semisimple operator and the dimension of its eigenspaces be a constant o n M . T h e n The eigenspaces Si, corresponding to the eigenvalues Xi, are orthogonal with respect to w and have even dimension. If none of the functions X i is nowhere constant; that is, there is no open subset V c M such that XiJv = constant, then the forms dXi are independent and are in involution. The corresponding vector fields, i x j w = -dXj, belong to subspaces Sj, pointwise. If, for every p E M , dims, = 2; that is, if every eigenvalue is doubly degenerate, and if these eigenvalues are nowhere constants, then (a) The set {Xi, i = 1 , 2 , . . . ,n } is a complete set of functions in involution and each vector field Xj is a completely integrable Hamiltonian system. (b) The diflerential %form w can be expressed in the following way: w = CE1w j , w i EE w(s,,wj = dXj A r j , where vi are some differential l-forms o n M . If we denote by yi the vector fields corresponding to rj(-ri = iy,w), then Xi,yi span the subspaces Sj. If Xi,yi are chosen in such a way that = 0 , then LxiT = 0. (c) If the eigenvalues Xi have no zeroes o n M , then the digerential ,%-fomsw,, = w o Tn,for n = 0 , 1 , 2 , ... , are again symplectic structures o n M .
Chapter 10
The Orbits Method
10.1 Reduced Phase Space An action of a Lie group G on a symplectic manifold ( M ,u ) is a differentiable map
satisfying the following requirements: @(.lP) = P ,
@(f,@ b , P ) ) = @(fg,p)
1
Vf,g E G,VP E M
*
The action (10.1) is said to be symplectic if the diffeomorphisms
are symplectic; i.e. @;w = W
For every
,
Vg E G .
< E 9, the map @€ : ( t , p ) E
92 x M
-+ @ ( t , p ) = @ ( e t E , p E) M 271
(10.2)
The Orbits Method
272
is an action of the additive group (8, +) on the manifold M . Thus, with every element in B we can associate a vector field on M defined by
J @ ) )= (i<M6)(p)
9
’P
M,
is a momentum map and cf, is a Poissonian action. Indeed, since @ leaves B invariant, then
L,cM6= 0 , i.e.
dicM6 -+ icMd6 = 0 In addition, from Eq. (10.8), we have
di= (
0
~
~
~
pW) )* p
(
~
The bilinear form R, is not degenerate.
Indeed, if
Qp(P,W )= 0 for every , '8l the representative vector V should be orthogonal to all the vectors of GP1(p),so that it should belong to G(G * p ) and, by Eq. (10.16), to T ( G , p ) . Therefore, by Eq. (10.18), we have
v = ( ~ p ) *=p0 .( v ~ Q The daflerential2-fom Clp is closed.
The Orbits Method
280
In fact, because dw = 0, we have
Thus, from
we also have dfl, = 0 , since (A,)* is a surjective application. For more details on reduction processes, see Refs. 106, 41, 153, 128 and 107; last reference also containing an example of noncommutative reduction in the context of noncommutive geometry. 12133
10.2 Orbits of a Lie Group in the Coadjoint Representation In the previous section we have seen how, given a symplectic manifold and a symplectic action of a Lie group on this manifold, which admits a momentum map, under appropriate conditions, we can define a symplectic structure on the reduced phase space. In this section we are going to see how, for the cotangent bundle 7 ' G of a Lie group G, we can define a symplectic action and a momentum map, such that the reduced phase space coincides with the orbit of the group in the coadjoint repre~entation.~~ Let G be a Lie group and consider the action of G on itself given by the left translations L, @ : (9,h)E
G x G + @(g,h)= gh E G ,
that is, by setting Qg
= L,,
Yg E G.
By using Eq. (10.19), we can introduce an action $, of G on T * G
(10.19)
Orbita of a Lie Group in the Coadjoint Representation
281
where ah is an arbitrary point of T G , that is a differential l-form on the tangent space to G at the point h, and ( L g - , ) f ; h: T
G -+
T;G
(10.21)
is the transposed operator of the derivative of L,-I at the point g h ,
(L,-I)*~~ : GhG -+ ThG
(.
Therefore, Eq. (10.20) gives $'(el
ah) = LZ(Qlt) = a h 3
$J(fl$J($, ah))= $(f, Llf-i(Qh))= (L;-i
Llf-I)(@h)
= L;-lj-l(uh) = qfg)-lQh =~ ( ~ g , Q h ) .
By Eq. (10.21), we can see that $J maps the differential l-form ah on ThG to a differential l-form on 7ghG. The diffeomorphisms (10.21) preserve the canonical differential l-form 8 on the cotangent bundle. Moreover, because LE-~u= -Llf-id8 = -dLi-,O = -d8 = u , V g E G, where u is the canonical symplectic form, the action (10.20) on the cotangent bundle is a symplectic action, This allows us to define a map J for $' as in Eq. (10.10). Let be an element of 9 and consider the action (10.19) of G into itself. The map
= (Riag)([)T
VtE9
that is J(ag)= Ria,.
(10.23)
Every point p in E* is a regular value for the momentum (10.23); that is, for every ag E J - ' ( p ) , the map J*a, : Z , ( T G )
-+ 7,E*
is surjective. Indeed if Y E T,G* and p ( t ) is the integral curve of
Y with
P(0) = I.1,
by applying to 1 . 1 ( ~the ~ operator Ri-l, we obtain a curve in P G t h r o u g ~ag at t = 0. This is so because
Rp, = /.L,
v a , E J-'(I.1)
and
which says that, for every Y E 7,4*,there exists a vector X E T,,(TG) such that
J*a,(Xf = Y . If for every g f G, to the element p in r?* we apply the right translation R;-l, we obtain a right invariant differential 1-form on G, a,(g) = R;-ip*
(10.24)
By letting g vary in G , Eq. (10.24) defines all and only the points of J - l ( p ) , because of Eq. (10.23). From Eq. (10.24), it is evident that the action of L:
Orbits of a Lie Group in the Coadjoint Representation
283
on the cotangent bundle maps points of J - l ( p ) to points of J - ' ( p ) for all g b e ~ o n g ~to~the g subgroup G, defined by
G p = (9
E G :Adi-Ip = p } .
(10.25)
From the relation (10.26)
it follows that G, can be also expressed in the form
G, = (9 E G : L;-1ap = a,}.
(10.27)
fiom Eq, (10.26), we can define an action of GPon J-'(p), which coincides with the action (10.20),when it is restricted to G, x J - ' ( p ) . This action is a proper action: in fact, if aP(hn) is a sequence of points in P1(pL)converging to a point of J-'(p), we have lim a,(hn)= a,@).
n4+m
By the continuity of the map (10.24), we have
so that
lim h, = h
n++w
with h E G,, since G , is closed. Let us suppose that the sequence L*-la,(hn) converges to a point of J - I ( p ) Sn for n + +m. We can thus write n++m lim Li;lap(hn)
= n++w lim ap(g nh )=a,
where obviously, gn E G,, V n E N. SOthe sequence (9nhn)nE.N converges to point f of G,, because G, is closed. Therefore, Iim h,=h,
n++m
lim g h - f ,
n++w
n -
a
The Orbit8 Method
284
and the sequence (gn)nf~ converges to a point of G,. The orbit of the point a,(la) of J - I ( p ) under the action of the group G, is the set
G, a,(h) 1
= (Lg-,a,(h): g € G,}
.
(10.28)
Thus, we have shown that (a) p is a regular value of J; (b) G, acts properly on J - ' ( p ) . From what has been said in the previous section, conditions (a) and (b) are suffici~ntto affirm that the set J - I ( p ) / G , , that is the set of the orbits of the points of J - ' ( p ) under the action of G,, is a symplectic manifold. This manifold is the reduced phase space and, of course, can be identified with the orbit of fi under the coadjoint action of the group G, that is G * p=Z { A d i - i p : g E G ) .
Indeed, from Eq. (10.26), the action of G, on the points of J - I { p ) reduces to the left translation of the points on the base. Therefore, with every orbit G, a,(h) of a point in J - I { p ) under the action of the group G,, we can associate the orbit G, . h of the point h in G under the action of G,,
G,*a,(h) 4 G p . h . In this way the reduced phase space is diffeomorphic to GIG,, so that
(10.29) To every orbit G, . h of GIG, we can associate a point of the orbit of p in
g8 under the coadjoint representation G,*h+Ad;l-ip,
so that
(10.30) and then
Orbita of a Lie Gmup an the Coadjoint Representation
285
Thus, by Eqs. (10.29)and (10.30),the reduced phases space J-l(p)/Gp can be identified with the orbit G p under the coadjoint action. Hence, the orbit of p under the coadjoint representation is a symplectic manifold. Now let us find the expression of the symplectic form 0, on the orbit G .p; for this purpose let us introduce the map
-
C*0, = W
(10.31)
.
Since
where ap(etEg)is a curve in J-'(p) through a p ( g ) , so that (ap)*g((Rg)*e(t))E Zp(g)J-l(pL) i Vice versa, if V
E 7a,(g)J-1(p), its
VtE
integral curve is
a,(a(t)) with a(0) = g. Furthermore ,
a(t)= a(t)g-'g = 7 ( t ) g = Rg7(t) , where ~ ( tis) a curve in G passing on e at t = 0, so that
and where
-
286
we have
where Ad* is the coadjoint action; i.e. A d * ( g , p )= A d i - , p .
The Orbits Method
Orbits of a Lie Gmup in the Goadjoint Representation
d
= --Ad*(etcg, dt p)
287
d
= -Ad;-,Ad;Ad*(ettg, dt p)
d dt
= -Adl-lAd*(g-letfg, p)l t=O
d dt
< p ( p ) = -Ad*(etc,p
Finally, we can write
In this way, we have obtained the formula which defines the symplectic form on the orbit
The above relation, of course, also holds for any other point V = Ad,*-,p of the orbit. Now, since
Eq. (10.32) can be written in the following definitive form:
Qcl(ad&4, ad;(P)) = (P, [& 111) *
(10.33)
The Orbits Method
288
10.3
The Rigid Body
In this section, we analyze the rigid body motion about a fixed point, in the absence of external forces. The rigid body represents a simple example of Hamiltonirtn system, whose configurations space is a Lie group. We shall see how, on every orbit of the coadjoint representation, the Euler equation is Hamiltonian, the Hamilton function being given by the kinetic energy. A rigid body is a system of particles subject to the holonomic constraint defined by the condition that the distance between any two points of the system is constant. The configuration space of a rigid body is the six-dimensional manifold R3 x S 0 ( 3 ) , where SO(3) is the group of the orthogonal matrixes 3 x 3, if in the considered rigid body there are at least three not-aligned points. Let us consider the problem of determining the motion of a free rigid body. This system is invariant under translations and thus there exist three first integrals which are the three components of the total moment^. Therefore the motion of the centre of mass is a free motion and we can thus choose an inertial system in which the centre of mass is at rest. In this frame a free rigid body rotates about its inertial centre as if it were bound to a fixed point. Thus the problem of the free motion of a rigid body is equivalent to the problem of the rigid body motion about a fixed point with three degrees of freedom. The configurations space is simply SO(3) and the position and the velocity of the body are defined by a point of the tangent bundle TSO(3). The system is invariant under rotations about the fixed point and, by Noether’s theorem, there exist three corresponding first integrals which are the three components J,, Jar and J , of the angular momentum. Besides these three integrals there is the total energy of the system, E , which has onfy the kinetic part. The four first integrals, J,, .Iv,J , and E, are functions defined on the tangent bundle
rsop). We can define an action of SO(3) on itself with the left translations
L, : h E SO(3) + L,(h) = gh E S 0 ( 3 ) , where gh denotes the matrix product. The tangent bundle TSO(3) is isomorphic to SO(3) x 7,S0(3), xS O (3) denoting the tangent space to SO(3) at the identity e; i.e, the space of 3 x 3 antisymmetric matrices.
The Rigid Body
289
There are two isomorphisms of TSO(3) in SU(3) x 7 3 0 ( 3 ) : the first is defined by the derivative of L,-i as foflows:
X : 9 E TSO(3) -+ A(&) = (g, (L,--i)*,&)E SU(3) x 7,S0(3),
(10.34)
where g is a tangent vector to the group at the point g; the second, by the d e r i ~ t j v of e Rg-l, the right trans~ation:
The tangent space 7',S0(3), on its turn, is isomorphic to the Euclidean space r#3, the jsom~rphismbeing given by
z:
(-", : ") -b
-c
E TeSU(3)t (-c,b,-a) E
R3*
(10.36)
0
The inner product
provides the space X S O ( 3 ) of a Lie algebra structure; if the internal product in is chosen to be the usual vector product, the map (10.36) is a homomorphism of Lie algebras. 10.3.1
The s p c e and the bad@aPtgular welocities
The velocity of the rigid body g is a tangent vector to the group at the point g: then the vector Ws
Z=
(207bQP>(&),
(10.33)
where 7r2 : SO(3) x ZSO(3) -+ZSO(3) is the projection map, is the angular v ~ ~ owith c i respect ~ ~ to the spuce, while the vector
is the apbgzllar velocitg with respect to the body. In fact, the e~ementg in SU(3) represents a position of the rigid body obtained by applying the motion g; that is, the left translation L,, to an
The Orbits Method
290
arbitrarily chosen initial state (e.g., the unit of the group). The angular velocity vector, w3, of the rigid body with respect to a fixed system, is given by w3
=q r l ) ,
9 E ZSO(3)
and, for every t E 3,eqt is a rotation with angular velocity w3. Since, under an infinitesimal rotation eq7(T Y
(13.10)
with a E T:M, X , Y E 7 , M , is called the Nijenhuis torsion of T . Thus, Nijenhuis' condition (13.9) is expressed by
N;.=O.
(13.11)
A consequence of Eq. (13.11) is that the vector fields of the sequence An+, = TAn , (A,
-US),
V n2 1
dose on an Abelian Lie algebra of s y m ~ e t r for ~ ~ KdV, s and KdV being a
~ a m i l t o ndynamics, i~ the sequence
is a sequence of gradients of conserved functionals, In other words, Eq. (13.11) ensures that the endomorph is^ ih generates a sequence of closed l-forms, in the sense that (6a=0,
6Ta=0)-6(Fna)=0,
Q E T M , 'dnZ1.
General Stmctwes
342
In our case, the functional 1-forms are exacts; that is, they are exterior derivatives of functionals which, since T is A-invariant, are first integraIs of KdV.
13.2
Strongly and Weakly Symplectic Forms
At this point, it is advisable to spend some words about the definition of symplectic form on an infinite-dimensional manifold, since in this case, a distinction between strongly symplectic forms and weakly symplectic forms must be introduced. We say that a differential 2-form w,on an infinite-dimensionalmanifold M , is a strongly symplectic structure, if (a) w is closed, that is dw = 0; (b) V p E M,wp : 7 , M x 7,M i.e. the map
-+ R
is a nondegenerate bilinear form;
Z:7,M-+7,rM,
(13.12)
which with every vector X E 7,M associates the differential 1-form Z(X) on 7 , M , defined as below
(I(X))(Y= ) wp(X,Y)
VYE
"&4
7
is injective and surjective. In other words, Z is an isomorphism between the spaces 7 , M and Tp*M
If the map { 13.12) is only injective, then the differential 2-form w is said to be a weakly symplectic structure. Such a distinction has not been done in finite dimensions, since an injective map between two finite dimensional vector space, with the same dimension, is also surjective. In infinite dimensions, the distinction is instead important. Indeed, let us consider a locally Hamiltonian vector field X and a strongly symplectic form w;then
Cxw
= 6ixw = 0.
If i x w is also an exact differential form; i.e. ixw = - 6 H ,
(13.13)
Invariant Endomorphism
343
the vector field X is a globally Hamiltonian vector field and H is the Hamilton function. Vice versa, if H is a differentiable function on M
H:M-+R, there exists a vector field X on M such that Eq. (13.13) holds, since the map (13.12) is an isomorphism; but, if w is only weakly symplectic, the vector field X cannot exist.
13.3 Invariant Endomorphism
All the evolution equations, introduced at the beginning (p. 267), apart from the Burgers' equation, are Hamiltonian systems with respect to a symplectic structure. Actually, many of them are Hamiltonian dynamics with respect to two symplectic ~ t r ~ ~ t namely ~ r ew1 ~ and, w2.~ ~ ~ ~ ~ ~ ~ For instance, 0
in the case of KdV equation, we have
w(X, Y) = ( X ( u ) D-'Y(u)) , with
D-'
Ix I"
1(
=2
-a7
dx
-
dz)
w2(X, Y) = ( X ( u ) ,EL1Y(u))
,
E k =
2
ax,, + -.3a,
1 + -u,, 3
where the bracket denotes the Lz scalar product. The Lenard sequence of gradients of conserved functionals is established, in terms of the operators D = d/ax and E k , as follows: (a,.)
DGn+1 = EkGn ; in the case of the sine-Gordon equations wxt
+ sinw = 0 ,
we have Wl(X,Y) = ( W w ) , D Y ( 4 ) 1 §Here x, t denote light-cone coordinates.
w 2 F ,
Y) = ( X ( V ) ,K ' Y ( 4 )
7
*
~
~
Geneml Strztctures
344
where
Indeed,
E, sin Y = D-' sin Y ,
so that the sineGordon equation can also be written in the f o ~ ~ ~ i n g form:'38 vt
+ E, sin v = 0.
Thus, a Lenard's type recursion of gradients of conserved functionals can be written as follows:
Many of the previous systems, including the Burgers' equation, admit, in conclusion, an operator; i.e. an endomorphism on the module of vector fields, namely rr*, which is invariant under the dynamics and responsible for the construction of (infinitely many) Abelian symmetries {vector fields) or, for the Hamiltonian ones, of infinitely many conservation laws. Thus, the endomorphism p, or its associated tensor fieid
T ( a , X )= ( a , r f X ) appears to be the most interesting object in the analysis of integrability of field theories. In fact, as it has been shown in Part 111, it is possible to characterize the complete integrability of systems with finitely many degrees of freedom (Liouville integrability) in terms of mixed tensor field T satisfying suitable conditions.
Example 37
The sine-Gordon equation v,t
+ sinv = 0,
Invariant Endomorphism
345
with
v = 3(.\/--iD2 - ux,
- 3v,D)
and where the tilde indicates that the transformation 3 u = -(u; 2
+ &uxx)
(13.14)
has been performed. Then, T, and T k are the same tensor field referred to two diflerent coordinate systems and K d V equation corresponds, in the same reading, to the Harniltonian dynamics generated by the second conserved functional of sine-Gordon eq~ation.'~' It follows that the conserved functionals of the sineGordon equation can be obtained from the ones of K d V equation simply by using the transformation (13.14). For instance,
and so on.
Example 38
The Liouville equation a,t
+ exp a = 0
admits the invariant endomorphism
TL = D2 - Da,D-la,
+ a2 ,
a
=
lim a,, X++CO
which is related t o the one p k of K d V equation by the similarity transformation
TL = J f k J - ' with
J
3
3(-D2
+ azx + uxD),
and where the tilde indicates that the following transformation 21
has been performed.
3 2
= --(a;
+ 2a
21
- a2)
346
GenemE Stmctures
Then TL and T k are the same tensor field referred to two different coordinate systems and KdV equation corresponds, in the same reading, to the Hami$ton~andynam~csgenerated by the second conserved funct~onalof Laouvilk’s equation.’80
Example 39
The Burgers’ equation ‘1Lt = 2uu,
4- u,,
admits the invariant endomorphism PB =
D i-DUD-1,
w ~ i c hgenerates an Abelian sequence of s y ~ m e t r i e sof the dynamics.
The next sections will be devoted to analyze the properties of our phenomenological tensor fields. 13.3.1
~ ~ n f l ~ ~~n c~ fal ~ l ~ance
Because of the Lenard’s sequence and of the bi-Hamiltonian structure of (some) evolution equations, the first relevant property of the tensor field T is given by
This characterization of the dynamics is very suggestive because of the similitude
Dynamics Symplectic
w
Geodesical
I’
Killing
g
HamiEtonian A Liouville Lax
fl T
Invariant structure a not degenerate, s k e ~ s y m m e t r i c ,closed tensor field a connection 2- form a symmetric, not degenerate tensor field a skewsymmetric tensor field, f u ~ l l i n gJacobi a ~ e ~ t i t ~ a volume form a (:) tensor fieEd with vanishing torsion
(3
(20)
(i)
347
Invariant Endomorphisms and Liouville 's Integrability
13.3.2
Nijenhuis torsion
The second relevant property, coming by the Lenard sequence, is if CY is &closed and dpclosed; that is, if 6a = 0 and 6(Ta)= 0. We know that such a property is ensured by
NT(%x,Y ) = 0
6(p"a)= 0
1
where159,110,96,97,160
NT(%x,Y ) ( a ,' f l T ( X ,Y ) ) and
' f l ~ ( XY ,) 5 [(Lq+xT)"- f'(LxT)"]Y. 13.3.3
Bidimensionality of eigenspaces of T ( K d V and sG)
Since T is a (1,1)-tensor field, we can put a corresponding eigenvalue problem for the associated endomorphism T on A(&):
TGx = XGx . It is not difficult to see that for each X there exist two (generalized) eigenvectors, namely G:, G: such that
TG; = XG; , TG: = AG: t- G;
;
this corresponds to Jordan's normal form for a finite matrix. Explicitly, we have d
G: = e2eJ[f2(ikj,z)]2,G: = e2ej-dkj [f2(ikj,~)I27 where f(k,z)are the Jost solutions of the Lax operator L
L2f
= -k2f
, k2 = - A .
13.4 Invariant Endomorphisms and Liouville's Integrability It has been shown that the properties
CAT=O 0
N - = 0,with N~(cr, X , Y,) -= (a,[(LpxT)" - f'(LxT)"]Y)
348
*
d = dim (eigenspaces of T ) = 2
seem to be verified by all evolution equations integrable by the Inverse Scattering Method. We recall that in Part 111, by using the first two properties but assuming diagonalizability instead of the third p r ~ p e r t y a, geometrical ~ ~ ~ ~ ~ integrability scheme was constructed according to which it was stated that:
A dynamicat vector field A which admits an invariant (,CAT= 0 ) mixed,
d~agonali~ab~e tensor field T , with vanishing Nijenhuis tensor field (NT = 0 ) and dou&Z~degenerat~e i g e n ~ a l u ~Xs w i t h o ~ ts t a t ~ o n apoints ~ (6.4 # 0 ) , is s e p ~ r a ~ lintegrable e~ and H ~ m ~ l t o n i ai.ne.~ a sep~ra~Le completely ~ n ~ ~ g r a b ~ ~ m i l t o n i as~stem.'O n The proof was performed by showing that NT = 0 implies the F'robenius integrability of the eigenspaces of T . 0 CAT = 0 implies the separability of A along the eigenmanifolds in dynamics with 1 degree of freedom, each of them with a first integral. The construction of a symplectic form (actually infinitely many), with respect to which A is a Hamiltonian vector field, was then easily accomplished. In spite of the relevance of the diagonalizable case, the third property is a characteristic feature of soliton theories. We want to state here an a priori separability criterion, based on this new spectral hypothesis and worth using for soliton equations. As far as solitonic dynamics is concerned, integrability is proven without further hypotheses, while for background-radiation dynamics, a compact a priori integrability criterion is, to date, lacking. The present results should naturally lead to the corresponding ones in terms of Lax pairs (these are considered in the context of bundles just based on phase manifold),a1 once the relationship between them and the above operator, now only analyticaily understood, will be translated into clear geometrical terms. We can prove the following integrability criterionE3:
A dynamical vector field A which a d m ~ t san in~ariantmded tensor field T , wit^ vanishing ~ z ~ e n htensor ~ i s Ni.and b ~ d i m e ~ s i o neigenspaces, a~ com~lete~y separates in 1 degree of ~ e e d~namics. ~ o ~ The ones a s ~ o c i ~ ~toethose d degrees of freedom, whose corresponding eigenvalues X are not stationary, are integrable and Hamiltonian. Indeed, denote by X i the generic discrete eigenvalue of T and assume that the continuous spectrum of T consists of the real semiaxis R+. Then the
Invariant Endomorphbms and Liouvrille 's Integnrbility
349
vanishing of the Nijenhuis torsion N;., associated to T , means that for all cy E h(')(M)and X,Y E T M ,
Ni.(~r, Y,X )
(0,
[(LTxT)- T(CxT)]Y)= 0.
(13.15)
According to our assumptions, a basis
of TM exists, such that
Now introduce the corresponding dual basis
of A 1 ( ~ that ) is a basis, for which
(13.16)
where i, j == 1,. , .,n,;:,:6 = 6;6(k - h). The relations corresponding to Eqs. (13.16) in terms of differential 1-forms read
26, = X i 6 i + 72, 21-i= xiri, i = I , 2,. . . ,n , 2 ~ ~ i ( k=) kyW) , 2 = 1,2, k E R+ , where denotes the transposed of 9.As it will be shown, no more ingredients are needed to prove the separability in 1 degree of freedom dynamics, and (except for nowhere stationarity of the Xi's) integrability of the discrete part of it. The analysis starts by observing that an explicit transcription of condition
General Structures
350
(13.15) is the following:
(13.17)
As a matter of fact, it is easily seen that Eq. (13.17) are equivalent to7 ,-a p, 6 ~ = ' T~
f,&$i A 6.9' = y ' ~ ( ~p,fr2,fk) 6 ~ ~ , (=k0)
(13.18)
this implying, by the F'robenius theorem, that without loss of generality, the T ' S , 0's and y's can be considered to be closed differential forms, or equivalently, the basis {ei, sir f l , ( k ) ,
f2,(k)
1
i = 172, * '
-
9%
,
f
R+)
can be chosen to be a holonomic frame. On the other hand, the first line of Eq. (13.17) is equivalent to 6Xi = (LEiXi)ri, this implying that
P'SX'
= Xi&Xif
(13.19)
It particularly means that the
7%can be chosen as equal to the GXa's if, as will be assumed, the Xi's are nowhere stationary. Furthermore, holonomicity implies that the set of functions A', X 2 , . . ,,A" can be completed to form a coordinate system!!
(A', A',
. . . ,An5 cp',
p2,.. .
rpn,
$ J l Y k$ J ,2 1 k ,
k
E %+)
in such a way that
7Here and in the following, 6 denotes the exterior derivative and A the usual wedge product. IISome of them may not be global but only periodic ones.
Invariant Endomorphisms and Liouwille 's Integrability
351
Just to fix our ideas, the tensor operator T acquires the following canonical form:
It is now easily proved that for such a TIthe A invariance, namely LAT = 0 , gives (a) (b)
(6Ai, A) = 0 , 6 -(6cpi,A) = 0 , 6$d
6
(c)
@m(&pi,A) = 0,
(d)
(Xi
(e)
6
-(6$'1('),
6pi
6
- A')z(6pi) A) = 0 ,
A)
(13.20)
=0 ,
from which separability and integrability follow. More specifically, Eq. (13.20(a)) means the vanishing of "A components" of A; Eq. (13.20(b)) the independence of the 'p components on the p's; Eq. (13.20(c)) the independence of the continuous coordinates; and Eq. (13.20(d)) just means that each cp component can only depend on the corresponding A. On the other hand, Eq. (13.20(e)) shows that the continuous components cannot depend on discrete variables; and Eq. (13.20(f)) that each continuous component can only be a function of the continuous variables with the same continuous index. The most general form of A is then
Geneml Structures
352
The dynamical equation then decouples in the following second-order systems for the continuous degrees of freedom (background radiation dynamics):
dl(k)
= Al,k($1s(k), $z!(k))),
q)2,(k)
= A 2 r k ( $ l , ( k ) , $2>(k))
,
and the following trivially integrable ones:
@ = Ai(Xi) , ..
= 0,
for the discrete part (soliton dynamics). Incidentally, the discrete part of the dynamics is Hamiltonian with respect to all symplectic forms i
for the discrete part of the spectrum, f ’ s being arbitrary nonvanishing functions. R e m a r k 23 The vector field A is not supposed to define a Hamiltonian dynamics. Its Hamiltonian structure arises from the supposed bidimensionality of eigenspaces of T and the requirement bX # 0. Recursion Operators i n Dissipative Dynamics
13.5
We have seen that a nonlinear evolution equation ut = A[.]; i.e. the equation defining integral curves of the vector field A, is integrable once that a mixed tensor field T on M exists satisfying the following conditions: a
a a
T is A invariant; i.e. LaT = 0, T satisfies Nijenhuis condition; i.e. [LTxT- TCxTJY= 0, for any two vector fields X and Y , T is diagonalizable with doubly degenerate eigenvalues X without stationary points.
These assumptions on T not only ensure generic integrability, but also the existence of symplectic forms with respect to which dynamics is Hamiltonian and integrability is the usual one in terms of action-angle variables. On the other hand, there are many physically relevant cases in which the dynamics is not Hamiltonian, and nevertheless a suitable generalization of the above
Recursion Opemtors in Dissipative Dynamics
353
geometrical scheme could still be useful. The aim of the present example is to explore the possibility of using invariant mixed tensor fields to analyze dissipative dynamics. In order to do that, it is natural to begin by removing only the last condition on T,as it is the one leading to the existence of constants of motion, An instance of a dynamics which admits an invariant mixed tensor field T which satisfies Nijenhuis condition, but which is not diagonalizable without complexification and whose eigenvalues are trivially constant, is given by Burgers’ equation. This equation is just the simplest one combining both nonlinear propagation and diffusive effects, and it can be used as the working example for our anaIysis,
13.5.1
The Buqers’ h ~ ~ ~ r c ~ y
It is w e l l - k n o ~ nthat ~ ~ ~the * ~Burgers’ ~ equation can be linearized through the transformation u = - vx
(13.21)
V
where v satisfies the heat equation vt
= v,
.
It can easily be shown7’ that the Burgers’ equation is a member of a whole hierarchy of nonlinear evolution equations which linearize, by using the same transformation (13.21), to equations of the type vt = D”v,
n = 1,2,... ,
(13.22)
with D denoting 2 derivative. The even elements of Eq. (13.22) obviously define dissipative dynamics, while the odd ones are integrab~eHamiltonian evolution equations with respect to the following symplectic form: (13.23) where
354
with Hamiltonian functionals given by 1
Np = 2
f_, ( D p v ) 2 d z . +OD
(13.24)
In order that Eqs. (13.23) and (13.24) make sense, some assumptions on the functional space M must be made, for example to assume that M consists of fast decreasing infinitely differentiable functions. Then clearly
TIvJ = I) is a Nijenhuis A-invariant tensor operator for the heat equation hierarchy. In the present geometrical approach, Eq, (13.21) plays the role of a coordinate transformation, and thus a N~jenhuisA-invariant tensor operator for the Burgers' hierarchy is readily obtained from f"u] byg4
which easily yields
T[u]= I)+ DUD-' .
(13.25)
The Burgers' hierarchy is then obtained by repeated applications, on the translation group generator A0 = u,, of the tensor operator expressed by Eq. (13.25), Ah = TkAO.
(13.26)
The first fields of the hierarchy are A0
= Us,
+
A1 = 2 ~ ~ 3U: X A,
=;
(3u3
~ ,
+ ~ U U -+, u,,), .
This hierarchy is just the transcription in the new coordinate frame of the linear one and, apart from some technical points on the phase manifold N , one can translate what has been said for Eq. (13.22) to the Burgers' hierarchy. More precisely, Eq. (13.26) splits into the following two sub hierarchies:
*
~ ~ s s ~ ~ ~ u t ~~ ~ e e
~
u
TAo,T3A0,.. . ,Y'2n+1Ao i . . .
~ 9
~
~
~
355
Recursion Operators in Dissipative Dynamics
which are, respectively, a sequence of dissipative and Hamiltonian vector fields. The foregoing statement can be understood by examining the spectral properties of If, whose b l ~ c k~ ~ a ~ a is ~ a ~
~~~~’
where the vector fields
such that
6 ’
e(&)=; 6qW
e’
6 --
- &A&)
‘
In the b i d i ~ e n s i o ~integral a~ manifold of {e(k),eik)), the operator T can be projected to
General Stmctwes
356
where
are action-angle type variables. Then, field of the type
f' transforms
a dissipative integrable
into a Hamiltonian one
and vice versa. This alternating character of T is responsible for the splitting of hierarchy (13.26) into two subhierarchies. F'urthermore, we observe that
T has a bidimensional invariant spaces, but is not diagonalizable without complexification.
Pz,which characterizes the
Hamiltonian subhierarchy, is diagonalizable with doubly degenerate constant eigenvalues,
Thus, for none of the subhierarchies one can use the integrabi~itycriterion to establish their integrability. However, we observe that the projections of dissipative dynamics on the bidimensional invariant spaces simply are 1 degree of freedom dynamics, while for the Hamiltonian ones, the existence of a functional J('))(U], which is not trivially conserved on each bidimensional space, ensures its integrability. It is worthwhile remarking that this same functional J(kj[u]obviously plays the role of a Ljapunov**functional for the projection of the dissipative dynamics on the bidimensional invariant ~ubmanifoId,thus ensuring the asymptotic stability of the solution J(k)[u]= 0.
The ~
a
~
~s u l ~ ~t ~ue r~ a ~~ c ~ ~ ~
We discuss in more details the ~ a m i ~ t o n i character an of subhierarchy (13.22). In order to do that, some care is needed for the appropriate choice of the **Alexander Ljapunov was born in Jarosiav (central Russia) in 1857 and died in S. Petersburg in 1918, He has been professor of mathematics at Kharkov University and after, member of the S. Petersburg Academy of Science.
Recursion Operators in Dissipative Dynamics
357
functional space M on which dynamics is defined. The most natural one would be to take M as the functional space whose elements u go to a constant as t -+ &too, as it is the space on which there lies the typical solitary wave of Burgers’ hierarchy. However, with such a choice it would not be possible to introduce a Hamiltonian structure on M . This can be understood easily by going back to the linear hierarchy for which M becomes, via the transforma~ion(13.21), the space of functions which as I): + f o o behave like exp[lct), and the Hamiltonian becomes meaningless. One is then tempted to restrict M in such a way, that both symplectic structures and the Hamiltonian one be well-defined. This can be accomplished by considering only function .(I):)tending to some nonvanishing fixed constants as I): 4 f c o or, equ~valently,functions .(I):)vanishing as 2 + fcm,whose integral has fixed value. More precisely, as for what refers to tangent spaces, the derivative of the Hopf-Cole map is a bijection 6v -+ 6u between S(R);i.e. the space of all fast decreasing test functions, and the space of functions which are derivatives of elements of S(%),this ensuring the existence of a symplectic structure with respect to which the subhierarchy is Hamiltonian. The previous analysis shows the role played by the spectral hypothes~son the invariant mixed tensor field T in characterizing dynamical systems. The violation of the diagonalizability hypothesis allowed the inclusion of dissipative dynamics into the geometrical scheme. Moreover, the example shows that even if the eigenvaiues of T 2are trivially constant, sequences of constants of motion can be constructed by it.
Chapter 14
Meaning and Existence of Recursion Operators
Some confusion exists in the literature about recursion operators. This chapter will be addressed to clarify the meaning and the existence of recursion operators for completely integrable Hamiltonian systems. In previous chapters it has been shown that completely integrable Hamiltonian dynamical systems may admit more than one Hamiltonian description. It has been also shown that, usually, with these alternative descriptions, one associates a (1,l)-tensor field which can be used under suitable conditions, as a recursion operator, namely as an operator which generates enough constants of the motion in involution. It seems to be an open question whether it is possible to find a recursion operator for any completely integrable system. In the hypothesis of nonresonance, it has been shown that a recursion operator can always be constructed, even for some infinite dimensional systems.80 Some authors claimed however that this is not always the case. So it seems to us that it is of some interest to further comment on possible meanings of recursion operators and to show that, in condition of nonresonance, any integrable system can be reduced to a linear normal form via a nonlinear noncanonical transformation. For these normal forms, it is straightforward to construct recursion operators.130
359
Meaning and Existence of Recursion Operators
360
14.1 Integrable Systems
Let M be a smooth Z~-dimensionalmanifoId. Let us suppose we can find n vector fields XI,. . . ,X, E X ( M ) and n functions PI,.. . ,F, E T ( M ) with the following properties:
[xi,xjl=0 ,
(14.1)
CxiFj=O,i , j E {l,...,?z].
(14.2)
Let us suppose also that, on an open dense submanifold of M , we have
XIA * . * A X n$ 0 ,
(14.3)
dFIA*-.AdFn #O.
(14.4)
We shall show that any dynamical system A on M , which is of the form n
A = E v i X i , d = v i ( F 1 ,... ,Fn), 2=
(14.5)
f
is explicitly integrable on the submanifo~don which Eqs. (14.3) and (14.4) are satisfied. We assume finally, that the level sets of the submersion
F :M
.. , ,F") --+ 32" , F = (P,
(14.6)
are compact. Then the vector fields X i are complete on each leaf F-l(a), a E W, and they integrate to a locally free action of the Abelian group Rn. Moreover, each leaf is parallelizable and we can find closed differential 1-forms d,. . . ,a", dai = 0, such that
d ( X j ) = a;,
i , j E (1,.. * ,n) *
(14.7)
With all previous construction, the vector field A in Eq. (14.5) can be explicitly integrated in a neighborhood of each leaf F-'(a), where we take as coordinates the functions {Fi, (8) with d@ = a$, The equations of motion of A are given by
@ = vi(Ff,. ..,F") ,
pz = o .
(14.8)
Integrable Systems
361
Therefore, the corresponding solutions are cPi(t) = t.i(F(P*))
Pi@)= &(Po)
+d(P0)
1
(14.9)
7
with po E M the initial point. We see that the functions Y' play the role of frequencies. We stress the fact that up to now we have not used any Hamiltonian structure. For an algebraic characterization of complete integrability, see Refs. 77 and 126.
14.1.1
Alternative Hamiltonian descriptions for integmble systems
We shall now investigate under which conditions a dynamical system, which is integrable in the sense stated before, admits infinitely many alternative Hamiltonian descriptions. With the n-functions F 1 , .. . ,F" obeying the condition expressed by Eq. (14.4), we can define a closed differential 2-form by Wf
=Cdfi(F') A d
,
(14.10)
i
which is nondegenerate as long as dfi A- .Adfn # 0 . Any one of these symplectic forms makes the action of 9" a Hamiltonian one. Indeed, by construction of +
Wf ix,wf=-dfj,
j € { l ,...,n } .
(14.11)
As for the vector field A in Eq. (14.5), we shall have that iawf = -
uidfi.
(14.12)
i
A necessary condition for ~ A W Fto be exact is that it is closed, namely that
C d v ' Adfi = O .
(14.13)
i
All sets of solutions of this equation for f l , . . . ,f " satisfying df1 A . A df,, # 0 will give alternative Hamiltonian descriptions for the dynamical systems A
Meaning and Existence of Recursion Operators
362
in Eq. (14.4). Moreover, any such A will be completely integrable in the Liouville-Arnold sense, the functions f l , . . . , f n being constants of the motion (by assumption of Eq. (14.2)) in involution, { f i ,fj}A
(xi, xj)= c X , f j = 0 .
= uf
(14.14)
There are two limiting case where it is easy to exhibit solutions of Eq. (14.13). The constant case All the frequencies vi are constant numbers so that dui = 0 and Eq. (14.13) is automatically satisfied. Any differential 2-form in Eq. (14.10) is an admissible symplectic structure, and the corresponding Hamiltonian function is given by Wf
(14.15)
=p f i . i
An example of system for which this happens is given by the n-dimensional harmonic oscillator written as i
1 a A . - ___ pi - - a -
&7&
34.2
q
i
a
, no sum over i ( 14.16)
Here mi and ki are the mass and the elastic constant of the ith oscillator. Now the functions Fi are just given by the partial Hamiltonians F i = I2 ( gmi +kiq:),
i ~ { l. ., . l n } .
(14.17)
The nonresonant case None of the frequencies ui is constant and we have that dv' A Adu" # 0. In this case we may think of the ui as "coordinates" and of the f j as €unctions of the v'.
Integrable Systems
363
In this second case, very simple solutions of Eq. (14.13) are given by linear functions fi = CjAijuj, i E ( 1 , . . . n } , Aij E 9.The corresponding Hamiltonian description for A can given with quadratic Hamiltonian functions by (14.18)
(14.19)
Moreover, any other symplectic structure of the form Wf
= xdfi(ui) A
d,
(14.20)
a
in which any fi depends only on the corresponding frequency ui,will be admissible ELS long as wf is nondegenerate; that is, as long as d f i A . .A dfn # 0. The associated Hamiltonian functions depend on the explicit form of the functions f i . For instance, if fi = (8Gi/8iji)(iji), the corresponding Hamiltonian can be written it5 (14.21)
A simple example for these case is given again by the n-dimensional harmonic oscillator written as
A=
CF ~ A ~ ,
(14.22)
i
where Fiand Ai are given by Eqs. (14.17) and (14.16), respectively. Now the partial Hamiltonians F iplay the role of frequencies. The intermediate cases are more involved. For further comments on them we refer to Ref. 80. It is worth stressing that there may be admissible Hamiltonian structures for A that cannot be derived by using the previous construction.
14.1.2
Recursion operators for integrable systems
We shall now show how to construct recursion operators for the integrable systems that we have considered in the previous sections. As we have seen,
Meaning and Existence of Recursion Operators
364
given the dynamical system expressed by Eq. (14.5), we can construct infinitely many Hamiltonian structures given for instance by Eq. (14.10) or Eq. (14.20).
The constant case: dwi = 0,Vi E ( 1 , . . . ,n}. Two possible alternative symplectic structures are obtained from Eq. (14.10) as
k
ij
(14.24) k
ij
with the condition dfl A ’ . . A dfn tensor field T on M by
T
=
# 0.
OW;’
Given them, we can construct a (1,l)-
=
C f k ( F k )I k ,
(14.25)
k
where I k is the identity operator on the kth bidimensional “plane” of T’M with “coordinates” (dF” a k ) .
The nonresonant case: dw’ A . . . A dvn # 0. In this case two possible alternative symplectic descriptions are obtained from Eq. (14.20) as (14.26) k
ij
6i.j f z(yi)dvi A
wf =
d
with the condition dfl A .. . A dfn a (1,l)-tensor field T on M by
T = Wf
=
fk(Wk)Wk
,
(14.27)
k
ij
0
# 0. Given these structures we can construct Wc’ =
C
f k ( V k )Ik
,
(14.28)
k
where I k is the identity operator on the kth bidimensional “plane” of P M with “coordinates” (dwk,a k ) . From the way they are constructed, one sees that T in Eqs. (14.25) and (14.28) are invariant under A, have double degenerate spectrum with
Znt egmble Systems
365
eigenvalues without critical points, and vanishing Nijenhuis torsion NT. Therefore they are recursion operators for the dynamical system A. 14.1.3
Liouville-Arnold integrable systems
Assume the dynamical vector field A on the symplectic manifold ( M , u o ) has n constants of the motion H ' , . . . H", which are in involution (with re spect to the Poisson structure associated with W O ) , functionally independent, d H 1 A . AdHn # 0, and generate complete vector fields XI,. . . , X,. We have then an action of R" on M that is locally free and fibrating. In this situation, angle differential 1-forms a', . . . , a" can be found, such that
d d =0 .
a i ( X j ) = c$,
Given any function F of the H j , (or dF A dH' A the condition
.
a
A
dH" = 0 ) satisfying
the differential 2-form
is an admissible symplectic structure for the R" action. In particular, if
F =1 - C H , ~ , 2
i
we just get back the {Hi} as Hamiltonian functions. With a set of action-angles variables (Jk,p k ) ,we have that
Meaning us$ Existence of Recursion Operators
366
where vk = aH/dJk, k E (1,. . . ,n> are the frequencies. In the nonresonant # 0, * we can cuse when dv’ A * A dv” f 0, or equi~lently,det(’6u~/aJk~ use the vk as variables and write the ~ m i s s i b l esymplectic structure
-
wV = z d v k k
with Hamiltonian a quadratic function
1
H” = 2
X(*”,”. k
By using the analysis of the previous section, we obtain that a not resonant complete integrable system has infinitely many admissible symplectic structures, some of them having the form i
with the condition df’ A - - Adf” # 0. However, in general, we may not obtain wo in this way. Moreover, such systems do admit recursion operators given by Eq. (14.28). a
Example 40
Let u s consider the folZow~ng2-degrees of freedom, com~letely integrable system. Tuke M = x T 2 = ((2, y , 8,q ) } with sympbctic structure wg = d x A di3 + d y A dq. The d y n a ~ z c u lsystem is described by the ~ u m i ~ ~ ~ n ~ H = x3 y3 x y . The c o ~ e s p o n d i ~dg y ~ a ~ vector ~ c a field ~ i s given by
+ +
vfi = 3 x Z + y , vq = 3 y 2 + x t . .
(14.29)
From what we have said ~ e ~ o rthis e , s y s t e ~a ~ m z ~ ns ~ n ~ tmealnyy u ~ t e ~ a t z v e in the dense open s u b m a ~ ~ € ocharacterized ~d by due A ~ ~ m ~ l t o descriptions n~an duI, # 0 , namely by 36xy-1 $. 0, which coincides with the submanifold on which W is n o n d e g e ~ ~ r u ~Two e . such s t ~ u c t ~ r are e s given by ~1
==
dve A d8
+ dv, A d q ,
*This is also equivalent to the nondegeneracy of the Hamiltonian function.
(1~.30~
Integmble Systems
367 ~2
= f (vs)dvfiA d.9
+ g(vv)dvv A d q ,
(14.31)
where f and g are any two functions such that df A dg # 0. The corresponding recursion operators, T = w2 o w l ' , are given by
(14.32)
We stress the fact that wo is not among the symplectic structures constructed in Eq. (14.31) and that our recursion operators (14.32) cannot be 'ffactorized" through W O . We shall make some more comment on the meaning of recursion operators and on their use in the analysis of complete integrability.78>80*'85>145 Let us suppose we have a dynamical vector field A E X ( M ) and a compatible (1,l)-tensor T field, namely LAT = 0, so that the functions trTk, k 2 1 are constants of the motion. By applying powers of T , we obtain vector fields A k = T k ( A ) which , are symmetries of A. The Lie algebra { A k , k 2 0) is Abelian if NT = 0. If F E T ( M )is a constant of the motion for A, we say -that T is an F-weak recursion operator if NT = 0 and d ( T ( d F ) ) = 0. If T is an F-weak recursion operator, one can prove that d ( T k ( ( d F ) ) = 0, Q k > 1. Locally, one finds functions F k E 3 ( M ) by dFk = T k ( d F ) , which are constants of the motion for A. It is worth stressing that a given operator T may be a recursion operator for the constant of the motion F and not a recursion for another constant of the motion G. Moreover, it may also happen that the tensor T is an F-recursion operator but T k ( d F ) A d F = 0, Q k 2 1, so that one cannot use T and F to generate new constants of the motion. This is what happens for instance with the Kepler problem if one starts with the standard Hamiltonian f ~ n c t i 0 n . l ~ ~ However, it is always true that T[d(l/k)trTk]= d ( l / ( k l ) t r T k + I ) . If w is an admissible symplectic structure for A, namely LAW = 0, we say that T is a w-weak recursion operator ift JILT = 0 and d ( T ( w ) ) = 0. If T is a w-weak recursion operator, one proves that d ( T k ( w ) ) = 0, V k > 1. All differential 2-forms wk = T k ( w ) are then admissible symplectic structures for A,
+
tAgain, we use the same symbol for the extension of T to differential forms.
368
Meaning and Existence of Recursion Operators
It is worth stressing that given any two admissible symplectic structures A, it need not be true that they are connected by a recursion operator. Moreover, it may happen that Tk(w) A w = 0, 'dk 2 1, so that one does not generate new symplectic structures. If A is Hamiltonian with respect to the couple (w, H ) , namely ~ A W= -dH, we say that T is a strong recursion operator if it is both a H-recursion operator and an w-recursion operator. If this is the case, any vector field A k is a Hamiltonian one with respect to w with Hamiltonian function Hk as well as with respect to w k with Hamiltonian function H . Moreover, the constants of the motion H k are pairwise in involution with respect to the Poisson structure constructed by inverting anyone of the symplectic structures w k , k 2 0.
w1 and w2 for
Chapter 15
Miscellanea
15.1
A Tensorial Version of the Lax Representation
In this section it is shown that the Lax representation (LR) can be regarded as the vanishing, along the dynamics, of the covariant derivative of a section of an M-based bundle.*l Although the Hamiltonian structure of nonlinear field theories leads to an extremely simple method for the construction of sequences of conserved functionals and to a geometrical interpretation of scattering data, it has not l in the construction of the LR. On the other hand, played a ~ d a m e n t a role although a deep and effective interpretation is to consider the LR m a linear problem whose integrability condition coincides with the original nonlinear evolution equation, it is not clear how the existence of an LR, in this sense, qualifiw the vector field and the manifold. In spite of its connection with the , ~ Lax formulation i s then powerful method of the inverse spectral t r a n s f ~ mthe lacking of a clear-cut geometrical interpretation; that is, the Lax dynamics is not defined in terms of a geometrical structure it preserves. Preliminary interesting answers to the problem are given by the loop groups approach (see, for instance, Ref. 163). The present geometrical approach is motivated, first of all by the interest per se of the possible g ~ ~ estructures ~ r ~underlyjng c ~ the Lax re~resentation for a dynamical vector field on a manifold, on the other hand, by our belief 369
Miscellanea
370
that a geometrical understanding can be of help in the extension to more space dimensions. Once given a vector field A on a manifold M ,
A:M+TM,
TMOA=IM,
where TM is the natural projection of tangent bundle ‘TM, our aim is to translate in geometrical terms the problem of looking for a Lax pair L, B; that is, for a pair of operator fields on M such that
L = [ B ,L] . The structure of the Lax equation naturally suggests two simple and appealing geometrical readings. First of all, one can think of it as the explicit form of the equation
C*L=O,
(15.1)
once L has been interpreted as a section of the linear frame bundle. In fact, once fixed a frame, L and A can be written as
L = L:e,
@ 6’
, A
= Ape, ,
and an equation of the form L = [ B , L ]is obtained by imposing Eq. (15.1) with
Bj
= ie3dAi- Aki[e,,e316i.
(To be specific in notation here and in the following, except for an infinite dimensional example, M is supposed to be a finite-dimensional differential manifold with Rn as a local model.) On the other hand the Lax equation can be read as the explicit form of an equation of the type
DaL=O,
(15.2)
where the covariant derivative is taken with respect to a prescribed connection on a fiber bundle based on M , not necessarily the linear frame bundle. To illustrate this possibility, consider the case in which the mentioned fiber bundle coincides with the principal fiber bundle of the structural group GL(n,32); i.e. the linear frame bundle. In such a case the connection form w can be written as w = (w!), p l A = 1,.. . , n ,
A Tensorid Versaon of the hRepresentation
371
where the wx’s are real-valued I-forms, and
D L = (dLf;+ wrLz - wzL:)e, 8 19’. By contracting with A and imposing Eq. (15.2),we obtain
where the dot denotes the iAd operator. In more compact notation
L = IB, L ] , where
i.e. Bj = -AaI’k,,, I”s being the connection coefficients. As it has been shown in the previous chapter, equations of the first type CAT = 0 are satisfied by (1, 1)-tensor fields associated with completely integrable n o n l ~ n efield ~ theories and play, in connection with symplectic structure and, under some special assumptions, a relevant role in their ~ t e ~ r a b j ~ t y properties, The “phenomenology” of integrable nonlinear field theory shows that two distinct operator fields play two different roles in them. One, let us call it T , which generates a sequence of conserved functionals, by its construction is surely an endomorphism of the module X ( M ) of vector fields on M (or by duality of X ( M ) * )and satisfies the equation CAT = 0. The other one, let us call it L , is the linear operator that is used in the inverse scattering method; it is not a priori an endomorphism of X ( M ) , and once we assume it to be an object of this type, it does not satisfy the equation LAL = 0. It is then natural to m u m e that the Lax equation must be read as an equation of the = 0. This assumption is confirmed by specific examples showing type that, while the equation LAT = 0 is typically a feature that the dynamical vector field shares with a large class of fields, on the contrary the equation DAL = 0, once chosen a suitable connection, is able to fix without ambiguity the direction field associated with A.
Miscellanea
372
The fo~lowingexample, though elementary, exhibits all the essential fe& tures of the exposed idea.
15.1.1
The LR of the harmonic oscillator as a parallel: transport condition
In a natural chart the dynamical vector field is
Once given the c o ~ ~ t i form* on
0
QdP - PdQ
Pdq - QdP
0
w = - (1
4H
+
),
(15.4)
where H is the Hamiltonian H = ( 1 / 2 ) ( p 2 q 2 ) , Eq. (15.3) implies
B=L( 2
1
-1
0
).
0
It is then stra~g~tforward to see that Eq. (15.2) is satisfied; that is, in a chosen frame
where L is the tensor field
quat ti on (15.2) can also be read, once given I, and w , as an equati~nfor A, and in this sense, not only is a property of A, but also defines as already mentioned, without ambiguity, the direction field associated with A, this being in contrast to the characterization induced by an equation of the type LnT = 0. In order to elucidate this point let us consider the following examples. 'The column vector
(3represents the vector field a(a/aq) + b ( a / a p ) .
A Tensorial Version of the b Representation
15.1.2
373
The A-invariant tensor field for the harmonic oscillator
The general solution in A of the equation CAT = 0, given 1 a T = - (2H q 2 E C 3aq d q + p 2 - C 3 da Pp ) , can be written in the following form:
with f and g being arbitrary functions. In coordinate notation, the equation C A T = 0 reads (15.6)
T. = [ A A 1 where
A=
[
On the other hand, and this is a general feature of Lax type equations derived by invariance of tensor fields, connections exist such that Eq. (15.6) becomes the coordinate transcription of equation DAT = 0. As matter of fact, the connection form w = - (1
3H
Qd9+PdP
9dPPd9
pdq-qdp
0
satisfies the relation A = - ~ A w . The general solution of equation DAT = 0, devised as an equation for A , is
(f being an arbitrary function), i.e. the harmonic oscillator up to parameterization. To avoid misunderstanding, we remark that the Lie derivative along
A of tensor fields sat~sfyingEq. (15.2) is generally a different zero. This is, for instance, the case for the tensor field given by Eq. (15.5). 15.1.3
The A-invariant tensor field for K d V
We recall that the evolution equation is ti+uux+uxox
=o,
and that a tensor field satisfying the equation
CAT = 0
(15.3)
is given by the operator field
T *= a,,
4-3u.f
u x /x 3
-dy,
--oo
whose adjoint is used for construction of the sequences of conserved functionals and is related by a Miura-like transformation, of tensorial nature, to the analogous operator for the sineGordon equation. Equation (15.7), explicitly written acquires the form ?’ = [ A ,T ] with A = a,,, - ua, - u,.
Remark 24
The ezistence of ~ - i n v a ~ aTn ti s so pecu~aa~ in ~agru~gian a A-invar~a~t T.
~ ~ ~ a q-equzualent ~ ~ c s ~ ,a g ~ a n g i u a~ l ~~ a~lead ~* s~ to
15.1.4
The A-covariant tensor field for Kd V
In order to consider the usual L for KdV we will again adopt the coordinate notation in terms of “local coordinates” u ( z ) :differentials du(z)and functional derivatives d/(su(z)),as formal elements of the “continuous natural bmis” of cotangent and tangent spaces, respectively. The vector field is then written as
1, M
A=
s
dzA(u)6u(x)’
It is easy to verify that the Lie derivative of the Lax tensor field,
corresponding to the Lax operator I, = ax,+ (1/6)u,does not vanish.
A Tensoda1 Version of the Lax Representation
375
If, on the other hand, we consider the connection form m
with
we obtain
ker(B) = - i ~ w [ z31 , = 4d'"(x - y)
+ ~(o)d'(~- y) - p1x S ( s - y) ,
and hence
B = 4axsx- ua,
1 - -24% 2
*
Therefore, the Lax representation for KdV equation can be written in the form DAL = 0. In order to illustrate the utility of the geometrical reading of the LR as a condition of parallel transport, consider the transformations of the Lax pair induced by transformations of field variables. This matter is relevant accordthat several integrab~enonlinear field ing to the general f~ling138~17g~180~105~11g theories are equivalent between them, up to inversion problems of transformations. This point of view has, for instance, led to connect the T operators for sineGordon and KdV and the T operators for Liouville' equation and KdV. To give a simple example of the tr~sformationmethod, consider once again the harmonic oscillator. To transform, for instance, the Lax pair
to action-angle variables ( J ,cp), L must be transformed as (l,l)-tensor field and B as the contraction of the c o ~ ~ c t i oform n (15.4)with the dynamical vector field. The transformation law for w from the natural basis in x coordinates to that in o' is Wf
=
( ~ ( ) ~ () ~ (g) ) . w
f
d
Miscellanea
376
Then, in the new frame
L=
(
-mcoscp
-2sincp
-msincp
&i7coscp
),
B=
( $)
,
-Jo
which are, of course, a Lax pair for the dynamics J = 0, @ = 1.
15.2
Liouville Integrability of Schrodinger Equation
Some years ago it was suggested176the use of complex canonical coordinates in the formulation of a generalized dynamics including classical and quantum mechanics as special cases. In the same spirit, a somehow dual viewpoint can be proposed151: rather than to complexify classical mechanics it may be useful to give a formulatioli of quantum mechanics in terms of realified vector spaces. By using the Stone-von Neumann theorem, a quantum mechanical system is associated with a vector field on some Hilbert space (Schrodinger picture) or a vector field; i.e. a derivation, on the algebra of observables (Heisenberg picture). In classical mechanics the analog infinitesimal generator of canonical transformations is a vector field on a symplectic manifold (the phase space). Therefore, if we want to use similar procedures, we need to real-off &(Q, C), the Hilbert space of square integrable complex functions defined on the configuration space Q, as a symplectic manifold or, more specifically, its a cotangent bundle. We shall see that it can be considered as T*(Lz(Q,IR)); &(Q,9)denoting the Hilbert space of square integrable real functions defined on Q. The approach is different from previous ones124@*71169 also dealing with the integrability of quantum mechanical system in the Heisenberg and Schrodinger picture. In order to make more transparent the geometrical and the physical content of the subject, difficult technical aspects (which are however important in the context of infinite dimensional manifold," as for instance, the distinction between weakly and strongly not degenerate bilinear forms, or the inverse of a Schrodinger operator and so on) will not be addressed. We shall limit ourselves to again observe that no serious difficulties arise working on an infinite
Liouville Integmbility of Schrodinger Equation
377
dimensional manifold whose local model is a Banach space, as in that case, the implicit function theorem stil holds true. Although in an infinite dimensional symplectic manifold, a Darboux’s chart a priori does not exist, for the Schrodinger equation
natural canonical coordinates p and q can be introduced. We introduce the real and the imaginary part of the wave function 20:
i
p(r, $1 = Im “ 1
t> 1
a@, t ) = ~ + ( rt>, >
and in this way L2(Q1C)is considered as the c o t a ~ e n bundle t of I&(&, In these new coordinates, the Schrodinger equation takes the form
sz).
where H I ,is defined by
and GH/Gp, G ~ / denote ~ q th0 components of the gradient of H [ q , p ] with respect to the real 1;2 scalar product. Our system is then a H a m ~ l t o n di ~y n ~ i c asystem l with respect to the Poisson bracket defined for any two functionals F[g,p]and G[q,p]by
What is less known is that the previous one is not the only possible Hamiltonian structure. Indeed, the ~ c h r o ~ n gequation er can also be written as
378
where HO is defined by
and 31 is the Schrodinger operator
It is then again a Hamiltonian dynamical systems with respect to a new Poisson bracket which, for any two functionals F[q,p ] and G[qlp], is defined bY
Thus, with the same vector field, we have the two following choices:
A phase manifold with a universal symplectic structure w1 := 12
S
dr(6p A 6q)
and a Hamiltonian funct~onald~pendingon the classical potential, o
A phase manifold with a symplectic structure determined by the classical potential wg
:= ti
s
dr(%-’Sp A Sq)
and the universal ~amiltonianf u n c ~ ~ o nrepresenting a~ the quantum probability. The two brackets satisfy the Jacobi ~dentity,as the associated differentia1 2-forms are closed for they do not depend on the point zz (p,q ) of the phase spuce. We have then the relation
+
(15.9)
where
LiouviEle Integmbilzty of Schrodinger Equation
379
and
-=[$
6H
6H
621
Since the tensor field T does not depend on the point 3 ( p ,q ) of the phase space, its torsion is identically zero, so that the relation (15.9) can be iterated to $J
-6 H= n p - 6HO . 6U
621
It turns out that the Schrodinger equation admits infinitely many conserved functionals defined by
They are all in involution with respect to the previous Poisson brackets:
It is worth stressing that for smooth potentials U ( z )in one space dimension, the eigenvalues of the Schrodinger operator Ifl are not degenerate, so that the eigenvalues of T are double degenerate. The eikonal transformation
The map p ( r , t ) = A(r,t )sin S(r,
6
is a canonical transformation between the ( p ,q ) and coordinates, since
'
(T
= S(2h)-'&,
x = A2)
Miscellanea
380
The Hamiltonian HI becomes
so that Hamilton’s equations
give
ax = --2ii div(XV.rr), at
2m
where P = x and J = hx(VS/m) represent the probability density and the current density, respectively. This transformation being nonlinear will transform previous bi-Hamiltonian descriptions into a mutually compatible pair of nonlinear type. Finally, it is worth to stress that the Schrodinger equation, in spite of its linearity, shows that the class of completely integrable field theories in higher dimensional spaces is not empty. Moreover, previous analysis appears to be interesting also in the formulation of variational principleslog for stochastic mechanics.
15.2.1
Comparison with the nonlinear Schrodinger equation
The two-dimensional nonlinear Schrodinger equation (15.10) has been analyzed by several authors. lS6* lZ2*
lS8tg5
Liouville Integmbildty of Schrodinger Equation
381
it takes the form
~ K I
z(:)=;(: dt
-:)(z) 69
'
where K1 is defined by
1 h'lkl,PI := 5
J
t.&E
{
ti2
-GI(a3P)2 4-
+ (P2 + 2 1 2 ) -
It is then a Hamiftonian dynamicai system with respect to the canonical Poisson bracket A1 defined by Eq.(15.8):
The previous one is not the only possible Hamiltonianstructure. As matter of fact the nonlinear Schrodinger equation can also be written ad22*138
where %!N is the Poisson operator
or equivalently,
Mdscel tanea
382
with Q: = ~
~
m and /
~
,
It is then again a Hamiltonian dynamical systems with respect to a new Poisson bracket+ given, for any two functionals F [ q , p ] and G [ q , p ] ,by
Once again, with the same vector field, we have two following choices:
* A phase manifold with the canonical symplectic structure
and a Hamiltonian functional accounting for the interaction. phase manifold with a symplectic structure determined by the interaction
*A
~2 3
AT1 := ti
s
d x ( ~ ~A 'Sy) 6 ~
and a free HaIniltonian functional given by the mean value of the momentum ?j = -iMS.
We have then the relation (15.11)
where
-2cYpL)-'q
I
It can be shown that the sum A2 -tA1 is again a Poisson bracket. This is equivalent to the vanishing of the torsion of the tensor field T N ,so that the tFor simplicity the proof of Jacobi identity for
A2
is omitted.
Liouville Integrability of Schrodinger Equation
383
relation (15.11) can be iterated to (15.12) Therefore, the nonlinear Schrodinger equation admits infinitely many conserved functionals. The first three hnctionals are
They are all in involution with respect to the previous Poisson brackets, i.e.
By observing that
the recursion relation (15.12) can be completed to
In terms of the operators T , TG,T K ,TN defined by
we have the general scheme on the next page.
384
Miscellanea
Schrodinger Hierarchy
I s-Gordon
Hierarchy
I
I TN
-1
I Nonlinear
Schrodinger Hierarchy
I
Integrable Systems on Lie Group Coadjoint Orbits
385
Remark 25 I t is interesting to observe that K-1 is a conserved functional both for the Schrodinger and the nonlinear Schrodinger equations. The same is not true for KO.This is due the fact that Schrodinger equation is not invariant under space translations and KOcorresponds t o the mean value (@)of the linear momentum fi = -ih&. I n other words the vector field associated to KO via the i s invariant for translation. canonical Poisson bracket It is worth finally to compare the recursion operators of the Schrodinger, with vanishing potential V ( x ) ,and nonlinear Schrodinger, with (Y = 0, hierarchies. It is easy to see that, in this case, T = T;.
15.3
Integrable Systems on Lie Group Coadjoint Orbits
Integrability is also analyzed, by several authors, using the Eulerian approach of coadjoint orbits of Lie groups. Let { e i } be a basis of a Lie algebra Q with [ei,ej]= cFjek and {Si}be the dual basis, in E* the dualbf Q. Moreover, let z be coordinates in Q* with respect to (Si), and 3 be the set
3={f ecm, f : Q * + R } . Let us define the bracket
where V f E 3 and V x,y E Q*,and the gradient V f of f is the element of Q defined by
The existence on of a not degenerate scalar product (. , which is invariant for the adjoint representation, allows the identification of Q with Q* according to a),
(G, 4 = ( Y ,
*
On the other hand, the property ([c,b],a) previous bracket can be also written as
+ (b, [c,a])= 0 implies that the
386
So for a dynamics generated in
B by a function H E 3,
we have
dx = [z, V H ]. dt This corresponds to the Euler approach for the rigid body dynamics, and to the Lax representation of KdV too, in reading the phase manifold of KdV as the coadjoint orbit of the m-dimensional Lie group of integral operator Fourier integrable on the circle S1 !1,60i*9
15.4
15.4.1
Deformation of a Lie Algebra ~
e
f
0
~
0
~
~
0
~
Let Q be a Lie algebra and X , Y any two elements in 8. A family of brackets
satisfying the Jacobi identity t f A E %, is called a ~ e ~ o ~ u of~ the ~ ~Lie n 8 ? algebra I;. Therefore, w has to satisfy the following conditions:
[ X ,w(Y, Z)]+ w ( X , [Y, 21) + cyclic permutation of X,Y,Z = 0 , w ( X , w ( Y ,Z ) ) + cyclic permutation of X , Y,2 = 0
I)
Such a deformation is a 2-cocycle w on Q, with coefficients in the adjoint representation, that defines a new Lie algebra structure. A deformation is called trivial if there exists an endomorphism T : I; -+ 8, such that the operator 1 AT is a morphism from the new Lie bracket ,-]A to the old Lie bracket I. , Thus, for a trivial deformatio~,we have
+ a].
[a
Deformation of a Lie Algebra
387
The above equality implies that, for arbitrary A, the following condition must be satisfied:
+
+
+
+
(1 A T ) ( [ X Y , ] Xw(X,Y ) )= [ X ,Y ] U [ X Y , ] A [ X ,TYI
+ A 2 [ T X TYI , ,
i.e.
w ( X ,Y ) = [ T X ,Y ]+ [ X ,T Y ]- T [ X ,Y ], T w ( X ,Y ) = [ T X ,TY]. Therefore, w is a coboundary of the cochain T with the property
H T ( X ,Y ) = 0 with H T ( X ,Y ) = [ ( L P ~ T f'(LxT)"]Y. )~ Moreover,
T [ X ,Y]T= [ T X ,TY], with
[x,Y]TE w ( x ,Y ). 15.4.2
LiiSNijenhuis and exterior-Nijenhuis derivatives
The Lie-Najenhuis derivative is defined on vector fields by
LZY = [ X ,Y]T = w ( X ,Y ) = [ T X ,Y ]+ [ X ,T Y ]- T [ X Y , ], and on differential forms, by defining the following exterior-Nzjenhuis derivative:
(dTf)(X)-=
Y
f
E
A(M)
7
( d ~ a ) ( XY ,) = d a ( T X ,Y )+ d a ( X ,T Y ) - ( d T a ) ( X ,Y ), cv E A'(M) . Indeed, the exterior-Nijenhuis derivative has the property [ d ~ ( df)l(x, ~ , Y )= d f ( H ~ ( Y x ,) ), so that
d$ = 0
HT
= 0,
Moreover, if T is invertible, the Poincare' lemma holds; that is, if d T a = 0, then locally a differential form ,6 exists, such that cr = dTp.
388
Miscellanea
We also notice that
dTd = - d d T . Tedious calculations show that
(d~ff)(Y X l)=: @&, y) - (f&,
x)f
(01,
CT,y) .
The vanishing of the N~jenhuisbracket
[ T X ,TY]- FIX, TY]- T [ T X Y] , + P [ X ,Y ]= 0, for a tensor field R satisfying the condition R2 = 1, gives the modified classical Yung-Baxter e q ~ a t i o n ~ ~ ~ ~ ' ~ ~
[ R X ,RYI
- R p x , Y)- R [ X ,RY]+ [X, Y ]= 0 .
In this case, the condition on u can be rewritten in the following form: # ( X , Y ) = [RX,YJ3- [X,RY]- R[X,YJ, W(X,
Y)= R[RX,RY]*
Finally, we observe63 that also the bracket defined by
+
[ X ,Y ] R = W ( X , Y) R[X,Y ]-= [RX,Y1+ [ X ,RY] I satisfies the Jacobi identity. It follows that, if R solves the modified classical Yang-Baxter equation, all the brackets
w ( X , Y)= [RX,Y ]
+ [X,RY]- R [ X ,Y],
+ txlRY], [ X ,Ylx = [ X ,Y I R f w x ,Y),
IX, Y I R = [EX,Y ]
satisfy the Jacobi identity.
Remark 26
DiRerent approaches to complete integrability of systems with i n ~ n i t e ~many y degrees of freedom exist, but a clear connection between them is, up to now, lucking. Perhaps a deeper un~erstundingmay provide new tools to tackle the r e ~ e ~ apn t~ o b ~ ofe ~n ~s n ~ ~ n quantum ear t~~0ry.113~s1
Chapter 16
Integrability of Fermionic Dynamics
There have been several attempts to analyze integrability of fermionic dynamical systems (see for instance, Refs. 31, 142 and 74) and to extend to such systems,75 in algorithmic sense at least, results and techniques used for Bosonic dynamics and based on the role of recursion operators. In particular, one would like to define a graded Nijenhuis torsion. In this chapter, we address this issues. We show that a mixed (1,l)-graded tensor field T can act as a recursion operator if and only if T is an even map.129 There are dynamical systems, like supersymmetric Witten's dynamics184 that allow a bi-Hamiltonian description with an even and odd Hamiltonian function and in term of an even and an odd Poisson structure, respectively, so that the dynamical vector field is always even.183i172This allows to construct an odd tensor field which could be a good candidate as a recursion operator. We explicitly show that this is not possible.
16.1 Recursion Operators in the Bosonic Case Here we briefly recall an alternative characterization in term of an invariant (under the dynamical evolution) (1,l)-tensor field T . We shall deal only with smooth; i.e. C" objects, and notations will follow as close as possible those of Refs. 1 and 41. In particular if M is an ordinary manifold (finitedimensional), we denote by F ( M ) the ring of real-valued 389
t n t e g ~ b ~ l of ~ tFermionic y Dynamics
390
functions on M, by ~ ~ theMLie algebra ) of vector fields, by X ( M ) *its dual of forms and by ql(M) the mixed (1,l)-tensor fields. It has been shown that the main property of the tensor field T , in the analysis of complete integrability of its infinitesimal automorphisms, is the vanishing of its Nijenhuis tensor NT = 0. It is then plausible that a suitable generalization of such a condition could play an important role in analyzing the integrability of dynamical systems with fermionic degrees of freedom. Moreover, it seems natural to think that such a generalization could come from a graded generalization of some of the following relations which are available in the Bosonic case: (a) NT = 0 +I m T is a Lie algebra. (b) NT = 0, d ( T d H ) = 0 =+ d(T'ddH) = 0. (c) NT = 0 dTOdT = 0 , where dT is the exterior-Nijenhuis derivative. (d) T =: AT1 o A2, NT = 0 +=+A1 A2 satisfies the Jacobi identity. Here A, and A2 are two Poisson structures. (e) w ( X ,Y ) =: [ T X ,Y ]-t [ X , T Y ]- T [ X ,Y ], T w ( X ,Y ) = [TX,TY].
+
One could expect that some, if not all, of the previous relations do not hold true in the graded situation. Before we proceed with the analysis of the graded ~ i j e n ~ ucon^^^^^^ ~s we u d ~~ ~ e er €aZcu~us e ~ ~ ~onas ~u ~ e ~ u n ~ ~ o shall give a brief review of the ~ that will be followed by the study of some simple examples. 16.2
Graded Differential Calculus
We review some fundamentals of supermanifold t h e ~ r y ' ~while J ~ ~referring to the literature for a mathematically coherent d e f i n i t i ~ n . ~In ~ ' the ) ~ ~following, by gmded we shall always mean Z2-graded. The basic algebraic object is a real exterior algebra BL = (BL)o&, ( B L ) ~ with L generators. An (m, dimensional s u p e r m ~ ~ f is o ~a dtopological manifold S modeled over the vector s u ~ e r s ~ u c e
B"'" L = (BLfO"x
(BL);L
(16.1)
by means of an atlas whose transition functions fulfill a suitable supersmoothness condition. A supersmooth function f : U c BY," + BL has the usual
Graded Dvemntial Calculua
391
superfield expansion
where the x's are the even (Grassmann) coordinates, the 6's are the odd ones, and the dependence of the coefficientfunctions f...(z)on the even variables is fixed by their values for real arguments. We shall denote by G(S) and G(U) the graded ring of supersmooth BLvalued functions on S and U c S, respectively. The class of supermanifolds which, up to now, turns out to be relevant for applications in physics is given by the De Witt supermanifolds. They are defined in terms of a coarse topology on Bryn,called the De Witt t o ~ ~ o g y ~ whose open sets are the counterimag~of open sets in R" through the body : B;'" + 9". An (m,n) supermanifold is De Witt if it has an map nm$n atlas such that the images of the coordinate maps are open in the De Witt topology. A De Witt (m,n)-supermanifold is a locally trivial fiber bundle over an ordinary m-manifold So (called the body of S) with a vector fiber.Ig7 This is not a surprise the fact that, modulo some technicalities, a De Witt supermanifold can be identified with a Berezin-Konstant ~ u p e r m a n i f o l d . ' ~ ~ ~ ~ The graded tangent space T S is constructed in the following manner. For each I E S, let g ( x ) be the germs of functions at x and denote by TxS the space of graded BL-linear maps X : B ( x ) + BL that satisfy Leibnitz rule. Then, TxS is a free graded BL-module of dimension (m,n),and the disjoint union UsesTxS can be given the structure of a rank (m,n) super vector bundle over S,denoted by TS. The sections X ( S ) of T S are a graded S;(S)-module and are identified with the graded Lie algebra DerG(S) of derivations of G'(S). Derivations (or vector fields) are said to be even (or odd) if they are even (or odd) as maps (satisfying in addition a graded Leibnitz rule) from O(S) -+ G(S). A local basis is given by (16.3)
Remark 27 ~ n e ~~ ~~stated, ~ s ~by using ~& ~a ~ ta r t~i ad ~~ ~ v a t we i v~ha~l n g left. In general, a ~ mean ~ a left a ~ e~~ i v u~t z vne ,a ~ e a~ dy e ~ v a t ~ vc ~t ~ from i f t i = (xj,@,"),when acting on any ~omogeneousj u n ~ t i o nf E B(s), Zejt and
Integmbility of Fernionic Dynamics
392
rdght derivatives are reEated by
In a similar way, one defines the cotangent space and bundle. TZS is the 4 3~and T*S = U,CsT,*S. T,*S space of graded BL-linear maps from Ts(S) is a free graded BL-module of dimension (rn,n), while T*S is a rank (m,n) super vector bundle over S. The sections X ( S ) * of T*S are a graded S;(S) module and are identified with the graded G(S)-Iinear maps from DerOfS) 4 Q(S).They are the differential 1-forms on S and are said to be even (respectively odd), if they are even (respectively odd) as n a p s X ( M )-+ B(S). In general, a p covariant and q c o n t ~ v u r i a ngraded ~ tensor is any graded x X(S)*+S;(S). S;(S)-multilinear map* CY : X ( S ) x * x X ( S )x X(S)* x The colbction of all rank ( p ,g) tensors is a graded S;(S)module. A graded p f o m is a skew-symmetric covariant graded tensors of rank p. We denote by P(S)the collection of all differential p forms. The ezterior diflerentiul on S is defined by letting X Idf Z= X ( f ) , V # E Q(S),X E K(S) and is extended to maps W(S>-+ Rp+l(S),p 2 0, in the usual way, so that
-
d2 =o.
If Xi E X ( S ) are homogeneous elements,
where
+
i- 1
a(i) = 1 i 4-P(Jcd) Z P ( X & )3 h=t
*With p X ( S ) factors and g-X(S)* factors.
+
Graded Differential Calculus
393
From definition, one has that p ( d ) = 0. The Lie derivative L(.) of forms is defined by
L(.): X ( S ) x W(S) + W(S),
Lx
=
x Iod + d x I, vx E X ( S ). 0
(16.7)
Clearly, ~ ( L x=)p ( X). The Lie derivative of any tensor product can be defined in an obvious manner by requiring the Leibnitz rule and can be extended to any tensor by using linearity. Suppose now that we have a tensor T E ql(S), which is homogeneous of degree p(T). Again we can define two graded endomorphisms of X ( S ) and X ( S ) * by the formulae (in the following two formulae, X,Y are homogeneous elements in X ( S ) , while a is any element in X ( S ) * )
P : X ( S ) +X ( S ), z1: X ( S ) * -+
X ( S ) *,
T ( X ,a ) =: P X Ia =: (-l)P(X)P(T)X IPa.
(16.8)
We could be tempted to define a graded Nijenhuis torsion of T by a relation analogous to usual one of the Bosonic case
GNT(X, Y ;a)=: G X T ( X ,Y ) Ia , (16.9)
It is easy to see, just by computation, that The map G 3 t :~X ( S ) x X ( S ) + X ( S ) defined in Eq, (16.9) is B(S)-linear and graded antisymmetric zf and only if p(T) = 0. When p(T) = 1, the map defined in Eq. (16.9) is not antisymmetric nor dinear also over even function, also when it i s restrict to even vector fields. Therefore Eqs. (16.8) and (16.9) define a graded tensor (which is an addition graded antisymmetric) i f and only if p(T) = 0.
Remark 28
Integrability of Fermionic Dynamics
394
16.3 Poisson Supermanifold We briefly describe how to introduce super Poisson structures on an (m,n)dimensional supermanifold S.59'33 For additional results, see also RRf. 68. As before, we shall denote by zi= (&,dk),i E (1,.. . , m n} the local coordinates on S. By direct calculations it can be proven' that If ( w i j ) is a n ( m n) x ( m n ) matrix, depending upon the point z E S, with the follouing properties:
+
+
+
the elements wij are homogeneous with parity p ( d ) = p(zi) + p ( z j ) + p ( w ) and p ( w ) not depending on the indices i and j ; 0
= -(_l)lP("i)+P(w)llP("j)+P(w)lWij
,
(16.10)
then, the follouing bracket e
d
(F,G} =: F - wii aza
-+
a G dzi
(16.12)
makes G(S) a Lie superalgebra (Poisson superstructure). We have two different kind of structures according to the fact that p ( w ) = 0 (even Poisson structure), or p ( w ) = 1 (odd Poisson structure). Indeed, one can check that the bracket (16.12) has properties:
Poisson Supermanifold
395
We infer from Eqs. (16.13) and (16.14) that, when thought of as elements of the Poisson superalgebra, homogeneous elements of Q(S)preserve their parity if p ( w ) = 0, while they change it if p ( w ) = 1. If the matrix (wid) is regular, then its inverse (wij),w,jwjk = 6 t l gives the components of a symplectic structure w = i d z i A dzJwji, namely, w is closed and nondegenerate with the properties P(Wij) =P
( 4 +P ( 4
+ P(W)
1
(16.15) and w is homogeneous with parity just equal to p(w). There is also a Darboux t h e 0 ~ e m . l ~ ~
Theorem 41
Let ( S ,w ) be an (m,n)-dimensional symplectic manifold with
w homogeneous. Then,
Proposition 42 0 I f p ( w ) = 0 , then dim S = (2r, n) and there exist local coordinates such that w = dqi A dpi
0
+'(d
(-:,): 1,
Ad 0, we have sin291 = sin& = COSLY. Thus, the integration goes 406
407
Concerning the Kepler Action Variables
from 81 = n/2 - a to 7r/2 to 8 2 = n / 2 + a and again back to 61; the sin6 goes from cosa to 1, then to COSQ. In this way, we obtain a - cos28 d 8 2x8
--
IT
sin2a
*/2
1
cos2r dr, 1 - sin2 a sin2 r
with r defined by cos 8 = sin a sin r
Therefore, with x
5
t a n r , we have
Jo=y+m[,,+ 1 1+x2
7r
=
27r# 7r
cos2a
1
2 d2 xcos2 a
(z--cosa ) 7r
7r
2
= 7rfi(l - cos a ) ,
and then, by using again Eq. (4.25),
The "r" integration requires the application of the method of residues. The roots r1 and r 2 of the equation
7ri
2mk --o 2 m E + -- - -
r
T2
are positive if E < 0 and correspond to the radii of turning points. In the complex z plane, the function
has two branch points at
Concerning the KepEer Action Variables
408
and a simple pole at z = 0, so that J, = i(R(2= 0) + R(z = + 0 3 ) ) .
Since R(z = 0 ) =
a
and R(z = +m) = r n k / e , we finally obtain mlc Jr --Z$
+ JZGiE.
Appendix D
Concerning the Reduced Phase Space
Appendix E
On the Canonical Differential I-Form
Let M be a differentiable manifold and T " M its cotangent bundle. The map
T:T*M-+M, which associates with every differential 1-form on TqM the point q E M , is a surjective differentiable map. Let Va, E Ta,(T*M)be a tangent vector on the cotangent bundle at the point ag E TPM; the derivative r* : T ( T * M >-+ T M
of the natural projection I maps the vector V, to the vector T m * , ( ~ a ~ which ~, is tangent to M at the point q. The map
defined by
is called the cunon~eul1-form on the cotangent bundle T*M. 410
O n the Canonical Differential 1-Form
411
If the manifold M is supposed to be a Lie group G, the diagram @;-I
T*G___) T*G
IT
.lG + G @g
ag
is a commutative diagram (here = Lg and, for every g E G, is the symplectic diffeomorphism of the induced action of G on the cotangent bundle T*G). Indeed, T ( @ ; - , ( a h ) )= r ( a g h )
Vah E
=gh,
T*G
and Vah
ag(7(ah)) = Qg(h)= g h ,
E T*G.
The diagram T*G ti-G
1
7 3
G
T(T*G)3TG T*
where (G and