Hamiltonian Dynamics
Gaetano Vilasi
World Scientific Publishing
Hamiltonian Dynamics
Hamil~onian
ynamics
Gaetano Vilasi ~ ~ ~ v e of r sSalerno ~~y
~ by
F
~
~
s
~
~
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.
~
~
I
L DYNAMICS ~ O ~
~
N
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 ,
For pho~ocopyingof material in this volume, please pay a copying fee through the Copyright Ctearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. in this case permission to ~ h o ~ o c ois~not y ~ q u j r e dfrom the publisher.
ISBN 981-02-3308-6
Printed in Singapore by Utc-Print
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<" and summing over A, we finally have
where the functions
are called the a f i n e connection coeficients. Since Eq. (1.3) represents the equation of a particle moving in a gravitational field, we are forced to interpret the affine connection coefficients as representing the gravitational force in the laboratory frame. We notice that no assumptions have been done on background mathematical structures as a vector space structure or a metric structure . An alternative version of the principle of equivalence is given by the socalled principle of general covariance which states that an equation, which is taken to describe a physical phenomenon, will be true if the equation holds in absence of gravitation, and moreover, it is form invariant for any coordinate transformation. Thus, this principle states that the mathematical expressions of the laws of the nature must maintain the same form regardless of our choice of a reference frame. Moreover, by extending the invariance of the laws of the nature to all types of motions of the reference frame, this principle marks the starting point for the possible relativization of acceleration.
A Primer for Various Formulations of Dynamics
1.1.3
11
Inertial f r a m e s revisited
At this point, we are in a position to revisit elementary mechanics, avoiding a lot of assumptions on the space of events or carrier space, as follows. We shall start with a 4-dimensional smooth manifold, for instance R4,as space of events, for the description of the evolution of particles. By using, in our laboratory frame, a coordinate system, say (xo,xl,x 2 , x3), it is an experimental observation that the evolution of states is governed by a second order differential equation of the type
We can now state what a free particle, and subsequently, a comparison dynamics are in this space. A motion of a particle is said to be a free motion, if there exists a coordinate system, namely (t0,t1,t2,t3), such that the equations of the motion can be written in the following form:
-d2ta =o dr2 Solutions of the above equation will define a vector space structure on the carrier space and will represent the world-line of a physical system; i.e. a really existing system, iff d t 0 / d r # 0. Thus, inertial frames are dynamically defined relatively to some chosen comparison system, avoiding any reference to dynamical systems arising in a specific gravitational theory. In this sense, Einstein's general theory of relativity is not a theory of invariance or covariance, as special relativity, which gives prescriptions about the choice of the reference frame, by requiring the parameter characterizing the frame (the velocity) not to appear in the transformed dynamical equations. Einstein's theory is a dynamical theory of the gravitational field, since it does not require the parameter characterizing the reference frame (the affine connection) to be absent from the transformed equations of the motion; it just prescribes how this parameter should appear in these equations. We conclude our revisiting by noticing that a system has to be physical with respect to all locally inertial observer, so that we shall call equivalent, two inertial frames, namely (to1 tl, t2,C3) and (tI0,c'l, < I 2 ,
The Lagrangian Coordinates
12
We finally mention the
Mach-Einstein principles of relativity. This principle tries to bring about the complete relativization of all kinds of motion, rotational and accelerated, as well as uniform. This is achieved by ascribing all the dynamical effects related to the acceleration and rotation of particle and electromagnetic systems, to motion, with respect to the universe as a whole. According to this principle, there can exist no observable difference between the rotation of a body with respect to the universe of stars and the rotation of the stars around the body. Thus, Mach’s principle constitutes an attempt to vindicate the kinematical principle in spite of the difficulties, of a dynamical nature, which had been the cause of its rejection. It was in part, with the intention to satisfy Mach’s principle, that Einstein elaborated the hypothesis of the cylindrical universe. This issue is still open.
1.1.4
The Lagrangian formulation of dynamics
Let us assume that the carrier space 9’ is endowed with the Euclidean metrics, so that the components of a generic vector coincide with the ones of the unique covector, naturally associated via the metric tensor. In terms of the components ( q I , q 2 , 4 3 ) of the position vector r’ and of the components ( q= 4 1 , w2 = 42,213 = q3) of the velocity vector 3, r‘=(ql,q2,q3),
(1.4)
(ul,VZ,u3),
Eq. (1.1) takes the form
{: -muh
= Fh ( t ,41, 42, q 3 , u 1 , u2, u3)
, h = 1,2,3,
(1.5)
z q h = ‘uh >
where Fh is the h-component of the force. The kinetic energy 7 = $mv2reads 1 2
7 = -m(q? + 4;
+ 4,”)
Therefore, by observing that
ar
d d d m a h = -mvh = -mQh = -dt dt dt a&
’
A Primer for Various Formulations of Dynamics
13
and thst 6 7 / 8 q h , the equations of motion (1.5) can be written in the following form:
In the conse~ativecase, there exists a function U(q1,q2, q3), the potential energy, such that
In such cases, by observing that aU/avh = 0, Eq. (1.8) or Eq. (1.5) can also be written in the ~ a ~ farm: ~ g ~ a ~
(1.10)
where the ~ a g r a n g ef ~ n c t i o nor simply the ~agrangianL(q, v, t ) is defined as the difference between the kinetic energy and the potential energy:
L=T-U.
(1.11)
Remark 2 In the cuse of Q generdked potentiat; i,e. in the ease in which the s ~terms d of a f ~ n c t i o nU(ql,q2,q3, ~ 1 , 2 1 2v, s ) , w ~ i c h force P can be e ~ r ~ s in beyond the coordinates q, also depends on the velocity 3,0 s Jih =
d a u -au -dtaVh
aqh '
it is possible to write the equations of the motion in the same f o r m Eq. (1.10).
(1.12) a5
in
tGiuaeppe Luigi Lagrangia waa born in Torino in 1736 and died in Paris in 1813. At the age of 19, he already was a professor of mat he ma ti^ at Artillery's School in Torino, and soon after, an associate founder of the Academy of Sciences of Torino. Author of the ~ ~ Analytique (Paris, 1788), Lagrange is considered aa one of the greatest mathematicians of the modern age. For a more extended bjography, see Appendix A.
~
The Lagrangian Coodinates
14
This i s the case, for instance, of a charged massive particle acted upon by an electromagnetic field (2,g).The force, in this case, is the Lorentz force:
p(?,v')= e ( g + $ A r \ ) ,
(1.13)
where e is the charge of the particle and the symbol A denotes the vector product. The Lorenta force can be derived from the following generalized potential:
-.
V(F,G) = e(cp - $. A ) ,
(1.14)
where ip and A' denote the scalar and the vector potential, respectively. Thus, the Lagrangian function describing the motion of such a particle is given by 1 2
C(F,,V'j= - m v 2 - e ( c p - G . A ) . 1.1.5
(1.15)
The Hamiltonian formulation of dynamics
In terms of the momentum vector p" ( p l r p 2 , p 3 ) ,Eq. (1.1) takes the form
(1.16)
The kinetic energy, on the other hand, can be written as follows:
(1.17) where the symbol * indicates that the velocity has been expressed in terms of the momentum by means of v' = p'/m. In the conservative case, the energy E = imv2+ U ( q l ,q2, Q ) , expressed in terms of momenta, is usually denoted by 3c and called the ~ a ~ i ~ t function on§ or simply the ~ a ~ z ~ t o n i a n ,
§William Rowan Hamilton was born in Dublin, Ireland in 1805 and died in Dunsik in 1865. He was a professor of astronomy at Dublin University and President of the Ireland Academy of Sciences. He invented the theory of quaternions and gave remarkable contributions to the Analytical Mechanics, in which, it successfully incorporated the theory of the light propagation.l1
A Primer for Various Formulations of Dynamics
15
Therefore, since 8U/bph = 0, the equations of motion can be written in the Hamiltonian form:
(1.19)
Remark 3 I t could seem to the reader that simple ,,ma1 manipu itions of the Newton equations have been done to transform to the Lagrange or t.0 the Hamilton equations. Actually, an additional structure, a scalar product, has been used. Indeed, when a vector space V (in our case R3)is supposed to be endowed with a scalar product defined by a metrics g (in our case the Euclidean metrics), g : (?I, 5) E
v x v + g(G, 77) = u’. v’ E R ,
with any function f , where
f
: (G) + f(?I) E
R,
we can associate a vector field, denoted by V f or 6f / h a , and called the gradient o f f , by
Therefore, a metric structure allows us to define a force if a function U exists, such that
F’
to be conservative,
F’=-vu. Under a change of coordinates, forces o n R H S of Newton’s equations transf o r m as accelerations, so that they are vector fields and do not require the use of any metric structure. These forces are measured with a dynamometer. O n the contrary, forces appearing in Lagrange’s equations are mathematically defined and physically measured in terms of the scalar function called work. A s a consequence, these forces are coeficients of a differential f o r m and cannot be identified with vector fields, unless a metric structure is at our disposal. Of course, this metric structure cannot be chosen arbitrarily but must be inferred from the results of experiments.
The Lagmngian Coordinates
16
Remark 4 I t could seem that the Lagrangian and the Hamiltonian formulations of dynamics do not bring in special advantages. This is certainly true in problems concerning particles not associated with other particles, f o r instance, as in a solid body or a fluid. I n more complicated cases, the Newtonian approach is still applicable, provided some proper precautions in the analysis of forces are observed. This force-analysis sometimes becomes cumbersome and it is dificult to give a unique answer to the problem. The analytical approach (Lagrangian or Hamiltonian) to the problem of the motion is much more powerful. According to it, for a general system of N particles, subject to k limitations, n = N - k special parameters, namely 91,. . . , qn, can be found such that the equations of dynamics assume the f o r m given by Eq. (1.10) or Eq. (1.19). Moreover, it will be shown that there is a unifying principle, the least action principle, which gives a meaning to the entire set of the analytical equations of dynamics (Lagrange or Hamilton equations). The statement of this principle is independent of any choice of the coordinate system and this implies that the analytical equations of dynamics are invariant with respect to any coordinate transformation. Unlike the Cauchy approach, which is local in nature, the unifying principle allows a global approach to the problem of the existence and uniqueness of the solution of dynamical equations.
1.2
Constraints
A particle is said to be constrained if it cannot take all possible positions in the space. In the following examples, the space is supposed to be endowed with the Euclidean metrics.
Example 1 Let us consider a particle P with coordinates (x,y , z ) linked to a f i e d point POwith coordinates ( X O ,yo, 20) by means of a rigid bar whose length is 1. Since the bar is rigid, the point P is free to move in the space taking only positions at distance 1 from the fixed point PO. I n other words, the point P is constrained to move along the surface of a sphere with center in Po and radius 1; in this way, it can fill only the positions whose coordinates satisfy the equation
(1.20)
Constmints
17
Example 2 Let us consider a particle P with coordinates ( x , y , z ) linked to a f i e d point PO with coordinates (20,yo, zo) by means of a flexible and inextensible wire whose length is 1. Of course, the particle P is free to mowe in the space inside the sphere with center PO and radius I ; in this way, it can only fill the positions whose coordinates satisfy the equation
+
(z - z0)2 (y - yo)2
+
(2 - zo)2
I 12.
(1.21)
The restrictions which impose a limitation to the mobility of P are called constraints or links. Equations (1.20) and (1.21) are mathematical expressions of constraints. A constraint is said to be two-sided if it is expressed by an equality, onesided if it is expressed by an inequality. Thus, the constraint (1.20) is two-sided while the constraint (1.21) is one-sided. More generally, a particle constrained to be bound to the surface, represented by the equation
f
(z, Y , 2) = 0 ,
(1.22)
or bound to the curve represented by the equations (1.23) is said to be subjected to two-sided constraints. Let us now consider the case in which the particle P is bound not to pass through the surface CT, represented by Eq. (1.22). If the surface 0 is supposed to divide the space into two regions, the one in which the particle is free to move is said to be the exterior region, and the remaining one the interior region. Of course, the left-hand side of Eq. (1.22), vanishing on 0,will be positive in one of the two regions and negative in the other one. Moreover, since it is possible to multiply the left hand side of Eq. (1.22) by a nowhere vanishing factor, we can arrange the equation in such a way that it will be positive in the external region; that is in the region in which the particle P is free to move. Therefore, the positions filled by P are all and the only ones satisfying the inequality f(X,Y,.)
L
0.
In this way, the constraint for P is one-sided.
(1.24)
The Lagrangian Coordinates
18
In the presence of such constraints, the possible positions of P are further distinguished in 0
ordinary positions
0
border positions - the ones satisfying the condition:
-
the ones satisfying the condition:
f(.,
Y,z ) = 0 '
(1.26)
In the last case, it is also said that the point is leaning on the surface u. Let us now extend the constraint concept to the case of an arbitrary system of particles. It is worth recalling that in considering the motion M of a system of particles S, it is usual, to speak of aptitude to indicate the ensemble of positions at a given time t , and of distribution of velocities v'p of points P of S at the same time. The aptitude is usually denoted by {P,v'p(t)}. Thus, for a system of particles S, any restriction on the positions or on the aptitude of S is called a constraint. The constraint will be called inner if such a restriction translates an intrinsic property of the natural body represented by S, outer if it originates from the presence of obstacles (other bodies) external to S. For instance, the rigidity constraint schematizes an intrinsic property of solids (indeformability), so that it is an inner constraint. While a particle is said to be free when it is not subjected to any constraints, a system of particles is said to be free when it is not subjected to outer constraints. For a system of particles S , it is important for further distinction to use terms introduced by Hertzv: holonomicll constraints and anholonomic constraints. 0
A constraint which directly imposes restrictions on the position of S,
0
is called holonomic. A constraint which directly imposes restrictions on the aptitude of S, is called anholonomic.
~~
TH. Hertz was born in Hamburg in 1857 and died in Bonn in 1894. A physicist and mathematician, he first detected the electromagnetic waves. IIl?rom Greek 6Aou (integer) and v6pou (law).
Constraints
19
In this way, the equality or the inequality representing a given constraint will contain only parameters corresponding to the position of S, if the constraint is holonomic; it will contain also the time derivative of such parameters, if the constraint is anholonomic, with the time derivatives specifying the aptitude of S. A classical example of anholonomic constraint can be given as follows: Let us consider a rigid sphere S on a plane T , rolling without slipping (see Remark 5 ) along it. According to the remark at the end of the section, this means that at any time t , the relative velocity of S with respect to T , in all contact points, is vanishing. As the characteristic velocity of rigid motions is
cQ= cP+ W ' A ( P - Q ) ,
(1.27)
where Q is a generic point of S, and w the angular velocity, it follows that the velocity of any point Q of the sphere S, at time in which v'p = 0, is given by GQ = W ' A ( P - Q ) ,
(1.28)
that is, the motion is a pure rotation. We can conclude that in such conditions, the aptitude of the sphere is a rotation around the instantaneous axis passing through the contact point P between S and K. Therefore, the requirement that a rigid sphere on a plane T rolls on it without slipping, constitutes an anholonomic constraint, since the aptitude of the sphere can only be a rotation about an axis passing through the contact point.
Remark 5 When the motion of a system of particles is observed f r o m two different frames, each one moving with respect to the other, it is a convention to assume one of them steady Tn, and the other one mobile TO,with fl and 0 denoting the origins of frames. The Euclidean space framed with Tn is called steady space, where else the one framed with To which moves with respect to Ta, is called mobile. Let us consider then a surface u, a border of a natural body, of the mobile space, and a surface u', a border of another body, of the steady space. W h e n during the time u and u' share points and tangent planes at those points, it is said that during the motion, u rolls o n u'. Let us suppose that during the motion, u rolls o n u', and let H be one of the contact points. The velocity of the point P of u, which at time t is laid upon H , is called slipping velocity of u with respect to u' at point H at time t. Finally, the circumstance that f o r all time t , the creeping velocity of u with respect t o u' at any contact point is vanishing, is referred to saying that during the motion, u rolls on a', without slipping.
The Lagmngian Coordinates
20
1.3
Degrees of Freedom and Lagrangian Coordinates
Let S be a generic system of particles constrained. The positions are that at time t , the constraints permit to S are said, “possible positions of S at time t” or also “compatible positions of S at time t.” Let Enbe the n-dimensional Euclidean space and let us suppose, at first, that S is composed of one particle P only. Let us assume that P is constrained to move on a regular curve y at rest in a Cartesian frame TO.The regular curve will be represented in the frame To by the parametric equations:
(1.29)
where cp, $, x are three functions defined in the closed interval [a,b] of R’, and X E [a,b] is a real parameter. The regularity of y means that the functions cp, +, x are supposed to be continuous together with their first derivatives in the interval [ a , b ] , and to satisfy the conditions below: 0 0
+
The function H ( A ) = dp’2 + $’2 x’2 is positive V A E [a,b ] . There is no pair (A’, A’’) of distinct values of A, such that, simultaneously, cp(A’) = cp(A”), $(A’) = $(A”), and x(A’) = x(A”).
As to the regularity of y, the possible positions of P are, at any time t , in a one-to-one correspondence with real numbers in the interval [a,b] c R’.
If a particle P is constrained to move on a moving regular curve 7, represented in the frame TO by the parametric equations (a family of regular curves):
(1.30)
at each time t , as the constraint depends on the time, the positions of P compatible with the constraint are in a one-to-one correspondence with real numbers in the interval [a,b] c !I? generally ’, changing in time. The fact that it is possible to establish a one-to-one map between the positions of P , compatible with constraints at a given time t , the set of
Degrees of Reedom and Lagmngian Coordinates
21
values that a parameter takes on an interval of R1 is usually expressed by saying that: In order to specify the position of a particle on a curve, only one parameter is necessary, or also that, the number of degrees of freedom of a point constrained on a regular curve is 1. Let us now assume that P is constrained to move on a regular surface 0 at rest in a Cartesian frame To. The regular surface will be represented in the frame To by the parametric equations:
(1.31)
where A1 and A2 are real parameters and the three functions p, +, x are defined in a simply connected bounded domain of R2 by ai
5 Xi 5 bi, i = 1,2.
(1.32)
The regularity of 0 means that the functions cp, +, x are supposed to be continuous, together with their first partial derivatives in [al,b l ] x [a2,bz] and to satisfy the following conditions: 0
the determinants
0
are nowhere vanishing, so that the function W(A1,Xz) = dA2 B2 C2 is always positive V XI, A2 E [a, b]; there is no pair ((A;, A;), (A:, A;)) of distinct sets of values (AI, Xz), such that, simultaneously, cp(XI,, A;) = p(Ay, A;), +(A',, Xl,) =+(A?, A!), and x(X;, A;) = x(Xl,l, A;).
+ +
As to the regularity of u, the possible positions of P are in a one-to-one correspondence with pairs of real numbers in the rectangle [a,b] c R'. W e come to the same conclusion when the particle P is constrained to move on a moving regular surface u'. In this case, of course, the rectangle, as the constraint, will depend on time.
22
The Lagmngian Coordinates
We shall say that only two independent parameters are needed to specify the positions of a particle on a surface, or that, the number of degrees of freedom of a point constrained on a regular surface is 2. In general, a system S of particles, however constrained, is said to have n degrees of freedom, if it is possible to establish a one-to-one map between the possible positions of S at time t and the values that n real parameters (q1 , . . . , qn) take in an open subset of Rn. The parameters (q1, . . . ,qn) are called Lagrangian coordinates of S . Of course, the choice of Lagrangian coordinates is not unique.
Examples a A particle P , constrained to lie on a curve, has 1 degree of freedom. It is possible to choose, as Lagrangian coordinate of P , a curvilinear coordinate. A particle P , constrained to lie on a plane, has 2 degrees of freedom. It is possible to choose, as Lagrangian coordinates of P , the Cartesian coordinates or the polar ones. a A free particle P has 3 degrees of freedom. It is possible to choose as Lagrangian coordinates of P , the Cartesian coordinates, the cylindric coordinates, the spheric-polar ones, etc. a A free rigid body has 6 degrees of freedom. Indeed, in order to specify the position of a frame Ta((,q,C)framed with the body, it is necessary to give the three coordinates of R and three of components of unit vectors along the (<,q,C) axis. It is possible to choose, as Lagrangian coordinates of the rigid body, the three coordinates of fl and a triplet of parameters in a one-to-one correspondence with three of the components of unit vectors along the (t,q,C) axis. a A rigid body with a fixed axis r has 1 degree of freedom. It is possible to choose as Lagrangian coordinate, the angle fl between two planes having T as intersection, one of them steady, the other framed with the body. a A rigid body with a fixed point has 3 degrees of freedom since only three parameters suffice to specify the position of a frame framed with the body and having the origin in the fixed point. a A system consisting of two rigid bodies which participate in a common axis (e.g., a compass), has 7 degrees of freedom. Indeed, six parameters are needed to specify the position of the first body and only one to specify the position of the second body with respect to the first.
The Calculus of Variations and the Lagrange Equations
23
The following definition will conclude the section: A se stern of ~ a ~ i c l is e scaZ~edh o l o n o m i ~i f at hus ~ n i ~ me ul ~~degrees y of freedom and i f it is submitted to holonomic constraints only. 1.4 The Calculus of Variations and the Lagrange Equations
By considering the evolution of a holonomic system with n degrees of freedom as a sequence of equilibrium states (d’ A ~ e m ~ e runder t) the action of all the forces (eflective, from constraints and inertial), and by applying the principle of the ~~u~~ works to ~ a T ~ e~q ~n aat ~~o nofs ~ ~ n a m i cthe s , equations of the motion can be written in the elegant and powerful Lagrangian form:
“ a‘ ----dt&h
dqh
- Q h h e {1,...,n } ,
where q’s denote the Lagrangian coordinates, q’s the Lagrangian velocities and Q’s the Lagrangian components of the “force.” In case the forces are conservative, a function U ( g / g / t ) (the potential energy) exists, such that Qh
d dU - aU = -dt aqh dQh ’
so that the Lagrangian equations can be written as follows:
where C = 7 - U is called the Lagrangian function. We do not report the derivation, as it can be found in almost all textbooks in classical mechanics. We just adopt here an axiomatic point of view according to which: 0
0
The state of a system is completely defined by specifying its coordinates and velocities (q/Q). The evolution; that is the sequence of states is completely determined by giving a function
defined on the sets of states, and two different configurations QA and of the system at two different time instant, t A and t g , such that
QB
qA
= q ( t A ) ,QB = Q ( t B , )
The Lagrangian Coonlanates
24
Among all close curves the action integral,
q h = qh(t)
joining A and B , the one for which
(1.33)
takes the least value, will represent the evolution of the system.
Remark 6
I t is worth to remark that this formulation holds for a small part of the trajectory. On the whole trajectory the integral S[q] may have just an extremum, not necessarily a minimum. However, the equations of the motion can be written by using only the extremum condition.
The axiomatic approach is called the principle of least action or the Hamilton principle. Integrals like those in Eq. (1.33) are defined on a space of functions
S:F-+%, and could be called functions, but for historical reasons, are called functionals. A few words on their use will be spent after a short historical comment. 1.4.1
Historical notes
The Newton problem The calculus of variations was founded simultaneously to the differential calculus (1686). In his Philosophiae Naturalis Principia Mathematica, Newton was the first to propose the problem of the body with the least opposition. It is a problem involving different concrete cases, as it concerns the best form of a body (submarine or a missile, etc.) in order to suffer, from the medium, the least opposition to its motion. So formulated, the problem is too difficult. For the sake of simplicity, the question was first proposed for a body with a form invariant for rotation about an axis parallel to the direction of the motion (equal inertial moments in the plane orthogonal to the velocity), and only for its head to avoid the difficult problem of vortices surrounding its tail. By k i n g the length and the height of the head the problem becomes: Given two points P and Q, find the planar curve joining them and generating, by rotation around the planar normal n' at P, a revolution surface while moving parallel to 6 , that suffers the least opposition from the medium.
The Calculus of Variations and the Lagmnge Equations
25
The solution clearly depends on the opposition law. Newton assumed that the opposition, on a generic element of the surface, was proportiond to the square of the projection of the velocity along 8. By taking an orthogonal Cartesian frame, with the x axis coincident with the rotation axis and by denoting with y = y(z) the unknown curve, it turns out that the opposition R is given by
where p , q are the x-coordinates of the points P , Q and k is a constant which depends on the velocity of the surface. The problem is then to find, among all curves y = y(x) joining P and Q, the one for which the previous integral takes the least value. The brachistochrone problem Ten years later, the following problem was proposed and solved by Johann Bernoulli: Given two points P and Q in a vertical plane, find among all planar curves joining them, the one which the time required for a particle to descend, without f i c t i o n , f r o m the origin P to Q , would be the least possible. By choosing a suitable orthogonal Cartesian frame with a vertical y axis, the time for a particle to descend from P to Q along the curve y = y(x) will be given by
where p , q are the x-coordinates of the points P, Q and g N 9.8 m/s2 is the modulus of the acceIeration due to gravity. Thus, the searched curve is the one for which the above integral takes the least value. Johann Bernoulli found, as a solution, the curve whose parametric equations are
I
x = k(8 - sin$), = k ( 8 - C O S ~ ,)
where k represents the ratio between the distances of points P and Q from the origin. His method of solution was strongly criticized, from a mathematical
26
T h e Lagrangian Coordinates
point of view, by Newton, Leibnitz, 1' Hospital and Jacob Bernoulli**who was able to find the same solution by using different methods. In particular, Jacob Bernoulli solved the problem by using a geometrical method, of larger applicability, based on the principle according to which, if a curve is characterized by a property of maximum or minimum, any part of it, no matter how small, is characterized by the same property, According to this principle, for instance, if a curve is a brachistochrone, any part of it is again a brachistochrone. It is, therefore, possible to replace the curve with a broken line in such B, way that the problem reduces to find its vertex by using ordinary differential calculus met hods. The approach used by Jacob Bernoulli, who must be considered the founder of the calculus of variations, were generalized and wisely extended by Eulertt to a large category of problems. He started with the classification of problems in two classes: find among all curves satisfying suitable boundary conditions, the one for which a given integral takes an extremum (a minimum or a maximum) value. find among all curves satisfying suitable boundary conditions for which given integrals take assigned values (constraints), the one for which another integral takes an extremum value (isoperimetric problems). He invented the z s o p ~ r ~ ~rule e t ~which c allows to reduce, at least on principle, a given problem of the second class to a problem belonging to the first one. A more rigorous treatment was given by Lagrange, who introduced the concept of variation, which allows this kind of problems to be treated with ordinary differential calculus methods. "Jacob Bernoulli was born a t Basilea (BBle) on December 27, 1654. We has been, €or many years, a professor of m a t h e ~ a t i c sat Basilea University. Supporter of Leibnitz' scientific ideas, he died at Basilea in 1705. Johann Bernoulli, the brother of Jacob, was born at B d e on August 7, 1667, and died there on January 1, 1748. He has been, for many years, a professor of mathematics a t Groningen from 1695 to 1705 and at B&le University, where he succeeded his brother, from 1705 to 1748. As an illustration of his character, it may be mentioned that he expelted his son Daniel from his house for obtaining a prize from the Acadernie de h n c e which he had expected to receive himself. ++LeonardEuler, born in Basilea in 1707 and died in St. Petersburg in 1783, was director of the Academy of Sciences in Berlin, and soon after, the Academy of Sciences in St. Petersburg. He was one of the most important and fecund mathematicians of all times, both in calculus and in its physical applications.
The Calculus of Variations and the Lagmnge Equations
27
Let zt = uo(z) be the curve which extremizes the integral I[%],among all the curves in the plane joining two given points P and Q. In order to analyze the properties of I[.], it is necessary to compare the value I[uO],taken by the integral on the given curve u = uo(z),with the one I[u]corresponding to a different curve. The difference for u ( z ) - uo(z) is the variation, for fixed z, in passing from the curve u = u ~ ( zto ) the curve u = u(x). If the two curves are %cry close,” this variation will be “very small,” similarly to the differential du that a given function u undergoes for a fittle change dx of 2,but very different in nature. In order to distinguish between them, and at the same time to underline their analogy, Lagrange introduced the symbol 6u for the variation and invented a calculus completely analogous to the ordinary one. In correspondence with the variation 6u of the curve u = ug(z), the integral I[uo] undergoes the variation
I[.]
- I[uol
which decomposes in different parts conformable to the decomposition given to a function f(z) from an increment of z. The different parts are given, save numerical factors, by succeeding differentials df, d 2 f , . .. . Similarly, the different parts of I[u]- I[uo],save numerical factors, are called first variation, second variation, . , . of the integral I and denoted by 61, 621, * * . . Lagrange was able to show that, similarly to the case of a function, the extrema of I satisfy the relation 61 = 0. In order to distinguish extrema, in minima and maxima, Legendre analyzed the second variation and introduced an elegant transforma~io~ of a2I Ieading to necessary conditions characterizing maxima or minima. The Legendre condition survived a deep criticism by Lagrange, who observed, among other questions, that the Riccatitt , equation leading to the Legendre transformation of the second variation @ I , does not always admit a bounded continuous solution in all the considered interval and that the transformation does not always exist. Finally, the difficult question was genially solved by Jacobi (1837) who gave a new necessary condition. In 1870, ~ e i e r s t r a s sobserved that, contrary to the case of f ~ c t i o n sthe , sign of the second variation does not ensure the existence of the maximum or minimum on the considered extrema. According to Weierstrass and Scheeffer, ZtJacopo Riccati was born in Venice in 1676 and died in “reviso in 1754. He was an aristocrat who studied mathematics privately. His famous equation is contained in Acta Eruditomn (Lipsia, 1722).
the reason for the difference lies in the complexity of a curve with respect to a point. Two curves, although “very close,” can strongly differ by their tangents at close points and the integral I also contains the derivative u‘ which characterizes such tangents. Weierstrass was able to give sufficient conditions for the existence of the minimum or the maximum in terms of the sign of a particular function which today is called the Weierstmss function. Weierstrass also criticized the representation y = U ( S ) for a curve, since this representation meets at most in a point, the parallels to the vertical axis, and then it greatly restricts the field of possible solutions. He developed its theory by using parametric representations
and replacing the integral I by
At the same time, Darboux found sufficient conditions for the minimum in the geodesic problem by introducing curvilinear coordinates, which allows us to write the considered integrals in a particularly simple form, where the minimum properties turn out to be evident. Very important contributions were then given by Kneser, Lindeberg, Gauss, Ostrogradsky, Delauney, Clebsch, Schwarz, Volterra and Hilbert . For more details, see for instance, Ref. 53. 1.4.2
A digression on the variation methods in problems with fixed boundaries
Continuous functionals
A f ~ c t i o n a Flu] l will be called c o n ~ ~ n u o uifs , to a “small change” of ~ ( z ) , there corresponds a “small change” in the functional F[u]. With previous definition, only a few functionals will be continuous, since in general the function in the integrals will depend on the derivatives of u. For instance, in the action integral, first derivatives are also included. Thus, it is natural to give the following definitions of close curves:
Two curves u(x)and u(x) are of zero-order proximity close, i f the absolute value of their diflerence Iu(z)- )I(. is small
The Calculus of Variations and the Lagrange Equations
29
Two curves ~ ( x and ) v ( x ) are of first-order proximity close, if the absolute va~uesof the d ~ # e ~ n c e{su ( x )- v(x)l and {a'(.) - v'(z)l are ~ ~ ~ 1 2 . Two curves U ( X ) and v(x) are of kth-order proximity close, if the absolute -v(~)(%)/, values of the di#erence~, /u(x)- v(x)t, lu'(x)- v ' ( x ) / ,. .. ,1dk)'(z) are small It is then possible to define the notion of distance CT between two curves. By assuming that u and v have continuous derivatives up to order k, the distance of order k: is defined as
and then close-lying curves would be curves with a small distance.
De~ivatiwesin a vector space 1. Strong derivative or Frechet derivative. Let U and V be two normed vector spaces and F a map from an open subset A of U into V .
F : A G U +V. The map F will be called difierentiable at point u E U , if a linear-bounded operator F' exists, such that
+
F(u h) - F(u) = F'h
+
b ( ~h) ,
,
where a is ~nfinitesima1with respect to the distance in V given by the norm:
The reader will recognize, by identifying U with Sn and V with a, the usual definition of the differentiability for a numerical function of n real variables. The linear part of the increment F'h, is called the strong diflerential (or the h c h e t di#e~nt%ul) of the map F at point u and the operator F' is called the strong de~wative(or the Frechet d e ~ v a t ~ vof e ) the map F at point u. It is easy to see that, if the map F is differentiab~e,the corresponding derivative
The Lagmngian Coordinates
30
is uniquely defined. Furthermore, the theorem on the derivative of composed maps can be easily proven.” 2. Weak derivative or Gateaux derivative. If G denotes a map from an open subset of U into V , the limit d DG(u,h) = -G(u dt
. G(u + th) + th)(t=o= ltm t t40
-
G(u) 3
where the convergence is considered only with respect to the norm of the vector space V , is called the weak differential or the Gateaux differential of G at point u. The Gateaux differential can be nonlinear with respect to h. In the case it is linear, for example, in the case a bounded linear operator G, exists, such that
DG[u,h) = G,h
,
the operator G, is called the weak derivative (or the Gateaux derivative) of G at point u. Let us observe that, in general, for the weak derivatives, the theorem on the derivative of a composed map is not true. It is easy to see that, if the strong derivative of a map F exists, then the weak derivative also exists and the two derivatives coincide. As a matter of fact, if F is strongly differentiab~e,then
F ( u 4-t h ) - F ( U ) = F’(u)(th)4-u ( u ,th) = t F ’ ( u ) ( h )+ a(u,t),
so that
The converse is not true, and generally never true in the finite dimensional spaces, as the folIowing example well shows.
Example 3
The function
The Colculus of Varlations and the Lagrunge Equations
31
is c o n t ~ ~ ~ on u ug2. s At the point (0,0 ) , its weak d~fferent~al exists:
Nevertheless, the weak differential is not the linear part of the increment of the ~ u n ~ tati othe~ point (O,O), since
The result of the previous exercise is not surprising, just like in the finite dimensional case, the existence of partial derivatives of a given function does not ensure its differentiabilit~,which can be ensured by the continuity of the partial derivatives. In this context, it is important the following theorem's proof can be found, for instance, in Ref. 28.
If the weak: d e ~ v a t ~ Fu v e of the map F exists in a n e ~ g h ~ o r hood of a point uo and represents a continuous operator at UO,then the strong derivative of F also exists at uo and coincides with the weak one.
Theorem 1.1
3, The gradient of a functional. Let U be a vector space of numerical functions of a real variable, namely u(z),endowed with a scalar product (. ,.) and let F be a ~ n c t i o n adefined ~ in U ,
F :u E U
4 F[u]E
8.
The gradient of F , or the functional derivative of F , with respect to the scalar product is the function of U,denoted by G = SF,/du, defined by the relation
6F E -dF [ u + ~ p ] I ~ = o dc =
(g,~) .
(1.34)
The second functional derivative of F is given by the weak derivative Guof G, which in this case, can be also defined as
The Lagrungian Coordinates
32
Of course, the operator G, is symmetric with respect to (. , .); i.e. G,=G,+, since
Example 4 Let U be the space of all Cw functions u ( x ) , defined on the interval I = ]a,b( of the real axis x and going to zero at a and b, together with all their x derivatives, and let F[u]be a functional,
1
6
~ [ u=]
f(u, u x ,U r x , *
*
unx,.
. .)dx
1
where f is a given function of u and of its x derivatives, which we are denoting with u,,t i z x , .. . ,u,,, . . . . According to the definition of Eq. (1.34), we have
and integrating b y parts all the terms, except the first,
Therefore,
so that
The Calcdvs of Variations and the Lagmnge Equations
33
In this way, in the simple case of a functional depending only on u,
Jd f 6
F1 .I
=
( U W ,
the f u n c ~ ~ o derivative ~al is given &y -SF =su
af au'
The functional derivative of a functional depending only on u and u x ,
1
b
Flu1 =
f(u,u x ) d x ,
is given by
For insta~ce,for square i n t e ~ a ~functions, $e we have
and the Gateaux derivatives of
are the following operators 1,
-ax,,
2u+axx,
ax
FE
d dx '
which are s ~ m ~ e t rwi cz t ~~ s ~tocthet L2 scalar ~ r o ~ u c t .
Example 5 The ~ n c t i o nus cannot be the gradient, with respect to the Lz scalar pmhct, of any functional, since its Gateaux derivative is the skewsymmetric operator a.,
The Lagrangian Coordinates
34
Example 6 tional
B y using the same procedure, it is easy to see that for the func-
we have
so that
The H a m i ~ t o np r i ~ c i p l e From the previous example, by identifying the coordinate x with the time t , the functions u’s with the Lagrangian coordinates q’s and the derivatives us’s with the Lagrangian velocities q’s, it follows that Lagrange’s equations
are equivalent to the vanishing of the functional derivatives of the functional
for functions q ( t ) vanishing at the instants t p and tQ. Thus, Lagrange’s equations are the equations for the extrema (or critical points) of S [ q ] . When some further hypothesis are assumed on the qft)’s, the Lagrangian equations give the curve on which the action integral takes its minimum value. For this reason, the Hamilton principle is also called the least action principle.
Remorki on Lagrange's Equations
1.5 1.5.1
35
Remarks on Lagrange's Equations E q ~ i ~ a l e~ n ta ~ a n g ~ a n ~
Looking at the action integral, it turns evident that the Lagrangian equations , differ by corresponding to Lagrangian functions C(q/&'t) and C ' ( q / g / ~ )which a term given by the derivative of a function depending only by the coordinates q and the time t , d
+ ZfW)
C'(4/4/t) = C ( q / d / t )
coincide. Indeed, the two action integrals,
differ by a constant term S'M - SbI = f ( Q / t g ) - f(P/tp)t
which does not contribute to the variation 6S"q] = 6S[q].
Two such Lagrangians are called equavatent. This is not, of course, the most general case and there exist examples of Lagrangian dynamics admitting more than one, not equivalent, Lagrangian. An exhaustive treatment can be found in Ref. 157.
1.5.2
Dgnamical similitzl.de
Let us start by observing that two Lagrangian functions differing by a constant factor give the same Lagrangian equations. Thanks to this circumstance, it is possible to infer some properties of the motion without integrating the corresponding equations. This happens, for instance, for simple systems with kinetic energy 7 = $mu2 = $m Chqi and a potential energy U ( q ) , which is a homogeneous function of the coordinates; i.e. a function satisfying the condition
W X Q ) = XaU(q)
9
where X is an arbitrary constant and a is the homogeneity degree of U.
36
By performing the transformation qh
(ri
= Xqh,
t 3 if =
the velocities 4h will be multiplied by the factor X/p
0
the kinetic energy 7will be multiplied by the factor (X/p)'
the potential energy U will be multiplied by the factor A"
u 3 X"U; the Lagrangian function C will change accoTding to
As a consequence, the Lagrangian function will be multiplied by a constant factor A* only if p = A"?,
To change the coordinates by a constant factor X means to pass from some trajectories to other ones geometrically similar to the first, the only difference lying in the linear dimensions (homothety). We finally arrive to the following conclusion: If the potential energy of a sample system is a homogeneous function of coordi~ates,the equataons of the motion a ~ g emo m e~ ~ r ~ c a~l l ~ m i trajec~ar tories for which the time interval t' and t , between corresponding points o n trajectories, have the ratio 1-4 t , (1.35)
"=(%>
Example 7 For small oscillations, the potential eriergy is a quadratic f m c tion of coordinates (a = 2). Equation (1.35) shows that the period P does not ~ e p e no n~ their amplitu~es~ ~ a l i l e ~i ~bs s e ~ a o~n ithe o ~c a n ~ e Z a b ~atm Duomo in Fisa).
Remarks
OR Lagmnge’s
Equations
37
Example 8
I n a homogeneous field, the potential energy is a linear function of coordinates (a = 1). Equation (1.35) shows that for a free falling body in the gravity field, the squares of time of falling are in the ratio of their initial heights.
Example 9
I n the case of the Newtonian attraction between two masses, or the Coulomb attraction of two charges, the potential energy is in the inverse proportion with their distance ( a = -1). Equation (1.35) shows that the squares of the revolution periods of planets, in their orbits, are proportional t o the third power of their linear dimensions (Kepler’s third law). The previous analysis can also be carried out, of course, by means of Newton’s equations. The reader is invited to do it by himself. 1.5.3
Electrical circuit analysis
The circuital relations, for a network of coupled reactive impedances in which a system of electrical currents i h is flowing, generated by electromotive forces vh, are
where Lhk = L k h are the mutual inductances ( h # k) and self-inductances ( h = k), ch the capacitances, Rh the resistive impedances and Fh = dVh/dt. They are the Lagrange equations associated with the Lagrangian f u n ~ t i o n , ~ ~ ~ ~
Chapter 2
Harniltonian Systems
Lagrange’s equations constitute a system of n second order differential equations in the unknown curves q h = Q h ( t ) . By writing
d 8C d2C d t a& - k= 1
a2L: q k + x p k=l %h%k
dk
-t-
d2L
%hat ’
they can be explicitly written in the form
where the force Fh is defined by
Thus, if the Lagrangian function is regular; that is, the Hessian determinant
3 of the matrix
is not vanishing, Lagrange’s equations can be written in the following normal form: 39
Hamiltonian Systems
40 n
h=l
Moreover, if the relations dqh/dt = v h are not interpreted any more as constraints, Lagrange's equations can be written, much more naturally, as a system of 2 n first order differential equations in the form
i.
--...---
d bC
aL
dtbvh
aqh
-O'
V h E { 1 , 2 ,...,n},
(2.1)
dtqh = v h >
and for a regular Lagrangian, in the following normal form:
2.1
The Legendre Transformation
The system (2.1) is not form invariant and can be transformed to a new system of 2 n first order differential equations in infinitely many ways. Among them there exists a remarkable tr~sformation,the so-called Legendre* t r u n s ~ o ~ u tion, leading to a particular system of 2 n first order differential equations, called a Hamiltonian system, possessing very interesting symmetry properties. The Legendre transformation is naturally suggested by the Lagrangian system and it consists in introducing n new auxiliary coordinates p, defined by ph =
ar:
-(q/~/t)l
V h e {1,2 l . . . , n } ,
avh
'Adrien Marie Legendre waa born in Toulouse on September 18, 752 and died in Paris on January 10, 1833. He was appointed professor a t the military school in Paris in 1777, and at the &Cole Normale in 1795. The i n h e n c e of Laplace was steadily exerted against his obtaining office public recognition, and Legendre, who was a timid student, accepted the obscurity to which the hostility of his colleague condemned him. Legendre's analysis is of high order of excellence, and is second only to that produced by Lagrange and Laplace, though it is not so original.
The Hamilton Equations
41
The p’s are called conjugate variables of q’s, or for their dynamical interpretation in some typical cases, conjugate momenta of 4’s. The above system, considered as an algebraic system of equations, can be solved, for a regular Lagrangian, with respect to the unknown vh in the following form: Vh = %(q/p/t)
-
(2.3)
By using Eqs. (2.2) and (2.3), the Lagrangian equations (2.1) become
where the symbol * indicates that the velocities V h have been expressed, by means of Eq. (2.3), in terms of Lagrangian coordinates q, their conjugate momenta p and time t. Thus, we obtain the nonnal system of 2n first order differential equations with the 2 n unknown functions q’s and p’s,
Of course, the above system is equivalent to the original Lagrangian system (2.1), since from one side, it follows from Eq. (2.1) by means of the described procedure ( J # 0), and vice versa, it is possible to go back to Eq. (2.1), replacing the momenta p in Eq. (2.5) with their expression given by Eq. (2.2).
2.2
2.2.1
The Hamilton Equations From Lagrange to Hamilton equations
It is remarkable that the RHS of Eqs. (2.5) can be expressed in terms of a unique function of q’s, p’s and t , called Hamilton function or characteristic function or simply Hamiltonian. In this way, the first order system will appear as simple as the original Lagrangian system (2.1), depending on the unique function C. The mentioned
Hamiltonican Systems
42
Hamiltonian function is essentially the function
representing, in the dynamical case, the total energy of the system. The only distinguishing difference is that it must be expressed in terms of q’s, p’s and t by means of Eqs. (2.2) and (2.3). In other words, the Hamiltonian function is the function W p / q / t ) = E(q/v/t)*= C P h W & / P / t ) - C ( q / 4 q / p / t ) / t )* h
(2.6)
In order to recognize that RHS of Eqs. (2.5) can be expressed in a very simple way, in terms of the function defined by Eq. (2.6), it is enough to apply the following classical procedure first introduced by Hamilton. By considering the q’s, p’s and t as independent variables and the ZI’S as function of them, given by Eq. (2.3), let us add, for fixed t , the arbitrary increments dp and 6q, to q’s and p’s. In this way, the function 3t will increase by the differential of I f l ( p / q / t ) , given by
On the other side+,from Eq. (2.6),
t A s for Eq. (2.3), the velocities v’s are not independent variables. However, as it is well known from differential calculus, the first differential of [L],=,coincides with the differential of L transformed with V h = a h ( p / q / t ) . This follows from the observation t h at the first order differential of a function f is form invariant; i.e. the transformed of its differential coincides with the differential of the transformed function: (df), = d(f), , where the symbol * denotes the transformation. This property, which does not hold for the second order differentials, will be more transparent after the geometrical considerations of Part 11.
The Hamilton Equations
43
Therefore, by comparison, the following equalities hold:
By using Eqs. (2.2) and (2.2.1), Lagrange's equations (2.1) give d
ac
ac
d dC (2.7)
or definitively
8%
,
V h e { 1 , 2,...,n } .
(2.8)
The above equations are called Hamilton's equations. Any system of the form (2.8), irrespective of how the function 3 t ( q / p / t ) has been chosen, is called a canonical system or a Hamiltonian system. The p's and q's are called canonical coordinates. No essential distinctions there exists between them, as the system (2.8) is invariant under the interchange: p t)q, 3t C) -3t. 2.2.2
fiorn Hamilton to Lagrange equations
It has been shown, in the previous section, that a given Lagrangian system reducible to the normal form; i.e. such that J' # 0, can be suitably transformed to a new system of 2n first order differential equations in the unknown p's and q's. Moreover, Eqs. (2.2),
ac
ph=--(Q/v/t), avh
VhG {1,2,...,n},
(2.9)
Hamiltonian Systems
44
once solved with respect to v's, have the form
(2.10) where 7-l denotes the Hamiltonian function defined by Eq. (2.6). It is, of course, possible to go back. Suppose that we start from Eq. (2.8) and that the Hessian determinant,
of 3t is nonvanishing, so that the equations
can be solved with respect to p's to give Ph = @h(q/V/t) 1
v h E {1,2, - * * 3 n>
Let us then define the Lagrangian function corresponding to 7-l by
L = x v h p h ( q / V / t )- x ( q / P ( q / v / t ) / t ) 7
h
or shortly
and consider its differential,
By using again the observation that (Sf)* = S ( f ) , , we obtain
The Hamilton Equations
45
Therefore, we have
(2.11)
By using the above relations, Hamilton’s equations (2.8) give -Ph
= --
d 813 --
[g]
dC
d =-
* = dQh
1
which are just the Lagrange equations (2.1). We finally observe that the Hessian matrices
of the Lagrangian and of the Hamiltonian, respectively, are inverse to each other, since
a2c SO
that
2.2.3
I?
a 2 ~
= 3-l.
Remarks on Hamilton’s equations
The virial theorem The time average f of a function f ( t ) is defined by the following limit:
Hamzltonian Systems
46
From the definition, it turns out that the time average of a function, which is the time derivative f = dF/dt of a bounded function F ( t ) , is vanishing. Indeed,
If the motion of a simple system, whose potential energy U ( q ) is a homogeneous function of coordinates, develops in a bounded region of the space, there exists a very simple relation among the time averages of the potential and kinetic energies. This relation is known under the name of the virial theorem. Let us start from Eq. (2.6) written in the following form:
The above relation can also be written as follows:
In the case the motion takes place with bounded “velocitied’ in a bounded region of the configuration space, by taking the time averages of both sides, we have ~
d3c
m = x q h - .
%h
For a simple system, the quantity
3c + C’ is twice the kinetic energy 7*,
so that we can write
We can finally argue that, for a simple system whose potential energy U ( q ) is a homogeneous function of coordinates of degree a , and for motion with bounded velocities in a bounded region of space, the interesting relation
holds.
The Poisson Bracket and the Jacobi-Poisson Theorem
Given that, for a simple system, 3t = 7" also written in the following equivalent form:
47
+ U ,the above relation can be
where E is the total energy. In particular, 0
for small oscillations: 7; = ii = (1/2)E, for the Newtonian interaction: 2 7 ; = -a, or E = -7,which given that F > 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 << ml, then p m2 and TI 2. The corresponding ~ a m i ~ t o n functionl ~an with m2 = m and ml = M , is given by
-
N
(4.16)
It is worth observing, by using the results of problems in the section devoted to the Poisson bracket, that the angular momentum L' is a constant of the motion, so that
{i13t)= 0 .
(4.17)
Of course the last property is shared by all central potentials; that is, by all potentials U ( r > depending only on the modulus r of the vector position F. Therefore, the trajectory lies in the plane determined by the initial values of position r f , and velocity 80,which is the plane orthogonal to the angular momentum E. 4.3.1
The Laplace-Runge-Lenr vector
For the Kepler potential, beyond the angular momentum i,there exist specific constants of the motion expressed by the so-called Laplace-Runge-Lenz vector given by (4.18) whose components are not independent , since
which simply says that the LaplaceRungct-Lenz vector lies on the plane of the motion. It is interesting to evaluate the Poisson brackets involving B. The reader is invited to verify that
The Kepler Dynamics
91
Moreover, the Poisson bracket
{g,g}= n ' A g , which has already been proven to hold for any vector in terms of components as follows:
g, can also be written
3
{Li,B j } =
EijkBk
*
k=l
More tedious, but important, is to verify that
It is well known that for a negative total energy, E < 0, the motion is bounded and the orbits are ellipses with the sun located in one of two focuses. In this case, we introduce the vector
so that the previous Poisson brackets can be written in the form
As a consequence, the vectors defined by
-.
1 2
J = -(L-A),
will have the following Poisson brackets: 3
(lh,Ik)=x&hkEIEj 1=1
3
(Jh,Jk)=ZEhklJl,
(lh,Jk)
==o,
t=l
in close analogy with the ones of the angular momentum. For readers familiar with Lie algebras, this shows that the Lie algebra of symmetries for the Kepler dynamics is twice so (31, or better, su (2) @ su (2), which is locally isomorphic with so (4).
The Integration Methods
92
More precisely, we observe that the Hamiltonian 31 can be written as
mk2 mk2 2(L2+ A2) - -4(12 + P ) *
%=-
In terms of the generators of S’O(4), La@= -Lp,(a,p = 1,2,3,4),defined by 3
Lhk = E E h & ,
h,k = 1 , 2 , 3 ,
d=l
Lh4
z=
--Ldh = A h , h = 1 , 2 , 3 ,
the Hamiltonian 3t becomes %=--
mk2 G1
where C1 = 4.3.2
L,ijLap is the first Casimir of SO (4).
The hydrogen atom
The SO(4) i n v a r i ~ c eexplains why the degeneracy of the q u energy~ levefs of the hydrogen atom is greater than what is naturally expected from the central symmetry (SO (3) invariance). Quantization rules roughly consist in replacing classical dynamical variables with self-adjoint operators in the Hilbert space of complex squared integrable functions, according to what follows:
-
where ti = 1.052’10-27 erg s is the Planckt constant (divided by 2n) and i the i ~ a g i n a r yunity. Thus, the quantum angular moment^ and the H ~ i l t o n i ~ operator corresponding to the classical Hamiltonian function 3c = (1/2m) tMax Planck was born in Kiel in 1858, and died in Gottingen in 1947. He was appointed t o a theoretical physics chair in 1880 at Kiel University and in 1884 at Berlin University. Revolutionary against his will, a t the -beginning Planck was persuaded that the discont~nuity concept, characterized by the so-called q u ~ ofn action ~ ~h, ~was a “purely mathematical lucky violence against the laws of classical physics.” It was really just the Erst example of the renormalization procedure, after systematically introduced in field theory to cancel the infinities. He was appointed t o a Nobel Prize in 1918.
~
93
The Kepler Dynamics
[p’. p’-
e2/r]will be given by
where V2 is the Laplace operator and e the electric charge of the proton. The Laplace-Runge-Lenz vector has to be written in the form
-
1 2m
B = -($
A
1: - L’ A 8- e2-r“r ’
in order for the c o r r ~ p o n dvector-operator, i~ which is called the Pauli vector,
to be a self-adjoint operator, Clwsical formulae will be replaced by the corr~pondingquantum ones 3
x & h i h
=0 ,
(&h,
81 = 0
h = 1,2,3 ,
h=l 3
3
[ki,L j ] = i f i x & $ j k & k , [kd)B j ] = i h x E i j k & & k=l
t
k=l
where the bracket [-, denotes the co~utator-operator.Moreover, we have
We can now restrict ourselves to the Hilbert subspace which corresponds to bound states; that is, to states with negative definite eigenvalues, E < 0, of fi. In this subspace we may define another self-adjoint vector-operator
(A,, A 2 , A):
The Integration Methods
94
so that the previous commutation relations can be written in the form 3
3
k=l
k=l
As a consequence, the operators defined by
jk
= ’(Lk
2
-Ah),
will satisfy the following commutation relations:
The Hamiltonian operator will thus be
and by observing that
it can be finally written as
H=-
me4
2(4f2 + li2)
The previous formula allows us to find the {quantum) energy spectrum En, and the corresponding degeneracy, of the hydrogen atom, by using only the knowledge of the irreducible representations of SU (2) @ SU (2), without any mention to the Schrodingert equation. %‘ErwinSchriidinger was born in Vienna in 1887, and died there in 1961. After his degree, obtained at Vienna University in 1906, he moved to Stoccard, Zurich and Berlin. After Hitler’s advent, Schrodinger moved to Oxford and Dublin. Finally, he returned to Vienna in 1956. According t o Niels Bohr, he was a ‘‘universal man”; indeed he was a scientist with large cultural interests covering physics, philosophy, politics, biology and classical Greek culture. He established the basic equation of nonrelativistic quantum mechanics, and was appointed, together with Dirac, to the Nobel Prize in 1933.
The Hamilton-Jacobi Integration Method
95
fn fact, since the operators fh satisfy the c o ~ u t a t i o nrelations of an an~ 2 gular momentum, f2 can be quantized accordingly: 1’ = ~ ’ +~l), (with integer or half integer, so that the energy levels of the hydrogen atom will be given by me4 - me4 2tia(21+ 112 which is, of course, Bohr’sl formula, with n = 22 1. The degeneracy of the energy levels will be
€3, = -
-m’ +
c(21f 1)
n-1
degE, =
= n2 = D
n-1 n-1
1=1
where D ( i , j >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
<
with E [0,+m[, 7 E [0,-t-oo[. The surfaces, defined by [ = constant, 7 = constant, define two families of revolution paraboloids having the z axis as a symmetry axis.
3. Find a complete integral of the Hamilton-Jacobi equation for the Hamiltonian with 3 degrees of freedom,
describ~ngthe dynamics of a particle in the field ( ~ e w t o n or i~ Coulombianll~generated by two particles, located at distance a at IICharles Augustin Coulomb, was born in Angouldme in 1736, and died in Paris in 1806.
He worked as an engineer and was a member of the Institute de Fraplce. During the last years of his life he has been general overseer of Paris University. His contributions to friction laws and to electromagnetism can be considered of basic importance.
The Liouville Theorem on the Complete Integmbility
positions ?I,
?2.
105
Use elliptic coordinates (t,q,cp), defined by
{
5=
uJ(p
- 1)(+
- 1)coscp,
y = uJ(t2 - 1)(q2 - 1)sincp,
=at77
where u is an arbitrary parameter and t E [l,+m[, q E [-1,1[. (Hint: choose u = a , ) 4.6
4.6.1
The Liouville Theorem on the Complete Integrability Reduction
The knowledge of a first integral f for a given dynamical system, described by the equations
id= X i ( z / t ) ,
vi E {1,2,. .., m } ,
simplifies the integration problem, since the relation
f ( z ( t ) / t )= constant 3 f o must be satisfied by any solution z(t) of the equations of motion; of course, for a suitable choice of the constant, which depends on the initial conditions. All m 1 hypersurfaces obtained by varying the constant fo will foliate the whole space, and each trajectory will belong to one and only one of them. The foliation is called regular, if the hypersurfaces have the same dimension; in this way each hypersurface is called a leaf. Furthermore, if an additional functionally independent first integral g is known, a given trajectory also belongs to the hypersurface
-
g ( 4 t ) l t )= go 9
defined by g. Each trajectory thus lies on the generically m - 2 dimensional intersection of leaves of the two foliations. It follows that the knowledge of m- 1, functionally independent first integrals, defining regular foliations, completely solves the integration problem, since the l-dimensional intersections of leaves just correspond to curves representing trajectories .
Remark 11 The previous picture is just the description of a virtual case and is given as a motivation for introducing the Lie theorem below. It almost
106
The Integration Methods
never realizes, even for a simple system, as the one of a harmonic oscillator with 2 degrees of freedom, described by the Hamiltonian
with W ~ / W Z ,an irrational number. A fine global coordinate analysis can be in which the notion of isolating integral (also less found in the Wintner correctly called uniform) is introduced. A n integral f (x)is said to be isolating i f it “can enable one to make predictions concerning the possible future ( o r past) positions of the points x = x ( t ) of the solution path which goes at t = 0 through xo (the case x ( t ) = xo of an equilibrium solution being not excluded).” Thus, it is naturally expected that, for a canonical system, the knowledge of 2n - 1, functionally independent first integrals, defining regular foliations, completely solves the integration problem. Actually, for a canonical system with n degrees of freedom (and with a 2n-dimensional phase space a), it turns out that the integration problem is completely solved knowing only n functionally independent first integrals fi, which are in involution; that is, such that
More precisely, the following remarkable theorem was proven by Sophus J,ie.134,25
Theorem 15 (Lie) I f f o r a canonical system of rank 2n, m functionally independent and involutive first integrals are known, which can be solved with respect to m of the p’s, the integration problem reduces to the integration of a new canonical system of rank n - m. In other words, while for a generic first order differential system the knowledge of m first integrals reduces the rank by m units, for a Hamiltonian system the rank is reduced by 2m units. It is interesting, for concrete applications, the case in which m = n;that is, the case in which the number of such first integrals is just equal to the number n of degrees of freedom.
The Liouville Theorem on the Complete Integrability
4.6.2
107
The Liovville theorem
Theorem 16 (Liouville) If for a Hamiltonian svstem, n functionally independent and involutive first integrals are known, which can be solved with respect to the p’s, the integration problem reduces to pure quadratures; that is, the equations of motion can be solved simply by evaluating integrals. The proof will be carried out by means of the Hamilton-Jacobi integration method. According to this method, in order to have the general integral of a canonical system
it is sufficient to find a complete integral V of the partial differential equation (4.26) Then the Liouville statement will be proven if we show that the knowledge of n first integrals f r ( p / q / t ) , with r E (1,. . . , n} of the canonical system, satisfyiig the following properties: (i) are functionally independent; i.e. Ahdfh = 0 Ah = 0, (ii) are in involution; i.e. {fp, fa} = 0, r, s E (1,. . . ,n}, (iii) and define an algebraic system f t ( p / q / t ) = solvable with respect to the n variables p,; i.e.
*
allows us to determine, by pure quadratures, a complete integral of Eq. (4.26). Let us first consider the case in which the Hamiltonian 31 and the functions fr do not explicitly depend on the time t. The equations fr(P/q)=nr
r~ {1l***ln}~
(4.27)
for fixed xls, define a submanifold M f ( = )known , as the level manifold, of the phase space a. By using hypothesis (iii), Eq. (4.27) can be solved with respect to p’s in the form pa = cpa(a/n), a E ( 1 , . . . ,n} I
The Integration Methods
108
Of course, on the level manifold M, the differences pa - cp,(q/n), and consequently, the Poisson brackets {pa - V a ( q / r ) , p p- cpp(q/n)} vanish identically. Furthermore, at the end of the section, we will show that the involutivity of the f's on the whole phase space @ implies the vanishing of the Poisson brackets { p m - p a ( q / n ) , p p- (pp(q/n)}on the whole phase space a, namely
On the other hand,
so that
which implies that the differential form n h=l
is closed, or which is the same, locally exact. In other words, in any simply connected part of AdT, a function W(q/n)exists, such that (4.28)
In this way, it has been shown that the involutivity of the f's implies that the solutions of the Eq. (4.27), with respect to the p's, can be locally expressed as follows:
Let us now define n new coordinates, namely X I , X Z , . . . ,xn, by
1
’
The Liouville Theorem on the Complete Integrability
109
Then, we easily check that the relations
implicitly define an invertible transformation between the variables b/q)and ( T / x ) , which turns out to be canonical for the results obtained in subsection (3.1.2). The invertibility of the transformation easily follows by observing that
that is, by noting that d2W/&h8Tk is just the generic element of the Jacobian matrix of the p’s with respect to the T ’ S , so that the Jacobian determinant det(a2W/8qh&rk) is the inverse of the Jacobian det(dfa/apj), which is supposed (hypothesis (iii)) to be different from zero. Therefore, we come to the new canonical system
(4.29)
with a new characteristic function K: given by
where the * indicates, as usual, that the transformation has been performed. The canonical Eq. (4.29) show that K really does not depend on the x’s, since ??h = 0. As a consequence, the derivatives a X / d ? r h will not depend on time t , so that the system (4.29) is trivially integrated in the following form:
where Vh
3
b x / d ? r h , and the 6’s are arbitrary constants.
The Integration Methods
110
The general case can be treated by introducing two additional auxiliary parameters PO,qo, and the double bracket
du dv
du dv
,
.
Of course, if u and v do not depend on PO,
and then
Furthermore, by choosing qo = t , we have
In terms of the double bracket, Liouville’s hypotheses on the knowledge of n involutive first integrals f r ( p / q / t ) , r E (1,.. . ,n } can be expressed as follows:
{{fp, fv))
=0,
P,W
.
E {0,1,. . ,n),
+
with fo = PO Ift . On the other hand, from the same hypotheses it follows that the algebraic system fr(P/q/t) = r r
7
p0+31=0, can be solved in the form pa - p a ( q / r / t ) = 0 ,
with
a E {0,1,. . ,n} , I
(4.30)
The LiouvdlEe Theorem on the Complete Integrability
111
As before, it turns out that the vanishing of {{fp, f v } } for all p, w
. . .,n}
E (0, 1,
implies that {{Pa-cpa,Pp-cpp)}
=o,
V a , P E {O,...,nL
or, more explicitly,
the differential form n
h=O
is closed or, which is the same, locally exact. In other words, in a sufficiently small region a function V ( q / n / t )exists such that
cpa(q/n/t)=
z, va
E {0,1,.
a9a
. . , n }.
It has thus been shown that the involutivity of the f’s implies that the solutions of the equations f,(p/q/t) = nr,with respect to the p’s, can be locally expressed as follows: f3V Pr = - ( 9 / n / t ) 8%
,
7-
E
(1,.. . ,n} ,
where V is such that
This is equivalent to the fact that V is a complete integral of the HamiltonJacobi equation
112
The Integmtaon Methods
Therefore, by defining n new coordinates, namely X I , ~
2
. . , xn1~ by
the relations
implicitly define an invertible transformation between the variables ( p / q ) and (r/x).For the results at the subsection (3.1.2)1this transformation is canonical and leads to the, trivially integrable, new canonical system kh
=o,
xh
= 0.
Lemma 17 (Involutive relations) Given n functions gr o n the phase space such that the algebraic system of equations
can be solved with respect to the p’s in the form Pa=(Pi(q),
then o n the whole phase space
and then o n Qi,
i E {ll...,n}l
a1we have
(4.32)
The Liouville Theorem on the Complete Integmbility
113
The last relation explacitly gives
Therefore, the partial derivatives aga/aqh can be expressed in the form n
ag.9 aqh
On the other hand, since
a(pj
i=l
- C p j ) / a p h = b j h , we have n
, so that
It thus follows that
The above relation can be written in the form (4.33)
with (4.34)
The Integration Methods
114
Since the Jacobian determinant d(g1, . . . , g,)/d(pl, from Eq. (4.33) it follows that {g,.,gs} = 0 ==+
xp = 0
. . . ,p,)
is nonvanishing,
*
Similarly, by using for Eq. (4.34) the same argument, we finally have
x,!~) = o =+{pi - qi,pj - pj} = 0 . 4.6.3
Remarks o n the Liozlville theorem
Let us consider a canonical system with a Hamiltonian function '? which I,does not explicitly depend on the time. It has been shown that the knowledge of n first integrals f , . ( p / q ) , r E { I , .. . ,n}, which are functionally independent; i . e . ~ ~ = l X h d f h = O ~ ~ ~ = O , i n i n v o l u t i o n ; i . e . { f , , f , } = O , r , s..., ~ { n1 }, , and such that the algebraic system defined by f,.(p/q/t)= ?rh can be solved with respect to the n variables p,., allows us to trivially integrate the equations of the motion. Arnold has given a global formulation of the theorem by requiring that the level manifold M , be compact and connected. This will be treated in details in Part 111. Here we shall limit ourselves to the following considerations. Let us first observe that W , in Eq. (4.28), is defined only locally. AS a consequence, the coordinates x are not uniquely defined on Mf(,). They will be continuous multivalued functions of the point p E Mf(,):
x : P E Mf(,) + P'
= X ( P ) E 8,
*
Therefore, to each point p E~Mf(,) we can associate a point p' E Rn whose coordinates are just given by x', x2,.. . ,xn. Really, as the x's are not uniquely defined, we can associate to p infinitely many points, one for any chosen determination of the x's: P E Mf(?r)-b P ' l , P ; , P L . .
.'
It is clear that, as the x's change continuously, all points p', associated with all the points p of Mj(,), will cover the whole space Rn. Let us investigate more closely the multivalued structure of the x's. A vector a' will be called a period if V 2, 2 and, 2 a" represent the same point in Mf(..). Of course, it will be independent on 2, as 2 and g+ a' are both solutions of Eq. (4.29). Moreover, the modulus la'/ of a' cannot be arbitrarily small since, in a sufficiently small region, 2 is single-valued.
+
The Liouville Theorem on the Complete Integrability
115
If a'l is a period with a minimal modulus, then mla'l, with ml E N , is still a period. Furthermore, any period which is parallel to a'l must be an integer multiple of a'1. In fact, if a'' = XZl, with X E (R - N ) , then by denoting with [A] the maximal integer lower than A, Z'- [XI21 = (A- [A])Z1 would be a period with modulus lower than Ia'11. As a consequence, any new period a' will have a component which is orthogonal to Zl. By choosing among them, the period Z2 whose component orthogonal to a'l has the lowest modulus, it turns out that the vectors mla'l+ m& are periods. Moreover, in the plane spanned by 21,a'2 there are no periods of different form. It follows, inductively, that all periods are of the form mla'l+
771222
+ + m,Z, ,
r E (0,.
. . ,n},mi E N ,
where r = 0 corresponds to the absence of periods and r 5 n, since all vectors cii are, by construction, linearly independent. Thus, each point p E Mf(,) will have just one image in each parallelepiped with sides Zi (if r < i 5 n, then cia = 00). The motion region is bounded if r = n and unbounded if r # n. If n = 2, three cases can occur: 0 T 0
T
0 T
= 0 + Mf(,) is topologically equivalent to a plane; = 1 + Mf(,) is topologically equivalent to a cylinder; = 2 + Mf(,) is topologically equivalent to a torus.
Only in the last case the motion region is compact. More generally, if dim Mf(,) = n, the compact hypersurface corresponding to r = n is called an n-torus T". In any case, the motion develops on Mf(,) c a, which is invariant. 4.6.4
Action- angle coordinate8
Let us consider more closely the case of the torus. By denoting with 71 the closed curves, which on Mf(,) are images of segments Xcil with 0 5 X 6 1, let us define
The J's will be first integrals, as they are functions of the f's and 1
Jl= -AlW, 2n
The Integration Methods
116
where AlW represents the variation of W along the curve 71. Moreover, they will be independent and involutive since
Therefore, starting from the very beginning with the J’s instead of the f’s, we can introduce their conjugate variables P h in the same way as the x’s were introduced as conjugate variables to the f ’s. Along a cycle T h , we will have (4.35)
According to the above equation, the ‘p’s are angle variables, since their variation is 27r along any closed walk, turning the torus just one time. Their conjugate momenta J give, apart from a constant factor the variation of the action W along a cycle in which all the p’s, but one, are constant. For this reason they are called action variables. The Hamiltonian function K = 31* will be function of the action variables J alone, and the angle variables satisfy the equations
whose integration give
The motion described by them is called a multiperiodic motion with frequencies V h . Let us finally observe that the action-angle variables are not uniquely defined, since any linear transformation of the (p’s, with integer coefficients and determinant of the associated matrix equal to 1,will again give angle variables, whose conjugate variables will still be action variables. 4.6.5
The action-angle coordinates f o r the harmonic oscillator
The Hamiltonian of the harmonic oscillator, with n = 1 degree of freedom, is given by
The Liouville Theorem on the Complete Integmbility
117
The system has just one first integral which, of course, is in involution with itself. Thus the system is completely integrable B la Liouville. The level manifold
is, in the phase space a, an ellipse having a = as semi-axis. The action variable
and b = ( l / w ) d m
must be evaluated along the curve 7 determined by the values of q* at turning points; i.e. by the values
whose corresponding momenta vanish. The corresponding integral can be easily performed. It can also be evaluated more simply by observing that, apart from the factor 1/2n, the action variable is the area nab of the mentioned ellipse. Thus, we obtain
J=
E -. W
The Hamiltonian of the harmonic oscillator in terms of action variables is then given by
IC=wJ. The angle coordinate can be evaluated by using Eq. (4.24). The same procedure, applied to the harmonic oscillator with n degrees of freedom,
leads to
The Integration Methods
118
4.6.6
The Kepler dynamics in action-angle variables
In the previous section, it was shown that the Hamiltonian function for the Kepler dynamics, in spherical-polar coordinates, reads
and that the corresponding reduced Hamilton-Jacobi equation has the form
It was also shown that, by using the method of separation of variables, a complete integral of the equation takes the following form:
where the functions W,.(r), W e ( d ) ,W,(cp) satisfy the equations
In the compact case, characterized by E < 0, we can introduce action variables J,, JG, and Jp by writing
I I
n2
2n
Since n, is constant and 0 5 cp 5 2n,we have
J, = IT,+, .
(4.36)
~
The Liouville Theorem on the Complete Integrability
119
The remaining two closed curves of integration are fixed by requiring the vanishing of the corresponding velocities, or better, of the corresponding momenta p.9 and p , expressed, of course, in terms of variables n.9 and xv. In this way, the integration limits are fixed by
Therefore, the “6” integration must be performed between the limits 61 and 6 2 given by the solutions of
where Eq. (4.25) has been used. Since 6 itself always lies between 0 and T , where sin6 > 0, we have sin61 = sin192 = COSQ. Thus, the integration goes from 291 = 1 r / 2 - ~to 7r/2 to 6 2 = 7r/2+a and again back to 61, so that the sin6 goes from COSQ to 1, then to COSQ. In this way, we obtain (see Appendix C)
The “ P integration, which is also performed in Appendix C by using the method of residues, gives (4.38)
From Eqs. (4.36), (4.37) and (4.38), we have
The above equation allows us to write directly the new Hamiltonian function K: = ‘H* as follows:
K(J) =-
mk2
+ J.9 + Jv)2
2(JT
The Integmtion Methods
120
4.6.7
The perturbations of integrable systems and the
K A M theorem There exist very few dynamical systems which satisfy Liouville’s theorem hypotheses. Generally, for Hamiltonian function not depending explicitly on the time t , there exists just the first integral given by the Hamiltonian. Some classical integrable systems are given by systems with a central symmetry, a particle in a Newtonian gravitational field generated by two fixed points, the spherical top in a Newtonian gravitational field. There exist some more dynamical systems with finitely many degrees of freedom, recently found in connection with integrability problem in field theory, where much more examples O C C U ~ . ~ ~ For the applications, it is interesting to elaborate methods which will allow us to study a given Hamiltonian dynamics by separating its Hamiltonian function ‘fl as the sum of two parts. The first part, namely %o, is required to be a completely integrable one, so that
‘fl = ‘flo
+ XZ1 ,
where X is a “small” parameter and is an analytic function of 2n variables. By using action-angle variables ( J , cp), the above equation can be written as follows: % ( J , ( P )= % o ( J )
+X’Hi(J,cp).
For X = 0, the phase space is foliated, according to Liouville’s theorem, in ), by Jh = ?Th, on which the curves, n-dimensional invariant tori M J ( ~ defined ‘Ph = v h ( J ) t
+ (Ph(0)
1
completely wound. It was common opinion, before 1954, that X # 0 completely destroys the foliation in invariant tori and the beautiful geometrical structure underlying integrable systems, giving rise to the ergodic behavior; that is, to orbits densely filling the submanifold 3t = constant. Therefore, the question is to know what remains of this geometrical structure when X # 0. This opinion was supported by the +fact that in the perturbative series there appear denominator terms like v’ k, where k = (k1,ka,. . . ,kn) are integer numbers. Therefore, when the ratios vi/vj are rational numbers, the series diverges. To say that the ratios v,/vj are rational numbers is equivalent to say that there exists a period TI which is a multiple of all period rj = l/vj, so that the orbit on the torus T” is closed. In such cases the torus is said to be resonant. Beyond this case, close I
~ * * ~
The Liouville Theorem on the Complete Integrability
121
to the resonance there will appear, however, terms too large (little divisors), since the rational numbers Q are dense in R. This happens, for instance, in the case of Jupiter and Saturnus, which move along their orbits each day by 299"l' and 120"5' degrees, respectively, so that 2vl- 5vz N 0. The existence of a strong perturbation, with a large period, of the motion of planets, connected with the little denominator 2vl - 5v2, was already known to Laplace. The presence of v' iin the denominators can be easily understood, by considering that the terms in the Fourier's expansion of 3c 1:
-
in a perturbative scheme, will be derived or integrated in t. Finally, in 1954, a positive answer about the applicability of perturbative methods and the role of the parameter X in the convergence of corresponding series was offered by Kolmogorov. His theorem, extended and formalized by Arnold (1963) and Moser (1967), is today known as KAM theorem. The theorem2 proves that, for sma2l values of 1A1 and nonvanishing Hessian of the Hamiltonian, only few invariant tori are destroyed. A large number of them are only deformed by the perturbation. On such deformed tori the orbits are still dense and almost periodic with n frequencies everywhere. Such invariant deformed tori correspond to unperturbed initial conditions for which
with a and b positive constants. It is shown that, for sufficiently large b, the constant a is of order O(X) and gives the measure of the lost tori. We will not go into more details and refer the interested reader to the literature (besides Arnold's book,2 see for instance Refs. 45 and 3). 4.6.8
The Poincard representation
It is possible to concretely see what happens by using the so called PoincarP map. Let us consider a Hamiltonian system with n = 2 degrees of freedom whose Hamiltonian function H does not depend explicitly on the time t. Let us also **Henry Poincar6 was born in Nancy in 1854, and died in Paris in 1912. He was a professor at Paris University and h o l e Polytechnique. An analysis, by Hadamard, Langevin, Boutroux, and Volterra, of his basic contribution to mathematics and theoretical physics can be found in La Nouvelle Collection Scientifique (Paris: Alcan, 1914).
122
The Integration Methods
suppose that the 3-dimensional manifold M E = {p E Q : % ( p / q ) = E } , defined by the first integral of the energy, is compact. We know that the existence of a second first integral, namely f , will ensure the complete integrability, and that the manifold ME will be foliated in 2-dimensional invariant tori T2. For a given initial condition, the motion will be represented by a helix belonging just to one torus and densely winding on it, never returning, provided the torus is not resonant, exactly at the same point. Let us now consider the 2-dimensional manifold C defined by the equation
representing the intersection of M E with the hyperplane defined by q2 = 0, and a point uo E C,which can be fixed by giving a point in the plane S 5 (PI, q1). By considering 00 as the initial condition at the time t o of the flow of a, there will exist a time instant, namely t l , in which the trajectory will again meet C in a point u1 E C. Thus, recursively, there will be a sequence of time instants t k in which the trajectory will meet C at points (Tk E C. Since the sequence of points U k E C is the image of a sequence s k = (pl(k),q1( k ) ) of points in the plane S, it is possible to describe the evolution by giving the sequence S k E s. The map
which describes the evolution, is called the Poincare' map. Furthermore, as it winds on the torus T2, the trajectory meets C on the 1dimensional submanifold determined by the ulterior equation f(p1 , p z , q1, q z ) = T ; that is, on a smooth closed curve, which is also the image of a similar curve of S. Therefore, for a not resonant torus, the points s k will dispose along a regular curve, while for a resonant torus the sequence will stop; that is, there will be an integer number r such that s k + r = s k , and so on. The above description can be also applied to noncompletely integrable dynamics. In this cases the trajectory will not meet C along a regular curve but in points (Tk covering a 2-dimensional region, which is an image of a 2-dimensional region of S (chaotic behavior). The KAM theorem predicts that, by increasing the value of the parameter A, it is possible to observe, in S, a transition from a picture composed by regular curves to a picture composed by a large part of previous curves together with extra points, and then to a picture composed by few curves and too many isolated points covering the whole interested region.
The Liouville Theorem on the Complete Integrability
123
Computer analysis gave, of course, exactly what was expected and was allowed to discover new remarkable completely integrable systems.
Further Readings 0 0
0 0
0
G. dell’ Antonio, Elementi di Meccanica (Liguori, Naples, 1996). A. Lichnerowicz, “Les variktds de Poisson et leurs dghbres de Lie associkes,” J. Diff. Georn. 12,253 (1977). A.Romano, Leioni di Meccanica Radonale (Liguori, Naples, 1990). E. C. G. Sudarshan and N. Mukunda, Classical Dynamic: A Modern Perspective (John Wiley, 1974). W. Yourgrau and S. Mandelstam, Variational Principles in Dynamics and Quantum theory (Dover, 1968).
Part I1
Basic Ideas of Differential Geometry
Differential geometry is the differential calculus that does not depend on the coordinate system and so, it is the best language f5r a science (Physics~ deputed to describe phenomena, which do not depend on the observer. Part I1 is devoted to a summary survey of useful geometric concepts, as differential manifold, tangent space, fiber bundles, Lie derivative, differential forms, and ezterior der~vative. ~ i ~ e ~forms e ~ describe t ~ a all ~ the relevant physical entities as the or^, the heat, the internal energy, the electromagnetic field, and so on. Fiber bundles are used, in elementary particle physics, to describe particles with internal structures (isotopic spin, etc.) and to construct instantonic solutions in field theory. ~ e n ~ f o r tthe h , sum over repeated, upper and lower indices is understood according to the Einstein convention: Caaibi =
127
Chapter 5
Manifolds and Tangent Spaces
5.1
Di~erential~ ~ i f o l d s
A manifold M is a separable topological space such that every point of it is representable at least on one chart; a chart (U, 'p) of a manifold M is an open set U
2 M , called domain of the chart, with a hom~morphism r p : U c M -+Ac sZn
from U to an open set A in Rn. A map f : U + V between two topological spaces U e V is called a homeomorpfbissm if it is one-to-one and both f and f - l are continuous maps. 9). The coordiLet p be an element in M representable on the chart (U, nates Cp'(p) of the image of the point p are called coordinates of p in the chart (U, v),or bcal coordinates of p . Thus, a chart is essentially a local coordinates system. A Ck atlas on a manifold M is a collection {(Um,vm)] of charts such aa the domains U, form a covering of M , and the homeomorphisms 'pa 0 'pi1 : P,(up
nUp) -+ cpp(u,
nu@)
are Ckmaps between open sets of Rn. Two Ckatlases are equivalent if their union is a Ckatlas. 129
Manifolds and Tangent Spaces
130
A manifold M with a Ck atlas equivalence class is called a C k differential manifold; its dimension is n. A manifold is said to be o ~ e n t u if~ an ~ eatlas can be chosen in such a way that for all a, p the Jacobian determinants det(bqh/aq$) have the same sign. An intrinsic definition will be given in Sec. 2.5. The sphere as a diflerential manifold The n-dimensional sphere S" is defined by
It is ;t separable topological space, and associated with U c S" and Y two charts (U, cp) and ( V ,$) can be introduced as follows:
c S",
UE~S"-N, VES"-S, where N and S are the north and south poles defined by
N
3
By using the notation 5 (3sxn+l)+ The map
(0,.. .,o, 1),
= (XI,..
,
s = (0,.. . ,o, -1).
,zn),a point p
E
U will have coordinates
is called the stereographic projection of Sn with respect to the north pole N . It is a one-to-one map that is continuous together with its inverse, which is given
bY
Similarly, the ste~ographzcprojection of S" with respect to the south pole S is defined as
@ : p E V + @ ( p ) = u+.=
2
1
+ xn+1 E R",
Curves on a Difleerential Manifold
131
and it is a one-to-one map continuous together with its inverse given by
Since the images of the intersection
u n Y = sn- { N , s) , for both
(p
and $, is
(Rn- (0)); i.e. p(U n V ) = $(U n V ) = Rn - (0)
,
the map cp 0 $--I
: G€
(P- (0))
-+
3t
=:
iz (p 0 .$-1)(d) = - E (R” - (0))
1.12
is a C’” map. Thus, the two compatible charts (U,cp) and (V,.$) are a C“ atlas for Sn, and ( t l , .. . t m ) ,(211,. ..,un)represent the local coordinates of a point p E Sn with respect to them.
Exercise 5.1.1. Show that the sphere S2 is orientable. 5.2
~
Curves on a Differential Manifold ~
~
efunctions r e on ~ a manifold ~ ~ a
~
~
~
If
f:M-+R is a tion on defined on a ~fferentialmanifc representable on the chart (U, p), the map f
0 9-1
: p(U)
M anL p is an element in M
c RZn-+
SR
transforms open set of !RZnto open set of R. As the coordinates p’((p) of p(p) represent the point p in the local chart (U,cp), so the map f” = f o p-l represents the function f in the local chart. The function f is said to be diflerentiable at p E M , if in a given chart (U,cp), f’ = f o p-l is differentiable at p ( p ) , or equivalently, if expressed in local coordinates, gives rise to a differentiable function.
Manifolds and Tangent Spaces
132
The given definition is independent from the choice of the chart. Indeed, if the point p is representable on two charts (U, p) and (V,$), we have
f 0q - 1
= (f 0 9-1) 0 (9 0 $ - I ) .
If (focp-') is differentiable at cp(p), then f o$-l is differentiable at + ( p ) , since p o $-l is differentiable at cp(p) and (p o +-l)($(p)) = cp(p). Let M and hl be two differential manifolds, with m and n dimensions, respectively, and fet f be a map
f :P E M
f
+ (PI E J f f
(5.1)
from M to hl. If p and f(p) are representable on the charts (U, p) and (V,$), respectiveIy, the function
9 0 f 0 p-l
: p(U)
c 92% -+ $(f(U))c !I?%
represents the map f in the local charts (U,-p) and ( V ,$), and we will say that f is differentiable at the point p E M , if f = $ o f o p-' is differentiable at P(Ph a bM~ eif it is differe~tjableat every The map (5.1) is said to be ~ i f f e ~ n t ~ in point p E M ; if f is one-to-one and f and f-' are differentiable, then f is said to be a diffeomorphism.
Cvmee on o manifold
A curve y on a differential manifold M is a homeomorphism y :T
EI
c !I? - + y ( ~E) M
from an open set I C R (open interval) to an open set in M . The curve y is said to be d ~ ~ e ~ e n t i uat blT e = 0 if the map 'pay: 7 f
Ic % z $ Q ( y ( T ) )E 92%
is differentiable at T = 0, where p is the homeomorphism of the chart on which p 5 ~ ( 0is) representable. if Two curves, 4 and y' on M are called e~uiva~ent
in a chart (U, cp). Of course, the relation (5.2)is true in any other
Tangent Space at a Point
5.3
133
Tangent Space at a Point
5.3.1
Tangent vectors to a curve on a manifold
Definition 18 It is called tangent vector X , to a differentiable curve y = y ( ~ on ) a manifold M , at the point p = "(TO), the directional derivative operator along the curve y = Y ( T ) , at the point p :
If p E Ua c M , T E I E R and f is a differentiable function on M , we have
on which p is reprewhere pa is the homeomorphism of the chart (Ua,qa) sented. Since
,f=
...,zn) E A C Rn + - f ( d l . . . l z " E) R, : T E % + T(T) = ( z ' ( T ) ,. . . ,~ " ( 7 )E) A Rn, (z',
(Pa o
we have
where Xi = dq/dTlTO. From the above formula it follows that every tangent vector to a given curve, at a given point, is a linear combination of the n partial derivatives { d / B x k } , which are linearly independent. Of course, every tangent vector is tangent to an infinite number of different curves through p for two different reasons. The first is that there are many curves which are tangent to one another and have the same tangent vector at p , and the second is that the same path may be reparametrized in such a way as to give the same tangent at p . So it becomes natural to give the alternative following definition.
Manifolds and Tangent Spaces
134
5.3.2
Tangent vectors to a manifold
The equivalence class of the curves Y ( T ) on M through p is called tangent vector to the manifold M at the point p. The set of all the tangent vectors to the manifold M at the point p , with the sum and the product by scalars defined by
(ax,
+ b Y p ) ( f ) = a X p ( f )+ b Y p ( f ) >
is a vector space called tangent space to M at p and it will be denoted with 7 , M . This space has the same dimension of the manifold, and the components of every tangent vector X , E 7 , M , in the local chart (U,cp) on which p is represented, are given by the relation
Remark 12 A tangent vector to a manifold M at a point p , could equivalently, be defined as a linear function from the space 3 ( U ) of differentiable functions, defined on a neighborhood U of p , to R:
w.4 + 9, satisfying the Leibnitz rule; i.e. X,(af
+ bg) = aX,(f + bXp(9),
X p ( f g )= f ( P ) X P ( d+S(P)X,(f) 1 with a, b of P.
E
8,and f and g differentiable functions defined on a neighborhood
The value of X , in f is
X,(f)=
(z)
x;,
-
(5.3)
P
where 2' = cpi(p) and X j are the components of X , in the given chart (U, cp). The previous formula is often written in the form
135
A Lligresaion on Vectors and Covectors
The tangent vectors
constitute a basis for the tangent space 7 , M , called the natural basis. Since a chart (2.4,~)at p induces an isomorphism; that is, an invertible linear map between the spaces 7,M and Rn,the dimension of 7 , M coincides with the dimension of the manifold M .
Transformation laws If p belongs to the intersection Ui nUj, then two sets of local coordinates can be introduced, namely (d, z2,.. . ,P)and (dl, x t 2 , .. .,x t n ) , and a vector Vp can be locally written in two different forms according to the chosen coordinate basis
(&)
p , . . . , (&)p
Or
( g i ) p l * * . .
(&)
*
P
Thus, we have
The above relation, once applied to the functions d k : R" familiar transformation law for the components of a vector
5.4 5.4.1
+ R,gives the
A Digression on Vectors and Covectors Vector space
Let us recall that a set E , whose elements we are denoting with capital Latin letters X,Y,2,.. , , is called a vector space (over the real numbers 8) if an internal composition law
+ : ( X , Y )E E X E H E can be defined, with respect to which
E is an Abelian group;
~ ~ ~ i and ~ oTangent ~ d sSpaces
138 0
a m u ~ t i p ~ ~ ~ abyt i real o n numbers c E 92 is defined to satisfy the folfowing properties: c -( X + Y ) = c . X + c . Y , (c1c2)
x = c1
'
(c2 * X ) ,
l*X=X. It is common use to drop the multiplication dot and the parentheses used in the previous properties. The elements of E are called vectors, and the vector corresponding to the identity element in E is denoted with 0. A set of vectors XI, X2, . . .,xk, in E is said to be linearly independent if k
c c i x i = 0 =+ ci = 0 , vi E {1,2,., , ,k}, i=l
and linearly dependent, otherwise. The set is said to be mmimal linearly independent, if the set, obtained by adding to it any other vector of E , is linearly dependent. This means that any other vector in E can be expressed as a linear combination of a maximal set, which for this reason, is called a basis for E. The number of vectors of a basis is called the d z m e n s ~ oof~ E. If {ej) denotes a basis for the vector space E , then any vector V can be written as
V =Viei, and the coefficients V i E 9 are called the components of V in the given basis {ei}Vi E R. 5.4.2
Rual vector space
The space of all linear maps from E to R ! is denoted with
E* = Lin(E, R) , and is called the dual space of E. The elements of E* will be denoted with small Greek letters a,P,y,. . . . For an arbitrary element a E E*, we have a:
x E E l - , .(X) E R ,
A Digression on Vectors and Covectors
137
+ Y ) = .(X) + .(Y)
vx,Y E 8 ,
, .(cX) = c a ( X ) , v c E R. .(X
It is easy to see that 0
E* is also a vector space, the internal composition law and the multiplication by real numbers being trivially defined as .(
+ P ) ( X ) = .(XI + P ( X )
( c a ) ( X )= c.(X)
0
9
.
The vectors of E* are called covectors in order to distinguish them from the vectors of E; E* has the same dimension of E. If el, e2, . . . ,en is a basis of E , a given vector X E E can be expressed &8
x = X'ei,
i E { 1 , 2 , . . ., n } .
Associated with the given basis { e i } of E , let us introduce the elements of E', namely d 1 , d 2 , .. . ,8",defined by
d'(X) =X ' ,
i E {1,2,.. . , n } ,
or equivalently d'(ej) = 6; , i , j E { 1 , 2 , , . . ,n}.
Since for any elements a E E* and X E E , we have
a ( X ) = cY(Xzei) = X i a ( e i ) = .id'(X), with ad = a ( e i ) E R, the element a can be expressed as a = .id2.
The set d 1 , d 2 , .. .,dn is then a basis for E*.
It is also possible to consider the dual E** = (E*)*of the dual of a given vector space E. The reader can easily prove that if the dimension of E is finite, then E** is isomorphic to E and they can be identified.
Manifolds and Tangent Spaces
138
Thus, E and E* are duals of each other; in order to underline the reciprocal duality, the value that a covector a takes on a vector X is also denoted by the bracket (., .), so that
a ( X ) = ( a , X ).
If { e : } and { e i } are two different bases and ( 1 9 ’ ~ ) and {Si}are their dual bases, respectively, a given vector V can be represented in two different forms. Then, we have
v = V’iei = V i e i . Thus, the value of a generic element 8” of {@} on V will be
#‘(v)= V’i19’k(e:) = Vi19’’(ei), and then
V t k= M ki V i ,
M denoting the matrix whose elements are 29”((ei). 5.5
Cotangent Space at a Point
As for any vector space E , we can introduce the dual T M of the vector space 7,M. The vector space 7,*M is called the cotangent space to the manifold M at the point p . Special covectors of T M can be associated with any differentiable function f defined on M , as follows: Iff : M + ?Ris a differentiable function at the point p E M , its differential (or gradient, or exterior derivative) dfp at the point p is the linear map
df, : 7 , M
-i R
of the tangent space 7,M in R defined by
(dfP,XP) = X p f
,
vx, E 7 , M .
More explicitly, let X , be a tangent vector to M in p , and $7) a curve, belonging to the equivalence class of curves through p , ($0) = p ) , which represents X,, then
Maps Between Manifolds
139
If p is representable on the chart (U, cp), we have
By choosing for f the coordinate functions z: : p E U
+ (pi@),we obtain
( d x i ,X,) = X i . Then, Eq. (5.7) can also be written in the form
or equivalently, in the form
df, =
(3)
dxi.
P
It follows that the set of covectors { d < } is the dual basis of the natural basis {a/ax:} of 7 , M . 5.6
Maps Between Manifolds
If 4 : M -+ N is a map from a manifold M to another manifold N,the image of a curve 7 on M will be a curve y' = q5 o y on N,and a tangent vector X p to 7 at the point p , will have, as an image, a tangent vector to y' at the transformed point &)I. Thus, # induces the linear map 4*p
: 7,M + 7 , ( p ) N
between the tangent spaces 7,M and %(,IN defined, formally, as in Eq. (5.6),
Manifolds and Tangent Spaces
140
However, here 4*,Xp is a vector belonging to T&)N. Indeed, if p is representable in the chart (24, cp), and q = 4 ( p ) in the chart ( V ,$J), then we
r=o
Vice versa we can associate with a given covector aqE
T N , the covector pp E 7p'M defined by ( P p , XP)
= ( a p , 4*,X,)
I
with q = + ( p ) . Thus, 4.,Xp is called the p ~ s h - f o ~ of ~ ~X,, r dwhile pP = #$a+(,)is called the pull-back of a + ( p ) . 5.7
Vector Fields
A vector field X on a manifold M is a rule which to every point p E M associates a tangent vector X p E 7 , M ; i.e. a map
X : pE
M -+ X ( p ) = X p E 7,M
is defined on M . In a local coordinate system, a vector field X can be written in the form
The vector field X ( p ) is said to be C hdifferentiable on a C kmanifold M , with h 5 k - 1, if the functions X'(p) are C hdifferentiable on the manifold M .
5.7.1
Holonomic and anholonomic basis of vector fields
The natural basis {a/&?} is not, of course, the sole possible basis for vector fields. By taking n, point-wise linearly independent, combinations ei of its elements; i.e. ei = a: (z)-
d
6x3
,
i = 1,.. . , n ,
Vector Fields
141
with det(ai(x)) # 0 everywhere, we obtain a new basis { e i ( x ) } . In fact, previous relations can be solved with respect to a / d x j in the form
a
.
q ( x ) e j , i = 1,...,n ,
where (e(x)) is the inverse matrix of (a{(x)). Thus, an arbitrary vector field X , that in the original basis was written as X = X i ( a / a x i ) , takes the form
x =Xtiei, where X'j = g ( x ) X a are the new components. In other words, a basis is given by n arbitrary point-wise linearly independent vector fields. If the matrix (a:(.)) is a Jacobian matrix, coordinates yi exist such that
a
ei = ayi '
i = 1,...,n.
Of course, such coordinates are the solutions of the differential system
which defines, then, a coordinate transformation between the x's and the y's. The natural basis, as well the ones related to it by a coordinate transformation, has the following obvious property:
[&,&I
=o,
i , j = 1,...,n
For an arbitrary linear combination, aa the one in Eq. (5.8), it will happen, generally, that
at least for some values of the indices i , j . If this linear combination is made with a Jacobian matrix, the previous property is again satisfied, since (ei,ej]=
-,[ [ a
lJ]
=O,
i,j=l,
..., n.
Manifolds and Tangent Spaces
142
It is true also the converse, namely: If the elements of a basis { e i } of vector fields fulfill the property
[ei,ej]= 0 ,
i , j = 1 ,...,n ,
then coordinates yi exist such that d e.--, a - dya
i = l , ...,n .
This can be understood by observing that the elements ei of a basis, however, this one is chosen, are linear combination (with functions as coefficients) of the partial derivatives d / d x i :
‘
a
ei = a i ( x ) - , dX.1
i = 1,.. , , n .
Thus,
Then,
(5.10) and, since det(af) # 0,
The Tangent Bundle
143
which is just the compatibility conditions to be satisfied, in order a differential system, like Eq. (5.9),admit solutions. In other words, the conditions
say that the differential forms dp = 3dx"
p = 1,.. . ,n
are closed; that is, functions yp, at least locally, exist such that 19, = dyp. Equation (5.10) shows that the commutation relations between elements of an arbitrary basis are of the form
[ei,e j ] = 4'eP = 0, V i , j, p , will be called holonomic (integrable), or A basis for which anholonomic (nonintegrable), otherwise.
5.8
The Tangent Bundle
The union U p E ~ 7 p M of all tangent spaces 7,M to a manifold M can be endowed with a structure of a differential manifold in a very natural way. An element of it is a pair ( p , X,) with p E M and X, E 7 , M . Thus, a sysx2,.. . , x", X',X2,. . . ,X")for U p E ~ 7 , M can tem of local coordinates (d, be introduced by using the local coordinates (d, x2,.. . ,x") of p and the ones ( X 1 , X 2 ,,.. , X n ) of X,. Regarded as a differential manifold, the set U p c ~ 7 & f is denoted with T M and it is called the tangent bundle of M , while a single tangent space 7 , M is called a fiber of TM. Of course, the dimension of T M is twice that of M . A curve on T M identifies a vector at each point of M , and so it defines a vector field on M . Such a curve; that is, a curve transversal to the fibers, is called a cross-section of T M . The names bundle, fiber, and cross-section refer, however, to general structures that our "special" differential manifold T M shares with a general class of differential manifolds called fiber bundles. A general fiber bundle consists of a base manifold, which in our case is M , and of one fiber attached to each point of the base space. If the dimension of the base space is n and the one of each fiber is m, the bundle has m f n dimensions. The points of each fiber are related to one another, while points on different fibers are not.
Manifolds and Tangent Spaces
144
The fibers need not be related to the differential structure of the base manifold M , In elementary particle physics, bundles are considered, whose fibers are isospin spaces attached to points in the space-time, which is the base manifold. Such a bundle will describe, besides the coordinates (t,2,y, z ) , also the isospin of an elementary particle. 5.9
General Definition of Fiber Bundle
Let E , M , F be differential manifolds and n a differentiable map
r:E+M from E to M . Let {Uj}j=l,...,, be a covering of M , made of open subsets of M , which are compatible charts' domains n
j=l
Let us suppose that, for every open set Uj,there is a homeomorphism from n-'(Uj) to the Cartesian product Uj x F of the form 'Pj : Y E X-'(uj>
-+
0
' P ~ ( Y ) = x(Y),G j
A
where ' P j : n-'(Uj) + F and the restriction
A 'Pj
(Y)
1=-1(~)
A
- 'Pj,p:T-'(P) =
E
uj
xF,
A
of 'Pj to n-'(p),
+F
+ j l =-1(P)
is a homeomorphism of n - ' ( p ) onto F , such that the diagram T-1
x d
Uj
(Uj)
L Pj .P Uj x F
commutes; i.e. T=P0'Pj,
p denoting the canonical projection from Uj x F onto Uj.
'pj
General Definition of Fiber Bundle
145 A-1
A
Of course, the set of maps ' P k , p o 'Pj,,: F + F for all p E U j n U k and for all j , k E { 1,. , . ,n} is a group G in a natural way. If this group is a Lie group and the maps
are differentiable, we will say that ( E ,M , T , F, G) is a differentiable fiber bundle; M is called the base of the bundle, F is called the typical fiber, and T - (~p ) is called the fiber in p . The group is called the structure group. Usually the bundle ( E ,M , T , F, G) is simply indicated by E . Locally, a fiber bundle is always a Cartesian product; i.e., T - (Uj) ~ 5 Uj x F. A fiber bundle E , which is globally a Cartesian product; that is, if E = M x F , is called trivial. For more details see RRf. 19. 5.9.1
More on the tangent bundle
Let us consider again an n-dimensional differential manifold M and the set of the pairs (p,X,) where p E M and X, E 7 , M . Such a set, denoted by the symbol T M ,can be provided of a differential bundle structure in the following way: as a base we take the differential manifold M and the map A is the projection T
: (p,X,) E 7
M +n(p,X,) = p E M .
The typical fiber F is the Euclidean space IR", and for every p E M , the fiber ~ - l ( p is ) 7 , M , the tangent space to M at p . The covering of M is made by the domains U j of compatible charts ( U j , $ j ) such that n
M
=
U U j .
j=l
The homeomorphisms ' p j are defined as follows: ' P j : ( P , x p ) ~ ~ - l ( U j- ), ' p j ( ~ , X p ) = ( ~ ( p i ~ p ) , ( ~ j ) * p ( ~ 2 ( ~ ~, ~ pu) ) )j
where r&, X,) = X,. The coordinates of the point ( p , X,) in T M are then (z' ,
. . .,zn,x,1, . . . ,X,"),
x
~
~
,
Manifolds and Tangent Spaces
146
where ziare the coordinates of p and
are the coordinates of X p in the chart (Uj,$ j ) . The linear map
is an isomorphism between the spaces 7,M and 32”. If p is representable on two charts (Uj,$ j ) and (Uk, $k), the map
is an isomorphism of %” onto itself. Thus, the structure group G is GL(n,8). As we have already noted, the space 7 M is usually called the tangent bundle. The cotangent bundle
The cotangent bundle is built exactly as the tangent bundle T M , by simply replacing the tangent spaces 7,M with the cotangent spaces T M . 5.9.2
Analysis of two bundles with S’ as base m a n i f o l d
The tangent bundle 75‘ of the circle S’ can be visualized as a cylinder, which globally is the Cartesian product S1x 8,and so it is a trivial bundle. A cylinder, once cut along a directrix, “becomes” an infinite rectangle belonging to %’; a part of it is a finite rectangle. Let a and a’ be the upper corners of the rectangle and b and b’ the down ones. Before the cut, a - a’ was identified with the same point of the cylinder and a’ - b’ with a different point along the same directrix. Thus, by gluing a with a’ and b with b’, we obtain an upper- and downbounded part of the cylinder, while by gluing a with b‘ and b with a’, we obtain the so-called Mobius band. Then, with the same base space S‘ and the same fibers R, we can build two globally different bundles. The first, 7 S ’ , is a trivial bundle, the second,
Geneml Definition of Fiber Bundle
147
a nontrivial one; both show the same local properties. Their difference, which is of global type, is described by the structure group.
The bundle 7-27' Let (l4j)j-l ,..., be an open covering of, S' = U,"=,Uj. Every Uj has, as coordinate system, a parameter rj along S1, and for p E U j , a basis of 7pS1c" F = 92 will be given by the vector d / d r j . Thus, a given vector V E 7pS1will be represented by v j d l d r j , where vj is a real number, and since j is fixed, there is no s u m over j. If p belongs to the intersection of two neighborhoods Uj and Ui,the vector V will have two representations, vjdldrj and vid/dri, where since r, and rj are unrelated, vj and vi are two nonzero real numbers. A
A-1
Since the typical fiber is 92, the homeomorphism o V j , p :R ! -+ !R maps v j in d,so that it reduces simply to the multiplication by the real number rij = v ' / v j . Thus, the structure group of T S ' , since rjj are nonzero arbitrary real numbers, is GL(1,R) 3 (92 - {0}, x); that is, 92 - (0) with the composition law given by the multiplication. We observe now that, for any j , the parameters rj can be chosen to be concordant; that is, in such a way that any two of them, namely rj and rj, increase in the same direction of S1(drildrj > 0 ) , in the intersection Uj nui. With this choice, rjj > 0 and the structure group reduces to R+, the positive real numbers with the composition law given by the multiplication. Moreover, the Jacobians drildrj could be chosen in such a way that, in the intersection Uj n Ui, we have drildrj = 1. Thus, the structure group reduces, finally, to (1, x ) , a trivial group &s trivial as the bundle 7s'.
The Mobius fiber bundle
It is easy to see that the Mobius strip is not an orientable manifold, so that at least one of the real numbers rij = v i / d will be -1. In this way, the structure group reduces to ((1,-l}, x). The bundle of frames and the principal fiber bundle The frame bundle FTM of an n-dimensional differential manifold M is a bundle having M as base space, GL(n, R) as structure group (the same of the tangent
Manifolds and Tangent Spaces
148
bundle T M ) ,and the set of all bases of R" as fiber. Since the set of all bases of '29 is homeomorphic to GL(n,R), the typical fiber F is just GL(n,R). Then 7 M has G = GL(n,FR) and F = R", while FrM has G = GL(n,R) and F = GL(n,R). The frame bundle FrM is just an example of a bundle in which the structure group (not necessarily G L ( n , R ) )is homeomorphic to the fiber. Such a bundle, that is, a bundle in which the structure group is homeomorphic to the fiber, is called a princzpal fiber bundle.
5.10 Integral Curves of a Vector Field As it has already been said, given a vector X, E 7 , M , there exist infinitely many differentiable curves on M which in p admit X , as tangent vector. Given two vectors, X, and X,,it is also easy to find curves on M admitting X, and X , as tangent vectors at p and q, respectively. It is also clear that the search for such curves becomes more and more difficult as the number of the given vectors increases. It is in a sense surprising that, given a vector field X on M ) and then the assignment of infinitely many tangent vectors (one in each tangent space 7 , M , V p E M ) , there exists always a curve p = ~ ( T ) , T~ ] a , b [ 112 C on M , whose tangent vector in a point po = TO) coincides with the value Xp0 of the field at the point, V T O€ ] a ,b[. As a matter of the fact, if xi = xi(.) is the local parametric representation of the unknown curve, the derivatives dxi/dT(,=, will represent the components of the tangent vector at T O , while, if X = X i ( a / a x i ) is the local representation of the vector field, Xi(p0) will be the components of the vector corresponding to the value of the vector field X at PO. Thus, the unknown curve will be the solution of the system of differential equations dxi dr
- = xi(xC), ~i
E ( 1 ~ 2 ,. .. , n ) .
We know that, for smooth X i , the above system always admits, locally, a unique solution xi = z ~ ( T assuming ), at T = TO a prefixed value xb. Such a solution is called an integral curve of the vector field X . Such curves are well known in physics as force lines, a name given by M. Faraday who introduced them for the electric and magnetic vector fields E and H.
149
Zntegml Curues of a Vector Field
Example 19 Let
be the harmonic oscallator vector field. Its integral curves will be given by solutions of the differential system
which are circles
x2 + y 2 = r2 of arbitrary radius r.
Example 20
The integral curves of the vector field Y a x x = ( x + - T) - -aY (Y+;)z,
a
where r = ( x 2+ y 2 ) I f 2 , will be given by the solutions of the differential system dx -=-(y+j),
-=
(x++) .
B y multiplying the first equation by x and the second by y, we obtain
y-= dr
yx+$
Thus, I d 2 dr
- -(X2 -k y2) = T , OT
~ a n ~ ~ and o~d Tangent s Spaces
150
which gives r = ~ + c ,
represents infinitely many spirals, one for each value of the constant c. We can summarize saying that, for every point p in M there exists an integral curve of X ; i.e. a curve on M
which satisfies the differential equations
More precisely, for every p E N ,there exists an interval I$,c !R and an integral curve T E I, -+ y(7,p) of class ChS' in Ip, such that y(0,p) = p. Moreover, this integral curve is uniquely determined. It follows that, if (T and T are elements of I p such that CT + T E Ip, we have y ( r 1 y ( 0 , p ) )= Y ( T $ - C , P ) ,
VPEM.
(5.11)
If po E M , there exist an open set U(p0)C M containing p0 and an interval
Im c 8 such that y is defined on Ipo x U(p0) Y : (T,P) E
I,, x W P O t + T'fT,PI E M
-
For every T E I$,,, the map YT : P
U(P0) -+ Y W = Y(7,P) E M
(5.12)
is a diffeornorphism between open subsets of M ; a point p in U ( p 0 ) goes to a point ~ ' ( p E) M aIong the integral curve of X at p. The position of yr(p) is determined by T. Let {U(po>} 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 ) = $1
X)+ Sz(a,X)
(a1
1
( k S ) ( a ,X ) = k ( S ( a ,X)), V k E R *
Exercise 6.1.1 Show that a basis of T1(E)can be giuen bg (8' an obvious ~ e ~ n ~for t ~this o ntensor product.
@ ej},
with
Previous examples exhaust the concepts of tensor of rank 2. More generallyl any ~ ~ l tm a~p ~ ~ ~ e ~ r
T : E x E: x x E x E* x E* x - * * x E* + R l -v e . 1
p times
q times
is called a tensor of ( p , +type. The tensor T is also said to be of rank p The space of all such tensors is denoted by
T ( E )= Lin(E x Ex,...x 5 x q times
,E* x E* ; ptimes
x dT13),
c q.
and c m be endowed naturally with a vector space structure defined by
+
(TI T2)(X,Y,. . . ,Z,a, P,
= Ti(X, Y,. Z,a,P, * . ,Y)
CT2(X,Y,..
a,P,
-12,
f
*
.,TI ,
( k T ) ( X ,Y, . *.,2,CX, 8,. . . ,y)= k(T(X,Y,. . . , 2,a , P I ., . ,y)), V k E 9 , A basis of
E and {$I*) be its dual basis. We have T(X,Y,.. ., Z , cl~,P,. . . ,Y)= X'Y'
* * *
ZkQa,Pq *
x (e%> 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
<
(6.38)
By adding the above equation to itself with the permutation (i -+j , j -+ i, k j ) of the indices, and subtracting the one with the permutation (i -+ j , j k,Ic --+ i}, we finally obtain
-+
-+
Differential Forms
184
where is a function of gij and its first derivatives, and is a function of gij and its first and second derivatives. Thus, once the metric tensor field g is given and the functions 4's and its first derivatives are known at a point p E M , the above equation allows us also to know the vaiue, at the point p , of the second derivatives of the ('s. In the same way, by successively differentiating the equation, we can determine all higher derivatives of the ['s at the point p. This suffices, if the manifold M is analytic, to determine the differential 1-form 5 everywhere. Since the assignment of at p determine, via Eq. (6,38), the symmetric part of the first derivatives &i/azj, we conclude that every Killing vector on M is determined by giving the values
at any point p E M. Therefore, since the number N of the parameters ai and 1 2
N = n + -n(n
bij
is given by
I - 1) = -n(n + I), 2
on an n-dimensional manifold M there exist at most (1/2)n(n + 1) Killing vectors, 6 for n = 3. It is worth observing that Eq. (6.36) may not admit solutions. An n-manifold M , endowed with a metric tensor field, is said to be maximalty symmetric if, on it, there exist (1/2)n(n 1) Killing vectors.
+
6.3.3
The Levi- Civita covariant derivative
On a differential manifold M there exist only three natural d e r i v a t ~ owhich ~~, are given by the Lie derivative, the interior product and the exterior derivative. Moreover, on differential forms they are not independent, because of the Cartan identity
Lx = i x d + dix , However, once a metric tensor field g is given on a manifold M , a new derivation, the L e v i - C i ~ t acovariant de~vativeV x with respect to the v e c ~ o ~ ~ e l d
The Metric Tensor Field on a Manifold
185
X , can be defined by (6.39)
where f, a and Y are a differentiable function, a differential 1-form and a vector field, respectively, and where X , is the vector field associated to CY via the metric tensor field 9;i.e.
or, symbolically
x, = g - l ( a ) . We notice that ( V x a ) ( Y )is the s u m of two terms which are antisymmetric and symmetric, respectively, under the interchange X t)Y . The Levi-Civita covariant derivative can be naturally extended to vector fields by the Leibnitz rule; that is, by (a,V X Y ) = LX(% Y ) - (VX% y > .
From Eqs. (6.39),it easily follows that the Levi-Civita covariant derivative is 3-linear; that is, the following property holds:
Vfx=fVx,
Qf
€3(M).
It is worth recalling that the same property does not hold for the Lie derivative; i.e. LfX
# fLx
1
unless when applied to functions. Let us evaluate the Levi-Civita covariant derivative in a basis We have
x = xiei,
Y =Yiej, a = adi
so that
X,
.. = gZ3ajei,
)
g =gijOa8
IJJ
,
{ej},
{di}.
so that
(6.40)
The Metric Tensor Field o n a Manifold
187
with hk 1 1 r$ = -9 (ej(gki)+ e i ( g k j ) - e k ( g i j ) - glicfik - gljcil, + gklcij) . 2
The above quantities, which are also called the ~ e v i - C ~ connectio~ ~ta coefficients, in a holonomic basis reduce to the Christoffel symbols given by Eq. (6.37). They are not the components of any tensor and have the PrOpertY
r$-r;i-c$ = o . Exercise 6.3.1
(6.41)
Show that, in a give^ basis
VXY = X i [ e i ( Y j )- T~;~Y=+~, so that
V,iej
= r;{eh.
The covariant derivative can be extended to any tensor field by the Leibnitz rule, so that Vx(S@TTj= ( V x S ) @ T + S @ V x T . Exercise 6.3.2 Show that the Levi-Civita covariant derivative of the metric tensor field vanishes; i.e.
vxg=o. Equation (6.40),for a = d h and X = e,, gives
V,@
=
-r;Q9j,
and can be taken as a starting point to define a more general covariant derivative, without use of any metrics, but we will not go on further on this subject. A purely algebraic formu~ationcan be found in Ref. 125. Equation (6.41) expresses, in a given basis, the vanishing of the (1, P)-tensor field defined by
T(a,x,Y ) = (a,VXY - vyx - [ X ,Y ] ), which is called the torsion of the connection V.
Dafferential Forms
188
Exercise 6.3.3
Show that actua~ly7 is a tensor field and that, an a given
basis,
Thus, the Levi-Civita covariant derivative has vanishing torsion and fulfills the property Vxg = 0. It can be shown that, given a metrics on a manifold, the only torsionless connection for which Vxg = 0 is the Levi-Civita c ~ n n e c t i o n . ~ ~ 6.3.4
The R
~
~ tensor ~ ~ field n n
The Riemann* (1,3)-tensor field R is defined by
where the R ( X ,Y ) is the curvature operator of
V,defined by
Exercise 6.3.4 Show that actually R,defined by Eq. (6.42), is a tensor field; that is, it is ~ - ~ ~ ~ ~ z Z ~ ~ e a r . Exercise 6.3.5
Show that, in a given basis,
Exercise 6.3.6 Show that the covariant derivative adent~t~
V satisfies the Jacobi
P x ,[VY, VZ]]+ IVY,[VZ, Ox]]4- [VZ, [Vx, Vy]]= 0
8
*Georg Friedrik Bernhard Riemann waa born in Breselenz on September 17, 1826 and died in Selasca on July 20, 1866. He studied a t Gottingen under Gauss, and subsequently at Berlin under Jacobi, Dirichlet, Steiner and Eisenstein, all of whom were professors there at the same time, In spite of poverty and sickness he struggled to pursue his researches. Riemann was one of the most profound and brilliant mathematicians of his time. In 1857 he was made professor at Gottingen.
The Metric Tensor Field on a Manifold
189
Show aka that in a coordanates basis the above equation reduces to the so-called Biancht? ~dentit~e§
0.3.5
The Rice; tensor and the ecalar czlruature
The RiccP tensor is the (0,2)-tensor field, which is defined in a coordinate basis by 71.j
=z!kj
3
=R j i
.
and it is a symmetric tensor field Rij
The scalar ~
u
~ is defined u t by ~
~
Exercise 6.3.7 Show that the Ricci tensor is a symmetric tensor field. Show also that the c o n t ~ a c t e~zanchz ~ i~en~it~es
t Luigi Bianchi was born in Parma in 1856 and died in Pisa in 1928. He has been a student of E. Betti and U. Dini in Pisa, and of F. Klein in Gottingen. He was a professor at the University and Scuola Normale Superiore in Pisa. He has been one of the most important Italian mathematicians in the last century. His works till up more than 10 volumes and concern mainly differential geometry and number theory. Both the original papers and the, now classical, books (on differential geometry, transformations groups, elliptic function) are written in a very transparent and elegant form. Bianchi was strongly loved by his students not only for the marked vis comfca (funny spirit), which was one of his characteristic features. tGregorio Flicci-Curbastro was born in Lug0 (Ravenna) in 1853 and died in Bologna in 1929. He atudied at Rome, Bofogna and Pisa Universities where he obtained his degree in 1875. He has been a student of F. Klein, in Gottingen, and teaching assistant of U. Dini in Pisa. He has been a professor of mathematical physics at Padova University from the year 1880. The main scientific ~ n t r i b u t i o nof Ricci has been the invention (together with its student Levi-Civita) of the absolute differential calculus, later an essential tool for the formulation of general relativity.
190
Diflerential Forms
The previous tensor fields play a basic role in Einstein§ general relativity in which space-time is represented as a 4-dimensional manifold with a metric tensor field g representing the ~ravitationalfield. The empty-space ~ravitationai field g is found by solving Einstein’s field equations
8.4
Endomorphisms Associated with a Mixed Tensor Field
Let T be a mized tensor field; that is, a tensor field of type (1, l), on the n-dimensional manifold M . In a coordinate basis it can be written as
(6.43)
TN
Let 7,M be the tangent space to the manifold at the point p and its dual. The tensor product T M @ 7 , M is isomorphic to L i n ( ~ ~ ,the~ ~ ) vector space of the linear operators on 7 , M , via the canonical isomorphism given by
1:a @ X E T M @7,M 4 Z(cr@X)2= L,&x E Lin(7,M,7pM), where La@xis the linear map
L,ax : Y E 7,M -+ L,,x(Y) = C U ( Y ) EX 7,M
I
§Albert Einstein was born in Ulm, Germany in 1879 and died a t Princeton in 1955. He spent his youth in Munich and after a period past in Milan, he moved, in 1896, t o Swiss. He obtained his Ph.D. at Zurich Polytechnic in 1905. After working a t Swiss Patent Office, he was appointed associate professor at Zurich University in 1909 and then at Berlin University in 1913. He wrote in 1905 six papers. The first of them, with the introduction of the photon, gave a basic contribution to the rise of quantum theory; the second and the third gave rise to special relativity and, with this, to the new concept of spacetime; the others explained the Brownian motion and, with this, introduced new methods to memure the dimensions of atoms. His paper on general relativity is dated 1915. He was appointed a Nobel Prize, for the photoelectric effect, in 1921. In 1933, with the introduction of racial Nazis laws, he moved to Princeton where he was appointed to a chair of professor of physics at The Institute for Advanced Studies, where he taught until 1955.
Endomorphisms Associated with a Mixed Tensor Field
191
Then, with the tensor (6.431, we can associate the endomorphism T on 7,JM
defined as follows (6.44)
Moreover, there is another ~somorphismbetween the tensor product
TJh/i@ 7,M and the space Lin(T,M, 7 , M ) that associates to every a @ X
E
T M @ 7,M the linear map L:,,
:p E
T M -+ ~ ~ , =~@(p)a ( ~E T) M .
Thus, with the tensor (6.43), we can associate also the endomorphi~m'f'
on
TM
defined by n
n
(6.45) i,j=l
k=l
By using Eqs. (6.44) and (6.45)' we find that ( P a , X )= ( c L ! , ? ~ ) . In fact,
Henceforth, when no confusion possibly arises, we will not d i s t i n ~ i s hbetween a tensor T and its associated endomorphisms T and f+.
Daflerential Forms
192
6.4.1 The ~ i j e n h bracket ~ ~ 3 of two mixed tensor field3
If S and T are the endomorphisms associated with two tensor fields of (1.1)type, and if X , Y are two arbitrary vector fields, the relation
+
23c$(X, Y ) = [ S X ,TYJi- [ T X ,S Y ] iS T [ X ,Y ] T S I X ,Y ]
- S [ X ,T Y J- S [ T X ,Yj - T ( X ,S Y ] - T [ S X ,Y ) is called the Nz~enhuisbracket of S and T . It defines a vector field ' ? f p ( X , Y ) which is antisymmetric under the interchange X t)Y . The (1,2)-type tensor field defined by
is called the Nijenhuis torsion of S and T . Let us observe that %$(X,Y ) can also be written in the form 1 2 The ~zjenhu2stors~onN;. of a mixed tensor field T with itself is called the Nijenhuis torsion of T ;it, generally, is not vanishing. In such a case, the previous relations become
31$(X, Y ) = -(LsxT + LTXS- SLxT - T L x S ) Y .
NT(~ X,,Y ) = (a, %T(X,Y ) f with
Exercise 6.4.2
Let us suppose that the tensor field (6.43) has a vanishing ~ ~ e n h u torsion. zs Thus, it sa~isfiesthe conditzo~
(6.46)
From this relation, it follows that if T is znvar2an~f o r a vector field A, it is ~ ~ ~ a rfor i a all n ~vector fields TnA, generated by repeated a p p ~ ~ c a t zofo ~T to A. Show that [TnA,TmA]= 0 V n ,m.
Endomorphisms Associated with a Mixed Tensor Field
193
The Lie derivative, with respect to A, of the tensor (6.43) is given by (see Eq. (6.8))
In local coordinates on M, we have
and
Then, the relation
(6.46) becomes
If 7 : T E R + M is a curve on M such that ~ ( 0 = ) p , and
we have
Thus,the relation (6.47) can also be written as ~ ' ~ Y ) ) (- ~~ ~ ,~X ) = )p[?'(p)(X, ~ ~ YY) - ~, ~ ) X ()Y ]. , (6.48)
Chapter 7
Integration Theory
7.1 Orientable Manifolds
A differential manifold M is said to be orientable if a nowhere vanishing continuous differential n-form R exists on it. Such a differential n-form is said to be a volume n-form. At each point p E M , the n-form R will define an n-covector, 0, E A"('&M),whose value Q P ( e l , e 2 , .. . , e n ) on a basis ( e l , e 2 , . . . , e n } in 7,M will be different from zero. So all the basis in 7,M will be divided in two classes according to the sign of Rp(el, e 2 , . . . , en). The two classes are independent from 0, because every nowhere vanishing n-form 0' will be proportional to 52 by a factor f # 0, and then it will take the same sign (depending on the sign of f ) on the elements of each class. The two classes will be called lefthanded and right-handed.* Thus, it will be possible to choose, continuously V p E M , a basis {el, e 2 , . , . ,en}, belonging to the same class. If the basis are coordinate basis, the Jacobian determinant in the transition from one basis to another will have, in the neighborhood of each point p E M , the same sign. The Mobius band is a not an orientable manifold.
'Which class has which name is a convention, since the sign of R' will depend on f, which is at our disposal. 195
Integmtion Theory
196
7.2
Integration on Orientable Manifolds
Let M be an n-dimensional orientable manifold and w be a differentia1n-form on it. In a given chart (U,cp), w will be locally r e ~ r ~ as~ n t ~ w = f ( ~ ' ., , , sn)dx' A )
* *
Ad ~ * I
The integral of w over U C_ M is defined by
s, i(u,. . , w=
f(x',
. xn)dx' . . .dz" ,
(7.1)
where p(U) C Rn is the ~ ~ u of g eU and the symbof dx' * * dxn denotes the measure for the usual integral of differential calculus. In order to show that the integral so defined does not depend on the coordinates, let us choose a different coordinate basis {a/ayi}in which w = f(y1,.
. . ,y*)dyl . . . dy" .
The equality ~ ( y ) d yAi * . * A dyn = f(x)dzl A evaluated on the basis {B/ayi}
gives
or
where J is the Jacobian determinant. Thus, we have
-
* *
A dxn ,
-
p Vectors and Dual Tensors
197
Then, our definition of the integral of w will not depend on the coordinates if
J,,,f(.’,
. , . ,sn)ds’
’
* *
0kn=
Leu,
Jf(Y’,’*’,yn)&’ *’*&J”.
As we know from differential calculus, the above equality holds only when J
> 0.
It follows that in the definition (7.1),an orientation for 24. must be chosen; that is, a requirement on the handedness of the basis must be added. This explains why, from the very beginning, M has been supposed to be an orientable manifold; that is, one for which it is possible to choose, continuously at every point p E M , a coordinate basis { 6 / a x k } with the same h ~ d e d n e s s . However, the integration theory of differential forms has been extended, by de %am, to nonorientable manifolds4 by introducing forms of odd parity, and this can have interesting physical applications.173 On the historical side we shall mention that they were introduced by Hermann WeylS6 and developed by S ~ h o u t e n , *and ~ called Weyl tensors. Synge and Schild refer to them as oriented tensors, while de Rham called them tensors of odd kind.7 Under a change of coordinates (x # x’), a ~ ~ ~ s~t e d~ ~f oetransforms ~~ n as folIows t ~ J 8xP 6x9 w;b...c = ---
IJ I
*.
a x f a ax’b
axr .&lc wpq”’r
where J is the Jacobian determinant and IJI its absolute value.
7.3 p-Vectors and Dual Tensors Completely antisymmetric tensors of type (p, 0), on a n-dimensional vector space E , are called p-vectors. A Grassmann algebra, can be, of course, constructed on them in complete analogy with that of p-covectors. The vector space of p-vectors is denoted with Vp(E). Its dimension is
dimVF(E~= and a basis is given by
(r )
=
( ) n-p
= d i m V n - p ( ~,)
~
~
Integration Theory
198
Thus, ~ ~dimA"(E). d i m V p ( E ~= dimVn-"(E) = d i m A p ( =
If {&) is the dual basis of {ei), the n-covector
X
.. . A 6"
1 -&iliz,.qi,~il A@A
. , . A din n! is a basis for the vector space A", and will be called a v o l ~ c~o ev e c ~ ~ r . By using the volume covector, we can associate, with any p-vector f), = 6' A 6' A
z
1 = -Xab''ceaA eb A
P!
. . A e, ,
the (n - p)-covector defined by
fz(X)
1
. . .
ixfz = -X'1'2'%E P!
A &+z i,az...i,@b+l .
A
.. . A d i m .
The above (n - p)-covector is called the f),-duaZ of X or also the Poincare' dual of A basis independent definition is given by
x.
ixfl(Y"fi,. . . , y n )= fz(xA Y P + ~A , .
. A Y-)
vyp3-1,.
. . ,ynE E .
It is also possible to make the inverse; that is, to associate, with any pcovector a , the (n- p)-vector
where
-
p Vectors and Dual Tensors
199
Then,
Let us next calculate the "dual of the dual" of a given p-covector a. We have
Summing up, we obtain
R - l ( i x R ) = (-1)P(*-P)X
vx E VP(E)
R(R-'(a)) = ( - l ) p ( n - p )V~a E h p ( E ) . Thus, the volume covector R and its inverse O-' provide the following mappings:
Integration Theory
200
7.4
Metric o Volume = Hodge Duality
In See. 6.3,it has been shown that a metric tensor g over a ~ - d ~ m e n s i ovector na~ space provides an isomorphism between vectors and covectors. It is easy to see that it also provides an isomorphism between p-vectors and p-covectors, since
is completely antisymmetric in the indices e', j , .. . ,k. Then, a metric tensor g allows us to complete the previous picture in the following way: eg4-
cgc-
\ d
sz
L
P $2-'
2 \
pl
F]pJ
tgt
[w
The composite map * = g o 51-I
which provides an isomorphism between p-covectors and ( n - p)-covectors, is called the Hodge dual. Its transposed W' o g provides an isomorphism between p-vectors and (n- p)-vectors, and is denoted by the same symbol. An example of this isomorphism is given by the so-caHed vector product of two vectors in the %dimensional Euclidean space (R', gij = S i j ) : Consider two vectors U ,V in R3; Take the associated covectors u = g(U, v = g(V, *) via the Euclidean metrics; Consider the 2-covector given by their exterior product: u A u; The volume dual of u A v is a vector which is called the vector product a ) ,
0
of u,
v.
Summing up:
Metric o Volume = Hodge Duality
201
Remark 14 If a manifold has a metric g , let {di} be a n orthonormal basis for diflerential 1-forms, and fl be the volume-form
f l = d1 A d2 A
* * '
A
8".
If { x k } is a n arbitrary coordinate, and A is the transformation matrix from { d x k } t o ( 1 9 ~ ) ;i.e.
,
di =
we have
fl = A: . A?. .. 3
. .. A d x k . A;. . .Az&"'kdxl A dx2 A . . . A dx" = = (det A)dxl A dx2 A * . . A dxn . A dxj A
O n the other hand, we also have gij
= @A$g(eh, ek) 9
where gij are the components of the metric tensor g in the coordinate basis { 8 / a x i } , and g(eh, ek) = d h k , since the original basis was orthonormal. Therefore, g E det(gij) = (det A)2
and
fl = m d x ' A d x 2 A
A
dxn .
&om Eq, (7.3) it follows that the components Rij'..k of fl-' ~12s.-
-
1
are given by
1 --
n12...n
*
However, in our metric manifold the inverse of R could also be defined by fiW..k =,pgjq
so that
. . .g k ~ ~ R p q .,. . ,
Integration Theory
202
If g is negative, f1212".nand flf12"'ndiffer b y a sign. I n special or general relativity, it is conventional to use fl"''.n in the dual relations. 7.5
Stokes Theorem
Let M be an orientable n-dimensional differential manifold and U be a region of M . We call boundary of U an orientable n - 1 dimensional submanifold of M , namely a U , which divides M - dU in two disjoint parts, U and CU,in such a way that any continuous curve joining a point of U with a point in CU contains a point of dU. Let us consider the integral of a n arbitrary n-form w over U
Let X be a vector field and U ( T )the images of U under the flow cp generated by
x:
u(T)= ( p T ( u )
'
From the analysis already performed in Sec. 4.1, it follows that
Lx L ( 7 )
=k
( T )
Lxw '
or
since the form w is obviously closed. On the other hand, by applying directly the definition of Lie derivative, we obtain
= lim T-+o
1
-
with U ( 0 ) = U and W ( T= ) U ( T )- U ( 0 ) .
Stokea Theorem
203
Let us consider a part aV(0) E. aU(O), to which the vector field X is not tangent, and a part 6V ( T ) of ~ U ( T representing ), a region between OU(0) and aU(.r) locally given by 6v(T)= av(0)X]o, T [ .
Then if ( 2 2 , x3,. . . ,x,) denote the coordinates for aV(O),we can introduce, with $1 = T , coordinates ( x ~ , x z .,..,z,) for ~ V ( T In ) . these coordinates the differential n-form w can be expressed as follows: W =
f ( x 1 , ~, z. . . , z n ) d ~ l A d x 2 A . " ' A d ~ ,
and
= lim
1J
7-tO-r
[Tf
( O , X ~ , .. . ,xCn)]dxzA
* *
- A dx,
BV(0)
denotes the restriction to dU. where the symbol The final equation
is independent from the constructed coordinates, but it requires that X should not be tangent to aU in V. For the whole boundary NJ, two cases can occur:
X is tangent to isolated points forming a submanifold of lower dimensionality.
Integration Theory
204
In such a case these points do not contribute to the integral. X is tangent to in an open region of it. In this case both sides of the Eq. (7.5) are vanishing and the equation still holds.
av
So, summing over all parts aV of aU, we obtain
from which
By comparing the two expressions, Eqs. (7.4) and (7.6), of the Lie derivative of the integral w , we finally obtain
su
and since i x w , as well as w ,is arbitrary, we conclude that
Theorem 24 (Stokes) For any (n - 1)-form manafold U, the following relation
cy
over a n n-dimensional
holds.
If U = [a,b] is an interval of the real line, and f : U function, then the Stokest theorem reduces to
J,” f ’ d x
+ 8 a differentiable
= f ( b ) - f(a) I
since aU = {a,b}. +George Gabriel Stokes was born in Skreen (Ireland) in 1819 and died in Cambridge in 1903. Physicist and mathematician, he has been a professor of mathematics at Cambridge University and is universally known for the results on the transformations of integrals, on the liquid waves and for his theories on optics, founded on the hypothesis of dragging ether.
Gradient, Curl and Divergence
7.6
205
Gradient, Curl and Divergence
On an n-dimensional orientable manifold M , endowed with a metric tensor field g, all properties concerning the volume duality and the Hodge duality can be point-wise carried over directly. This allows us to better understand the meaning of some familiar concepts in R3,such as the gradient, the cur1 and the divergence. 0
T h e gradient Consider a function f and take its exterior derivative d f . The vector field associated to the differential form d f , by means of the inverse of the Euclidean metric tensor 17, is called the gradient of f : V f = v - ' ( d f , - ) CL---L. i v f q = d f .
0
T h e curl Consider a vector field U , take the associated differential form a = q ( U , - )and its exterior derivative d a . The volume-dual of d a is a vector which is called the curl of U : V x
0
U
= Q-'(da),
with R = d x A d y A d z . T h e divergence Consider a vector field V , take the associated differential 2-form via the volume form R = d x A d y A d z . Its exterior derivative is proportional to R up to a multiplicative function called the divergence of V:
(V * V)R = d i v a . Moreover, if V = V x U = Q - ' ( d a ) , then
V * V x U = d i v , u R = din-l(dalR = d ( O ( O - ' ( d a ) ) ) = d(-l)3-'da
= d2a = 0.
Exercise 7.6.1. Use Stokes' theorem t o prove that, f o r every exact differential 2-form w on the sphere S2,
Integmtaon Theory
206
Proof. In order for w to be exact, a diflerential 1-form
Q
have to exist such that
w = d a . I n this case, Stokes’ theorem gives
s,,. s,, s,, =
da =
Q
= 0,
since S2 has n o boundary. Exercise 7.6.2. Use Stokes’ theorem to show that f o r the differential 2-form w = x’dx2 A dx3 o n S3,
where S2 is the unit sphere considered as a submanifold of S3. Answer. The differential3-form
dw = dx’
A dx2 A dx3
is the usual volume form, so that when integrated on the volume V of the sphere, gives
The result then follows by Stokes’ theorem. The above exercise gives an example of a closed differential 2-form on S2 which is not exact, since it does not satisfy the criterion of the first exercise.
7.7 A Primer for Cohomology Let P ( M )and BP(M) be the set of all closed differential p-forms and the set of all exact differential p-forms, respectively. Both sets have a natural structure of vector space and, moreover, BP(M) is a subspace of Zp(M).Then, we can introduce in P ( M ) an equivalent relation, namely M by declaring
aM p
H
( a - p) E BP(M);
207
A Primer for Cohomology
i.e.,
The set of all equivalence classes is denoted with W ( M )and is called the pth de Rham cohomology vector space of M . It is easy to show that, if M is any connected manifold, then
H o ( M )= Z o ( M )= %. Indeed, a zero-differential form is just a function, so that Z o ( M )is the space of functions f for which df = 0; i.e. Z o ( M )= 8. Moreover, B o ( M )= ( 0 ) ; i.e. the zero function, so that constants f and g are equivalent if they coincide. If M is not connected, then an element in Z o ( M )will be constant on each connected component of M , but the value of the constant can be different on different components, so that
H o ( M )= Z o ( M )= R", where m is the number of components of M .
Exercise 7.7.1. Show that for the n-dimensional open ball or any region U difleomorphic to it, W ( U )= 0 , p 2 1 . (Hint: All closed differential p-forms are equivalent to one another, and hence to the zero differential p-form).
Exercise 7.7.2. Show that H*(S*) # 0 ,
H*-l(P) = 0 .
It can be proven that51 H"(Sn) = 8 , HP(Sn)=O, O < p < n , HO(SD) = 8 .
Integration Theory
208
Remark 15
The dimension of Hp(M) is called the pth-Bettit number.
Remark 16 The given definition of HP(M) relied on the differential structure of M . However, it can be proven (see, for instance Ref. 55) that the cohomology groups depend, only o n the topology of M and not its differentiability. 7.8
Scalar Product of Differential p-Forms
Let M be an orientable n-dimensional compact differential manifold and let a, p be differential p-forms h p ( M ) . The Hodge dual * p of p is a differential (n - p)-form: * p E An-p(M),so that a A * p is a differential n-form. This allows us to define the scalar product ( a ,p) of a and p, by
Exercise 7.8.1. Show that the previous formula defines a scalar product o n AP(M).
7.8.1 Exterior codifferential By using the above scalar product, we can define a new operator 6, the adjoint of d , by
Clearly,
6 : hP(M) +AP-'(M),
6f = 0 for every f u n c t i o n f
15 p
5n,
.
The operator 6 is called the codifferential. It is worth to observe that it can be introduced only by using a metric tensor field defined on M . SEnrico Betti, born in Pistoia in 1823 and died in Pisa in 1892, has been a professor of mathematical physics a t Pisa University and Director of the Scuola Normale Superiore in Pisa. He gave deep contributions to algebra, topology, elasticity theory, and potential theory. An excellent teacher, and among his students were Luigi Bianchi and Vito Volterra.
Scalar Product of Dafierential p -Forms
209
Exercise 7.8.2. Show that for every daflerential p-form a
and that 6 is not a derivation.
Let us finally observe that, from the Eq. (7.8),it follows that J2 = 0. The Laplace-Beltrami operator
The Laplace-Beltrami§ operator A is defined by the relation A=do6+60d=(d+d)2,
and it is a self-adjoint operator, since
Exercise 7.8.3. Give the expression of A in !R3 by using the Cartesian coordinates and the spherical-polar ones.
A differential form w that satisfies the differential equation Aw=O
is called harmonic. Clearly, Aw=O.-t.dw
= 0 , 6 =~O .
Indeed
+
( A w , w ) = ( d d w , ~ ) (6dw,w) = (6w, aw)
+ (dw,dw),
with (6w, 6w) 2 0 , (dw,dw) 2 0. 5Eugenio Beltrami, born in Cremona in 1835 and died in Rome in 1900, has been a professor of algebra and analytical geometry at Bologna University and, after, at Pisa, Pavia and Rome universities. His research activity on the Newtonian potential and on the differential parameters can be considered of basic importance, and his Saggio s u l b anterpretuzione della Geometria non Eucladea is now considered as classical work.
Integmtion Theory
210
7.8.2
Hodge theorem
Hodge theorem is an important decomposition theorem that we quote without proof.
Theorem 25 (Hodge) Every differential p-form w can be written as w=da+6P+y,
where a is a differential ( p - 1)-form, P is a differential ( p + l)-form, and y a harmonic form. Moreover, the differential forms d a , 6P and y are unique.
Chapter 8
Lie Groups and Lie Algebras
8.1
Lie Groups
A finite-dimensional Lie group is a C F manifold G of dimension n, endowed with a group structure, such that the product ( g , h )E G x G H g h E G
(8-1)
and the inverse
are Cm maps. The diffeomorphisms
L, ; h E G + g h
E
G , R, : h E G + hg
EG
are called the left translation by g and the right translation by h, respectively. Let (U,cp) be a chart in G such that e E U and cp'(e) = 0, where e is the identity of the group. For every open set U containing e, there exists an open set V c U ,to which e belongs, such that V . V c U ,where V V = {gh : g , h E
-
V)* Then, the product p ( V ) x cp(V) is an open set in Rn x Rn containing the point (0,O). Since G is a Lie group, the map (8.1) is differentiable, so that, if g and h are two elements in V , the coordinates cpi(gh)= (gh)i of their product 211
Lie Groups and Lie Algebras
212
are differentiable functions of the coordinates xi = p i ( g ) of g and yi = pi(h) of h, and we can set ( g h ) i = fi(x1,. . . ,x",yl, *
. . , y")
or shortly, (&)a
=
fi@, Y) '
The group structure implies that the functions f i must satisfy the following properties:
The first property follows from the associativity of the group product; i.e.
((sh).)i = ( S ( W i >
Q g , h,u E
G =j fi(f(x, Y), ). = fih f(Y, 2 ) ) 7
for every choice of x,y,z in p(V), with zi = pi(.). The second follows from (geli = (eg)* = xi =+ fi(x,0) = fi(0,x) = x i .
Moreover, for the f i ' s the following expansion can be performed: fi(x,y) = xi
+ yi +
c
x:,x*yp.
(8.3)
*2l,P>1
To build the inverse 9 - l of an element g it is sufficient to solve the following system of equations with respect to y': fi(Z1,.
. . ,Z",y 1 ,. .. ,y")
= 0.
(8.4)
From Eq. (8.3), we have
so that the Jacobian determinant a(f', . . . , f")/a(y', . . . , y") at the point (0,O) in R" x !Xn is 1. Therefore, by continuity, the Jacobian does not vanish in a neighborhood of the origin and, by the implicit functions theorem, there exists an open set V' c V in e , such that for every g E V' the system (8.4) has a unique solution (y', . . . , y").
Building of a Lie Algebra from a Lie Group
8.1.1
213
Local Lie groups
A local Lie group is a local version of a Lie Group. Then, it is a pair ( A ,f), where A is an open set in Rn containing the origin of the coordinates, and f a differentiable map
f:AxA+R" satisfying, Vz, y, z E A , the following conditions:
(4 f(2,f(y, 2)) = f(f(2,Yh z>; (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
<
(10.3)
Then, every one parameter subgroup et( of the group G operates as a locally Hamiltonian flow on M . In fact, since the action (10.1) is symplectic*
C(,W
=0
and
L(,w
+ i<,&
= dit,w
= dic,w,
we have
dic,w
= 0,
that is, the differential 1-form &W is closed. Therefore, for every [ E B, a function Jc can be locally found on M such
that icMw = dJ6.
(10.4)
Furthermore, since
we have
or Jc+q
= Jt
+ Jq + Q
7
where Q is a constant. It is possible to choose this constant to be zero, so that Jc+V
= J(
+
Jq
.
(10.5)
~~~
*In this chapter the Lie derivative, with respect to a vector field X , has been denoted with the symbol Cx,instead of Lx,to avoid confusion with the left translation L,.
Reduced Phase Space
273
If, for every 5 E 0, & is a globally Hamiltonian vector field, then the Jt’s are globally defined on M and Eq. (10.5) allows us to define a map J:p~hi+J(p)~8*,
(10.6)
where J ( p ) is the element of Q* such that ((7
J(P))= J&>
1
vt EB
*
Following sou ria^,^^ the map (10.6) is called a momentum map. For every ( E B and p E M , let 7 ~be ,the~map 7€,p :9
E G -+ 7(,p(S) = Jt(@p,(p))- JAd,-l€(p) ,
whose derivative at the identity ( T € , ~ ) * ~once , evaluated on a vector q E gives (+/t,p)*e(v)
d = z7t9p(et’)/t=O
B,
The Orbits Method
274
or equivaiently, if the diagram
is commutative, then ( { J € , J v ) - J [ € , q ] ) ( P )= o ,
-
VE,rlEG, V P E M .
Therefore, the condition (10.7) assures that the linear map
‘5
JE
is a homomorphism of the Lie algebra of G in the Lie algebra of the Hamilton functions on M . In this case, the action (10.1) is called a Poissonian action. Of course, not all the symplectic actions @ on symplectic manifolds M are Poissonian. If the symplectic form w defined on the manifold is also exact; i.e. w = -d6,
(10.8)
and the action @ leaves B invariant, its to say
@;6=6,
then the map J : M
(10.9)
VgEG,
-+9*, defined as below
(t> 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<,B - i t M w = 0,
I
‘t
9
9
(10.10)
275
Reduced Phase Space
as to say
dJ6 = i c M w , that is, J is a momentum map. Now, because (see Appendix D)
(Adg-lOM (PI = ( @ g - 1 ) w D S ( p ) (<M ( @ g ( P ) ) ) 1 we also have
i.e. J(@g(P))
If p is an element of
= Adi-1 (J(P))t
VPEM,
Vg E G *
G*,the set Gp = { g E G : A d i - i p = p }
is a subgroup of G, acting on M . The group G usually permutes the sets of the type J- '( p ) = { p E
M :J(p)=p} .
In fact, if p E J - l ( p ) , then J ( p ) = p but Q g ( p ) ,with g E G, may not belong to J - ' ( p ) . Let p E J - l ( p ) and g be an element of G,, then
Ad;-i(J(p))= A d ; - i p = p , which, by using Eq. (10.7), can be written as
Adi-1 ( J ( P ) ) = J ( Q g ( P ) ) 9
The Orbits Method
276
so that J(@.,(P)) = P , v g E G,, V P E J - Y P )
7
that is, G, leaves J - ' ( p ) fixed. Then, by Eq. ( l O . l ) , we can define an action @ of G, on J - l ( p ) , @ : (SlP) E
G, x J - ' ( p ) -+ @(g,P) E J - ' ( p )
*
(10.12)
The orbit of the point p E J - ' ( p ) under the action of the group G, is given bY
G,*P=
{@g(P)
(10.13)
: 9 E G,}.
We can introduce an equivalence relation on J - ' ( p ) by defining two points of J - ' ( p ) to be equivalent if they belong to the same orbit (Eq. (10.13)). The set of equivalence classes, denoted as usual by J-l(p)/G,, is the set of the orbits of the points of J - ' ( p ) under the action of G,. The point p E G* is said to be a regular value of J if, for every p E J-' ( p ) , the derivative
JaP: 7,M
+ 7,B*
is a surjective map; in such case it can be proven' that the set J-'(,u) is a differential manifold. The action (Eq. (10.12)), @ : (SlP) E
G, x
J-Yd + @ ( g ) P )E
J-'(p)
1
is called a proper action if it satisfies the following condition:
If (pn)nE~ and (Qgn(p,)),E~ are sequences of points of J - ' ( p ) ) which converge in J - ' ( p ) , then (gn)nEN admits a subsequence which converges in G,. The hypotheses that p E B' is a regular value of J and that the action (10.12) is a proper action are sufficient conditions to assure that J d l ( p ) / G , is a differential manifold and that the map flp
:
J-W --t J - l ( P ) / G p
)
which associates with every point of J - ' ( p ) ) the orbit to which it belongs, and its derivative (flp)*p
are surjective maps.
: % J - ' ( p ) + T G , , T ( J - ' ( ~ ) / GI ~ )
(10.14)
Reduced Phdae Space
277
The ~ ~ n i f o IJ d- l ( ~ ~ / Gispcalled the ~ d ~ phuse c e space ~ and can be endowed with a natural symplectic structure. Indeed, let us consider two vectors and @ tangent to J - l ( p ) / G , at the point y, which is the orbit of a point P1(p) under the action of G,. Let us choose point p in this orbit: the vectors and @ tangent to the orbit at y can be obtained from some vectors V and W , tangent to P 1 ( p )at the point p , by wing the map (10.14). We can thus define on ~ - l ( ~ ) / aGbilinear p form $2, expressible in terms of the symplectic form u on M
that is,
0
The bilinear form a, does not depend by the choice of the point p in the orbit and of the vectors V and W of T J - l ( p ) .
Indeed, let
?(Gp
*
= {SM(P) : E
gp)
2
where &@) is defined in Eq. (10.3) and gp is the Lie algebra of Gp. We can now prove that 7 p ( G P * P ) = 7,@' P) n 7,J--'(P)
*
(10.16)
As a matter of fact if < ~ ( pbelongs ) to & ( G * p ) ,then Eq. (10.16) can be seen to be equivalent to saying that ( ~ ( p E) 7 p J - l ( p ) if and only if ( E Gp. Therefore, we may write
since J satisfies Eq. (10.7). Because J ( p ) = p for every p E J - I ( p ) ,
~ J - ' ( ( C Ls)ker J * ~ , Then, ( ~ ( pE) 7 p J - ' ( p ) = ker JSpif and only if
that is, if ( E Gp. Let V be a vector in 7 , M ; for every
E 9, from Eq. (10.4), we have
v,== ( ~ € ~ w ) ~=(d Jv€)f p ( V )
~p(<M(p),
*
If a(t) is the integral curve of V , then
so that, V E 7 p J - l ( p ) = ker Jwpif and only if
or equivalently, if and only if W p ( < M ~ ~ , V0, ~
v<
(10.17)
*
.
Therefore, the two tangent spaces %(J-I(pL)) and 7,(G p) are one the orthogonal complement (with respect to w)of the other. Now, because
Reduced Phase Space
279
we have
and ( Z g ) * p ( < M ( p ) )= 0 1
v< E G p .
(10.18)
In this way the vectors V and W , which correspond to the vectors and @ in Eq. (10.15), are defined up to a vector of G(G, p ) ; but the addition +
of a vector of this space to V and W does not modify the right-hand side of Eq. (10.15), because the spaces 7pJ-'l(p) and 7pG. p are "orthogonal". Concerning the independence of Eq.(10.15) from the choice of the point p on the orbit, this depends on the fact that the action CP is symplectic and on the invariance of J-'(,u). In fact, ( ~ ~ R p ~ ~ ~ (W') p )= ( v# ~/ g, ( p ) ( v /W') , 5
V'I W' E
Xis(p)J-'(ct)
?
so that, by using the relations
V' = ( W * P ( V )I
W' = ( @ g ) * p W )
, v, W E TJ-l(I.4 I
we finaliy obtain
( ~ ~ ~ ~ ~ W') ~ ~= #(~ gP( p ~ ) ( ( (@ ~V ) * P' ( v, )(,~ g ) * P ( W ) ) = #p(V, W >= (
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
<
QPE : ( t , g ) E R x G -+ @ c ( t , g ) = @ ( e f P E , E g )G defines an action of R on G. The vector field
is a right invariant vector field, because
@ ( e t t , g ) = etEg = Rg(etE),
and
The Orbits Method
282
From Eq. (lO.lO), we deduce that the momentum J is the map
J : a, E T * G -+ J(a,) E 8 * ,
(10.22)
defined by (Appendix E)
(c, J(ag))= a g ( c G ( g ) )
ag((%)*e(C)>
= (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 << l), (10.39) we have 77 = (Rg-l)*g(i) 1
from which Eq. (10.37) foilows. To the motion eq7g in L e fixed frame it corresponds the infinitesimal rotation efT in the body frame obtained by applying L,-Ito eq7g, ,ET
= 9- 1 er1T 9.
(10.40)
Of course, X(() = w, is the angular velocity with respect to the body. Therefore, we can write gecT = eqTg.
From Eq. (10.39), we have
so that
t=
G g - 1 )*g(Q)
from which Eq. (10.38) follows. Equation (10.36) allows us to simplify the notation, since we can use, instead of the expressions (10.37) and (10.38) for wc and w 3 , the simplified ones = (Lg-1)*g9 E6
(10.41)
w* = (Rg-l)*gb E 0'
(10.42)
wc
and
291
Rigid Body Equations
10.3.2
The space and the bod@ a n ~ ~~ ~~ a~ ~ e
n
t
~
The Lie algebra B = 7eS0(3) of the group SO(3) is the three-dimensional space of the angular vebcities of a11 possible rotations and the Lie bracket of such algebra is given by the usual vector product. If the tangent bundle 7 S 0 ( 3 ) is the space of the rigid body velocities, the cotangent bundle 'PSO(3)is the space of the angular momenta J. If the vector J lies in the cotangent space to the group at the point 9, in analogy with Eqs. (10.41) and (10.42),it can be transported to the cotangent space G* to the group at the identity, either with left translations or with right translations. Thus, we obtain two vectors
J, = L:J E Q*
(10.43)
and
J, = RiJ E g*
-
(10.44)
The vector J, is the angular momentum with respect to the body and J, is the angular momentum with respect to the space. Actually, the algebras 9 and I?' can be identified, since it can be easily proven that
where 6 and rl are elements of B, the dot denotes the Euclidean scalar product and T r the trace operator. The above equation defines an isomorphism between the spaces Q and Q*. We can thus consider the angular velocity and angulas momenta vectors as lying in the same space. However, in what follows we shall not make this identification, and we will continue to consider the angutar velocity vectors as belo~gingto B and the momenta vectors as belonging to g*. 10.4 Rigid Body Equations
As we saw in the previous section, the total angular momentum J, of the rigid body is a constant of the motion, so that dJs =o. dt
(10.46)
The Orbits Method
292
In Sec, 8.2.4, we introduced, for every 6 E Q, the linear operator adz whose action on an element a! in Q* is defined as follows: (ud~a)(77) = (ad+, rt) = (a,U d ~ 7 7 )= (a, ft)rtl)
(10.47)
for every 71 E Q. The above equation can be written in the form (It1 4
1
77) = (a,!&rtl) I
v t ,77 6 , v
E 8*I
where the bracket 1. ,.[ is defined by
14,a[= a d z a .
(10.48)
Once 8 is identified with Q*,the bracket f - , reduces, up a sign, to the Lie bracket of the algebra 8. If g ( t ) is a curve in S0(3),the relation a(t) = A d i f t ) - l a defines, for every a E Q*, a curve in G*. The following relation a[
d -d 4t t ) = - l ( ~ ~ ( t ) - l ) * ~ ( ~ ) g , a ( t ~ [
(10.49)
is proven in Appendix F. &om Eqs. (10.43) and (10.44)) we have
Jc(t) = A d i ~ t ~ J ~ ( t )
(10.50)
so that, by means of Eq. ~ 1 0 . 4 1Eq. ~ , (10.49) gives dJ, = dt
1WC) JCI
(10.51)
Equation (10.51) is called the EuEer equation. An important ~ ~ q of Eq, (10.50) is that the flow defined by the Euler equation maps points of a given orbit of the coadjoint representation to points belonging to the same orbit. It follows that the orbits of the coadjoint representation in the dual space of the algebra are invariant manifolds for the flow defined by the Euler Eq. (10.51). It is well-known that the vectors wc and Jc are related by the following equation:
Jc = z w , where
}
~
Rigid Body Equations
293
is the inertial operator. Since 5 is linear and symmetric operator, we can an on SO(3). Indeed, by setting define a ~ e m a ~ i metrics
where
which defines a metric tensor field on SO(3). The kinetic energy T is
Since we = S-' Jc, the kinetic energy can be expressed as a function of the angular momentum
1
1
1
T = ~ ( w C , Q w c=) Z ( W ~ , W = ~ ) 5(W1Jc,Jc) .
(10.52)
Thus, the kinetic energy can be considered to be a function defined on the dual space 9 ' of the algebra. Every tangent vector to the orbit V at the point J can be written as follows:
It,J L with 6 E 9. On the other hand, since
dT = w c , the right hand side of the Eder equation can be written in the form
IdT, J L where the differential 1-form dT, being the exterior derivative of a function defined on G*,belongs to the dual space of 8*,that is S .
The Orbits Method
294
The symplectic structure on V is given by Eq. (10.33). In terms of the bracket 1. ,.( here introduced, it can be written as
~
~
~(
or, 4,
d~ ~=~(It, ~ dT1, ~ ~ J >~= ~ ) ~) dT)) = d , ~ ( u ~> ~ ~ ~ )
where H denotes the restriction of T to an orbit of the coadjoint representation. From the above equation, it follows
L~~~~~~= 0 > which says that the Euler equation is a Hamiltonian equation, H being the Hamilton function.
Chapter 11
Classical Electrodynamics
Particles and fields are the fundamental concepts of classical physics. Particles are identified as point particles and fields are tensor-valued functions on spacetime. Given sources and initial cond~t~ons, fields are ruled by the Maxwell* equations.
11.1 Maxwell's Equations The pheno~enolo~ical equations of the ele~trodynami~s are given by the following MaxwelI equations:
iu
8 * <do = 0
c dt
L,
Js 2 .
ZdD = -
s,,
magnetic poles do not exist
I?. ti
Farada~'slaw
d -7idG = Q
Gauss' law
*James Clerk Maxwell was born in Edin~urghin 1831 and died in Cambridge iR 1979. He has been a professor of physics at Cambridge University from the year 1871. Famous physiciat and mathematician, he gave a mathematical expreasion t o Faraday's intuitive and experimental views in the classic k t i s e on Electricity and Magnetism (London, 1873) in which, as a consequence of his electromagnetic theory of the light, he foresaw electromagnetic waves later detected by Hertz and applied by Marconi. 295
Classical Electrodynamdcs
296
where 0 0
0 0
c is the light velocity in vacuum,
S is a regular surface in !R3, whic,. may change in time, with a given orientation defining the exterior normal 5, &” is the boundary of S with the orientation induced by the one of S, U is a regular submanifold (a volume) of !R3 with a given orientation and aU is the boundary (a surface) of U with the orientation induced by the one of U.
E and are the electric vector field and the magnetic induction vector field that can also be defined by the Lorentz force %, which acts on a particle with electric charge e and velocity ii: (11.1) Maxwell’s equations represent the synthesis of discoveries by Faraday, Gauss and Amp&re.t $Michael Faraday was born at Newington, United Kingdom in 1791. In 1813 he was engaged as laboratory assistant by H. Davy. Against the current idea on the action of forces, he introduced the concept of force lanes to explain the propagation of electric or magnetic effects. Today they are known as integral curves of electric or magnetic fields. In the year 1831,he discovered the electromagnetic indzlction phenomenon and constructed the Erst electric generator. He also discovered the effects of the magnetic field on the light plane polarization and, in chemistry, two fundamental laws on the propagation of the electric current in chemical solutions. In the first, he established the direct proportionality between the amount of transformed matter and the amount of electric charge passing trough the electrolyte; in the second, he established the proportionality between the amounts of different substances and their equivalent weights. In order to describe the experiments and t o explain the results, Faraday invented the words ion, cathode, anode, electrolyte. He died at Hampton Court in 1867. Karl F’rederick Gauss was born at Braunschweig in 1777 and died in Gottingen in 1855. From the year 1807 he has been professor at Gottingen University and Director of the Gottingen Astronomic Observatory. Founder of the differential geometry of surfaces, mathematician, physicist and astronomer, he was called princeps mathematicorum. In hie works, he adopted the motto pauca sed maturn (little and deep). Indeed, his works are celebrated also for the excellence of the form. However, the pauca fill up eleven big volumes. Anarc5 Marie Amphre was born at Lyons in 1775,and died at Marseilles on June 10,1836. Mathem~tician,chemist, physicist, man fascinated by the mystery, Amp-ire attempted to And in nature an answer to his need of universality. From the year 1809 he has been professor of mathematics at the &oEe Polytechnique in Paris. His papers, concerning the connection between electricity and magnetism, were written in 1820.
Geometrical Identification of Fields on R3
297
When a charge density p can be introduced, such that
the Maxwell equations take the form:
l,2 .
fidu = 0
(11.2)
where J’= pv’ is the current density. When S is a stationary surface and fields are sufficiently regular, we can use Stokes’ theorem to get the Maxwell equations in the following differential form:
I
1 aB’ r o t z = --c at divd = 47rp
(11.3)
11.2 Geometrical Identification of Fields on R3 Phenomenological Eqs. (11.2) are important to understand the geometrical meaning of fields ($,a,fi,B). Indeed, Eqs. (11.2) show that 8, b and J’ have to define differential 2-forms on S3,since they are integrated on a 2dimensional manifold aU,while 2 and have to define differential 1-forms on R3,since they are integrated on a 1-dimensional manifold as. Finally, p has to define it differentiable 3-form) since it is integrated on a volume U.
Classical Electrodynamics
298
Actually, because of transformation properties of electromagnetic fields, ~ ~ t ~ ~ have to define t ~ i s t e d rather determined on physical grounds, 8 and than even differential forms. Indeed, the transformation laws of the fields E’ and under space transformations
B,
t+t’=t, xi -+ X’a = f i ( x ) ,
or time inversion
t-+t’=-t, xi
-+
x/i
-x2, ’
may be obtained from the expression (11.1) of the Lorentz force. It is clear that, owing to the presence of a vector product in the expression of the Lorentz force, 8 must change sign under time inversion. Therefore, will be effected, by such a transformation, even if it does not depend on time. Clearly, if we consider only coordinate transformations with positive Jacobian determinant, the identification of electromagnetic fields with even differential forms would be possible. We can resume the above discussion saying that: Fields entering Ampkre’s law must be identified with twisted diflerential forms while fields entering Faraday’s law will be identified with even differential forms.
From Eqs. (11.2), we can try the following identification:
E = E,dx
+ E,dy + E z d z
B = B x d y A d z + B,dr
H= H x d x + H,dy
D = D,dy
A dz
A dx
electric field differential 1-form
+ B,dx
A dy
+HZdz
+ D,dr
A dx
magnetic induction differential 2 -form magnetic field differential 1-form
+ D,dX
A dy
electric induction differential 2-form
J = J x d y A d z + J,dz A da: + J z d x A dy
it current differential 2-form
R=edxAdyAdt
charge d i f l e r e n t i a l 3 - f o m
Geometrical Identification of Electromagnetic Field in Space- Time
299
which allows us to rewrite Maxwell’s equations in the following form:
c dt
S, S,, B =-
I dJ, D + c dt
E
p S,
J =
J,, H .
If S is stationary and fields are regular, we can apply Stokes’ theorem to get Maxwell’s equations in the following differential form:
dB=O 1 .
d E = -B C
d D - 4nR = 0 dH - -(B+ 47rJ) = 0 1
C
As a consequenceof transformation properties, in particular of the behavior
of 8 under time inversion, it seems more appropriate to consider electromagnetic fields directly on spacetime.
11.3 Geometrical Identification of Electromagnetic Field in Space-Time
By introducing the following differential forms:
F = dxo A E - B Faraday’s 2-form G = dxo A H
+D
Ampire’s 2-form
I = dxo A J - R charge-current 3-form
Classical Electdynamics
300
with zo = ct, ~axweIl’sequations can be written simply as follows:
dF = O ,
(11.4)
dG=I. From the last equation, since d2 = 0, we have the continuity equation
dI=O. 11.3.1
The vector potential and the gauge tranBfonnation
The first equation in Eqs. (11.4) says that F is a closed differential 2-form and, then, that there exists, locally, a differential 1-form A such that
F=dA. With the differential 1-form A we can associate, by using the metrics, a vector field, which is called the vector potential. The differential 1-form A is not uniquely defined, since we can add to A any exact differential 1-form df:
F
= dA = d(A
+ df).
The map
A -+A‘ = A + df is called a gauge transformation. Under space inversion
P : (2O,21 , 22 ,It:3 ) -+ ( 2 0 , - - & we have
we have
-22, -23),
Geometrical Identification of Electromagnetic Field in Space-Time
301
Thus, the potential A behaves like an ordinary differential l-form under parity t r ~ f o r m a t i o nand like a twisted ~fferential1-form under time reversal. These properties are in agreement with CPT theorem according to which photons are charge-odd particles; that is, the potential A is a twisted differential l-form under charge conjugation:
C : Aodx’
+ Aidxi -+
A&I?s” + A:ddi = -(AodzO +Aid&).
11.3.2 Const&diue equations Fields (J??,B,I?, 6 ) are not independent entities but are connected through pheno~enolo~ical relations
D=B(2,B],
H = H[#,d], which depend on the specific media under consideration.$ When conducting media are also considered, there is a generalized Ohm’s law
J = Jp3,dJ I
If we restrict ourselves to linear response media, the constitutive equations have the following form: (11.5) where E and p are invertible linear maps called dielectric map and magnetic ~ ~map, r~pectively, e ~ ~ ~ l ~ t ~ In the case of specific isotropic media, as the vacuum, Eqs. (11.5) takes the very simple form:
~
(11.6) where Z is the identity map. *The square bracket has been used to remind that the relations D = D [ e ,81 and H =
HfE,81 may not be local (hysteresis) and may not be linear.
Classical Electrodynamics
302
In order to put Eqs. (11.6) in a provisional geometrical form, we need a linear map between differential 1-forms and differential Zforms, from E to D and from H to B. If we think to E , B , H , D as differential forms on the manifold M = R3, the linear map, we are speaking about, can be given by the Hodge dual * : A(!R3) --+ A2(R3). Thus, the experimental relations (11.6) determine the Euclidean metric in B3 and can be written in the following geometrical form:
where the Hodge dual * is constructed out from the volume form S l = dx A d y A d z and the Euclidean metric g i j = Sjj. However, special relativity forces to think F and G as differ~ntial2-forms on the space-time manifold M = S4. Thus, the linear map is still given by a Hodge dual but, this time, constructed out from the volume form SZ = cdt A d x A dy A dz and the Minkowski metric gaj = q i j . Therefore, we obtain the constitutive equations in the following form:
In four dimensions, the Hodge dual
* :A2(R4) --+ A2(R4) determines the metrics up to a scalar function, so that the constitutive properties of the vacuum fix up the conformal Lorentzian structure of space-time.
Remark 22 the f o r m
It is advisable to observe that Maxwell's equations, written an dF=0,
dG=I, are invariant for any transformation in space-time. If we require that such t ~ ~ s j o ~ a t preserve ~ o n s the const~~utzve equations
then the symmetry transformation group reduces to the conformal group.
Geometrical Identification of Electromagnetic Field i n Space- Time
303
Finally, by using the codifferential operator 6, which, in M = R4,can be expressed as follows:
6 =-*d*, we have PO
PO
so that Maxwell’s equations in the vacuum can be written in the following form:
dF=0, 6~ =
-p* I,
&O
or
dF = 0 ,
(11.7)
11.3.3
The wave equation
By replacing F = d A in the second Eq. (11.7),we have
or
We can also write
where 0 is the Laplace-Beltrami operator which in the Minkowski space-time is given by a2 82 a2 a2 0= - -- -- ax; ax: ax; ax;.
Classical Electrodynamics
304
We, thus, find that the solutions of Maxwell's equations are closely related with the study of the wave equation.
11.3.4 Plane waves Let us look for a solution F of Maxwell's equations in the empty-space
d F = 0, d*F=O, of travelling wave type; that is, such that the fields @ and l? are functions of ( = x - ct:
E = E ( x - ct) , B = B ( x - C t ) . We have
d F = dxQA d E - d B , where
E
+ E,dy + E,dz
=Exdx
, B = B x d y A d z + B,dz
A
d x + B,dx A d y
Thus, d E = aEx -d[
A
at aE,
= -c-dt
at
dx + s d ( A d y
at A dx + s
+ Fd Ed ,t
Adz
d ( x - ct) A d y + aEz -d(z
- ct) A
at
at
dz ,
and
aBx d B = -d(
at
= -d(x
at
A
d y A d z + s d t ; A d z A d x + -d(
- ct) A
at
dy A dz
at
-c
A
d x A dy
s d t A d z A d x - c-dt
at
at
A d x A dy
Therefore,
dx' A d E = c-dt
at
A d z A d y + c-dt
at
Adz Adz,
.
Geometrical Identification of Electromagnetic Field in Space- Time
305
and
dF = dx’ A d E =
- dB
dE, aEvdt A dx A dy + c-dt
CxaBx - -d(x
=c
a<
->
A
dx A dz
- ct) A dy A d z -
(-aE, - aB,
A
dt A d z A d y + c
dz A dx - c-dt
($
a<
-k
3)
A
dx A dy
dt Ada: Adz
a<
In this way,
dF = 0 =+ B, = 0,
B, = E,,
B, = - E , .
Similarly,
d* F = 0
+ E,
=O,
B, = E , ,
B, = - E z .
It follows that a plane electromagnetic wave has transverse electric and magnetic fields that are determined by two independent functions, corresponding to the two independent polarization states. Further Readings 0
0
R. Aldovrandi and J. G. Pereira, An Introduction to Geometrical Physics (World Scientific, 1995). N . V . Balasubramanian, J. W. Linn and D. P. Sen Gupta, Differential forms on Electromagnetic Networks (Butterworths, London, 1970).
0 0 0
0
0
W. L. Burke, Applied Differential Geometry (Cambridge University Press, 1985). H. Flanders, Differential Forms (Academic Press, New York, 1963). C. Godbillon, Gbombtrie Differentielle et M2canique Analytique (Hermann, Paris, 1969). V. Guillemin and S. Sternberg, SympEectic Techniques in Physics (Cambridge University Press, 1984). B. Kostant, “Orbits, symplectic structures and representation theory,” Pmc. U.S.-Japan Seminar in Differential Geometry, Kyoto, 1965 (Nippon Hyoronisha Tokyo, Japan, 1965), p. 71.
306
0
Classical Electrodynamics
R. S. Ingarden and A. Jamiolkowski, Classical Electrodynamics (Elsevier, Amsterdam, 1985). P. Libermann and C.-M. Marle, Syrnplectic Geometry and Analytical Mechanics (Dordrecht, Reidel, 1987). A. Lichnerovich, “New geometrical dynamics,” in Lecture Notes in Mathematics (Springer, 1975), p. 570. G. M. Marle, Symplectic Manifolds, Dynarnicul Groups and Hamiltonian Mechanics (Reidel, Boston, 1976). J. Moser, “Various Aspects of Integrable Hamiltonian Systems,’’ in Progr. Math., Vol. 8 (Birkhauser, Boston, 1980). G. Pichon, Grovrpes de Lie: re‘ppresentation line‘aires et applications (Hermann Paris-Collection MBthodes). W. Thirring, A course i n Mathematical Physics (Springer Verlag, 1978). C. Von Westenholtz, Diflerential forms in Mathematical Physics (North-Holland, 1981).
Part IV
Integrable Field Theories
The last few decades have shown the exciting prospects of tackling nonlinear field theories (in two dimensions) nonperturbatively by exploiting their complete integrability properties.92193189J16 Let us recall that the concept of completely integrable Hamiltonian systems with finitely many degrees of freedom goes back to the last century.1359164 Some qualitative features of these systems remain true in some special classes of infinite-dimensional Hamiltonian systems expressed by nonlinear evolution equations; a short list of them being: , Korteweg-deVries equation ut ~ U z Uzxx = 0
+ + + 21% + uxxr = 0 iut + guzx + Iu12u = 0
modified Korteweg-de Vries equation
Ut
uxt
nonlinear Schrodinger equation
+ e" = 0
Liouville equation
+
Gt - d A (Gxz JF) = 0
Landau-Liftchitz equation
utt
- (1211~~ + azxz)z = 0
Boussinesque equation
uxt
+ sin u = 0
sine-Gordon equation
ut
- 2uuz - Huxx = 0
(Ut
Benjamin-Oto equation
+ UU, + uxxz), + 3a2uvy = 0
+ ar (ab > 0) iut + U,x + i((al2u),= 0 4 = feb
Ut
Kodamtsev-Petviashivili equation Toda potential derivative-nonlinear Schrodinger equation
- (U-1/2)xxx = 0
Harry-Dym equation
A further remarkable example of integrable evolution equation is given by the Burgers equation
ut
= 221212
+ uxx ,
(11.8)
which describes the heat diffusion and it is integrable by the Hopf-Cole transformation. It is worth observing that the evolution equation (11.8) does not correspond to a Hamiltonian dynamics but to a dissipative one. Nevertheless, also its integrability may be explained in terms of an invariant mixed tensor field. 79380
309
310
Classical Electrodynamics
A relevant progress in the study of these systems with an infinite-dirnensional phase manifold hi,was the introduction of the Lax ~ e ~ r e s e ~ t u t i ~ n , l 3 which played an important roIe in formulating the inverse s ~ t ~ ~ e e~ ~t g~ universally recognized as one of the most remarkable result of theoretical physics in the last decades, and of the “AKNS s ~ ~ e ~ e . This ” ~ ’ method allows the integration of nonlinear dynamics, both with a finitely or infinitely many degrees of freedom, for which a Lax representation can be given,l*lis this being both of physical and mathematica1 re1evan~e.I~~ On the other hand, the natural arena, for the analysis of the integrab~lity of dynamical systems, i s represented by the phase space that i s endowed with a natura~symplectic structure. In terms of this structure, the scattering data are interpretable as action-angle type variables. We shall see how the integrability of nonlinear field theories can be naturally explained in terms of mixed tensor fields, rather than symplectic structures, and how such tensors are linked to the Lax operators. The approach leads to a theorem of integrability that does not assume a finite number of degrees of freedom and, for a dynamical system with finitely many degrees of freedom, is equivalent to the classical Liouville theorem. As a working example we will use the Korteweg-de Vries equation ( K d V ) , which is the most-known completely integrable nonlinear field theory.
Chapter 12
KdV Equation
The equation ut
+ uu, + u,,,
= 0,
(12.1)
where u : ( z , t ) E R2 -+ IR is a numerical function depending on the variables 2,t , and the indices denote partial derivatives, was derived by D. J. Korteweg and G. de Vries120 in 1985 in order to describe shallow water waves moving in a channel without any change of shape,’’O and it is known as KdV equation, or simply, Kd V.
12.1 An Existence and Uniqueness Theorem
A travelling wave type solution of KdV, of the form u(2,t ) = s(x - C t ) ,
can be easily found in the hypothesis that s vanishes at the infinity together with its space derivatives. Indeed, setting ( = x - ct, KdV equation becomes
-cst
+ sst + sttt = 0, 311
KdV Equation
312
where the apex denotes the derivative with respect to (. Thus, integrating and using the boundary condition, we obtain 1 + -s2 + s/I = 0 . 2
-cs
A first integral of the above equation can be found multiplying by s', so that -CSd
1 + -S2d + s"9' 2
=0 ,
and integrating once again
+ A3 s 3 + d2 = 0 ,
-a2
whose solution, given by
is called a so2itay wave. Moreover, a uniqueness theorem can be easily proven'31 in the same hypothesis; that is, by assuming that u goes to zero at the infinity together with its space derivatives. Indeed, if u and v are two such solutions, ut+UUx+uxxx=0, 'Ut
+
'U'UX
+vxxx = 0 ,
their difference w = u - v will satisfy the equation Wt
+ UW, + v z w + W x x x = 0
Thus, multiplying by w, we obtain wwt
+
WUW,
4-w 2 21%
+ wwxxx = 0 ,
so that +W
(wwt
+
WUW,
+ w2vX+ wwxxx)dx= 0 ,
313
An Existence and Uniqueness Theorem
or
(12.2) Integrating by parts, it is easily seen that the fast integral vanishes
wwxxxdx= 0 , while
Therefore, Eq. (12.2) becomes (12.3) Thus, setting r+m
Eq. (12.3) implies d -E(t) 5 ~ E ( t ) . dt
Therefore,
E(t)5 E ( O ) P t , with
Lrn +W
E(0) =
+oo
1-
w2(x,0)dx 11:
[u(x,0 ) - v(x,0)l2dx.
It follows that ufx, 0) = v(x,0 )
* E(0) = 0 3 E(t)= 0 * u(x,t ) = v(x,t ).
We can resume saying that
For any given initial condition, U O ( X ) , an the class of C" functions defined on the real line and vanishing at infinity together with all space derivatives,
KdV Equation
314
there exists one and only one sol'ution u ( x ,t ) of KdV equation satisfying the initial condition, u ( x ,0 ) = uo(x).
12.2 Symmetries 12.2.1
S
~
~
es - ~t
~~ ~
~~
e
~
~
~
e
~
It is easy to see that KdV equation is invariant under Gaiilei transformation
{
x' = x + A t ,
t'=t.
Indeed, we have
au _.
-
au' ax' auiatt --
aut
where u' denote the composite function Ut(X',
t') = ufx'
- At',
t'),
Therefore, KdV becomes
or written in terms of the function zi = u'
+ A,
In conclusion, KdV equation is invariant under the transformation
[
z+z-t-At,
-+t, u+u-A,
(12.4)
315
Symmetries
12.2.2
Backlund transformation
Internal symmetries, as usual, also play a relevant role in the analysis of dynamical systems, as the following example well shows.
Example 36
Let us consider the Burger equation, given by Ut
+
= 22121,
Uxx.
I t is easy to verifg that, under the map (12.5)
Burgers’ equation becomes the heat equation vt = v,,
.
(12.6)
The map given by Eq. (12.5) is called the Hopf-Cole transformation, and can be used as follows. First, let us observe that from Eq. (12.5) we can obtain
and then
v,, = uxv + UV, = uxv + u2v = (u,
+ 2)..
(12.7)
Second, i f v is a solution of Eq. (12.6), then also v, is a solution of the same equation. The same is, of course, true for all higher derivatives. Therefore, starting with a solution v of heat equation, we can construct, at least, two solutions, namely u and a, of Burgers’ equation: vx
u=-,
-
u=-
vxx
vx
V
’
so that
-
vxx
(12.8)
uu=--. V
B y comparing Eqs. (127) and (12.8), we m a y write
-
u=-
u,+u U
2
Kd V Equation
316
which allows us to obtain a new solution f7om a given one, and constitutes an ezample of so-called Backlund t r a n s f o ~ a t a o n It ~ . expresses the invariance of Burgers’ eq~atzonunder translat~onsa ~ o the n ~5 axis. Similarly, Miura observed that, by performing the transformation
1 u = vx - -112 6 ’ KdV t r ~ s f o r m in s the so-called modified KdV equation (mKdV): Vt
1 - -v2vx + v x x x = 0, 6
which is manifestly invariant under the interchange
so that, if v is a solution of mKdV, then -v is again a solution. As a consequence, given a solution v of mKdV, we obtain two solutions u and ii of KdV: u=v,-
ii=vx+-v61
1
-u2,
6
2
F’rom previous relations, we may write li--=zv
’
iz+u=2ux,
so that (2
- u)x = -vvx 2 3
+
= &G@ u).
The above equation allows us to obtain a new solution from a given one, and consti~utesanother example of Backfund transformation. By performing a Hopf-Cole transformation
on the modified KdV, we have $22
+t f $xxx
- 3 - q$x
=0 .
(12.9)
Symmetriea
317
Then, by composing Miura's and Hopf-Cole's transformation, we obtain the map u = -6- $ x x
111'
for which the KdV equation becomes the more unpleasant Eq. (12.9). However, the look of Eq. (12.9) can be deceiving. Indeed, we stated that KdV can be written in the following form: ( u = - 6q 7 x x,
1+ $t
- 3-$x3
$X%
$xxx
= 0,
which ca.n be written as follows: 1
i(
.ttxx+jZ1$=O, 111t
+
4axxx
1 + uax + px) 111 = 0 ,
where the shorthand notation
has been introduced. Moreover, a change for the better can be done by using the GaIiIei invariance of KdV expressed by Eq. ( 1 2 . 4 ) : x+z+6Xt,
t+t,
In fact, it is easy to see that this change allows us to write the above system in the following form:
318
KdV Equation
so that, by introducing the operators
L = a,,
1 + -21, 6
1 2
- .a,
B = -4a,,,
- -21%
)
KdV equation assumes the following remarkable form:
L11,==W> ?j=B+.
(12.10)
~~nservation Laws
12.3
From KdV equation we have +m
lw +m
utdx = -
=-
(uuZ-t- u,,,)dx
so that
ddt J'"
udx = 0.
--oo
Thus, the functional 4-m
K ~ [ uz]
s_,
lLdX,
is a first integral of KdV. Another first integral is easily obtained by rnuItiplying KdV by u and by applying the same procedure. After an integration by parts, we obtain CW
uutdx = -
lw s_,
+
( ~ ~ 2 1 %uu,xx)dx
fw
=-
(u2uz - uxaxx)dx
Conservation Laws
319
Thus, we may write a second conservation law
A third conservation law is given by K3fUIf
2
1’” (g-
u:>
ds.
--m
Let us observe that the gradients Gi(u) = GKi/Su of previous functionals are given by
and that the following Lenard’s recursive formula holds131
a
-G,+I = Eke,, 8 X
n = 1,2,3,
(12.11)
where the operator Bk, expressed by
is ~ t i s y ~ e t rwith i c respect to the L2 scalar product. Moreover, Eq. (12.10) suggests that the eigenvahes of the Schrodinger operator
corresponding to a “potential” given by a solution u(s,t)of KdV, do not depend on time, so that these eigenvalues, considered as functionals X(uj of the potential u, are first integrals of KdV. The direct proof was given by Gardner et algebraically solving the eigenvalue equation u1,1011155
1
$x,
f p { x , t)$ = W)$
Kd V Equation
320
with respect to u: u
$ZX
=:
6X(t) - 6-,
$
and then by replacing the above expression in KdV equation. Thus, At$'
- ($Qz
- $zQ)x
=09
(12.13)
where
If II, vanishes when 1x1
00,
Eq. (12.13), integrated with respect to
2,gives
$2dx = 0, and then
X = constant.
12.3.1
&us ~
p ~ ~ e n ~ u ~ ~ ~ n
By taking the time derivative of the first equation of the system (lZ.lO), we can write
t$ + Ld = Ad, and by using the second equation, namely
L$
4= B$, we have
+ LB$ = AB$ = BL$,
so that we obtain
L$ = P,Lilt 9 where the bracket f., denotes the usual comm~tatorbetween operators. The reader can easily check that KdV is just represented by the equation a]
L=[B,Lj.
(12.14)
The above equation is called Lax representation of KdV and can be introduced for many other systems with an infinite-dimensionalphase manifold M .
Conservation Laws
321
The introduction of Lax representati~n’~~ has been a relevant progress in the study of integrable systems and has played an important role in formulating the inverse scattering method, ~iversallyrecognized aa one of the standard integration techniques.lol*8 Shortly, it consists in the fo~Iowing. Let M be some space of functions, chosen so that, for each t , the solution ~ ( tof) a generic evolution equation
G ( t } = A(%),
(12.15)
lies in M. Suppose that, with each function ZL in M , we can associate a self-adjoint operator L(u -+L ) over some Hilbert space, in such a way that, if u changes with t according to Eq.(12.15)jthe operators L(t),which also change with t , remain unitary equivalent to themselves:
L(t) = u(t)L(o)u(t)-l j
(12.16)
with U ( t ) denoting a 1-parameter family of unitary operators. By taking the “time derivative” of the above equation, we have
L= [B,L],
(12.17)
where the skew-symmetricoperator
B = uu-1 is the generator of the family U(t). The above equation, is calIed Lax representation of the dynamics given by Eq. (12.15}j and the pair ( B ,L ) is called a Lax pair. A consequence of Eq, (12.16) is that the eigendues of the operators L(t) do not depend on t . Indeed, let us consider the eigenvalue equation for the Lax operator at time t = 0:
(12.18)
Kd V Equation
322
or equivalently
where
Therefore,
j, = 0 , $ ( t ) = B $ ( t )
(12.19)
*
For this reason, Eq. (12.16) is called an isospectral flow. 12.3.2
The inverse scattering method
Let us show how, by taking advantage of the G e l , f a n d - L e v ~ t a ~ - ~ ~ ~ h e n k o formu1a102~143~118~14 Lax representation allows us to solve the given evolution equation (12.15). Let ua(z) be the initial condition; i.e. uo = u(x,O), and LObe the associated Lax operator. Suppose that we are able to solve the corresponding eigenvalue problem
Lo$@= k % p , that is to find 0
the free states (continuous spectrum); i.e. the states, corresponding to k2 > 0, represented by waves $O(x, k), whose asymptotic behavior is given by
$'(x,k) $O(x,
=+YmCo(k)exp[-ikx] , k) =+TmCO(k)exp[-ikz]
+ Cy(lC)exp[ilCx),
where
~ t the r e ~ e c t ~c o ~ e~c~e~t, are called the t r ~ ~ s ~ ic so es ~i c~i e~and respectively;
Conservation Laws 0
323
the bound states (point spectrum); i.e. the states, corresponding to Ic2 < 0, represented by eigenfunctions $:(q kn), with ki = (or better, k, = ix,, X, > 0 ) , whose asymptotic behavior is given by
--xi
$(:,.
i x n ) z + L e x p [ ~ n z7 ]
$(,:.
ixn)
where the coefficientsc:
z ~ c c:(xn) o
e x p [ - ~ n ~i ]
(xn) are called the normalization constants.
The set
so= {xn,&Xn),
RO(WV k E
is called the set of scattering data. Of course, before solving the evolution equation, only the form of the operator B generating the isospectral flow is known, but not its explicit dependence on (5,t ) , However, as it will be explicitly shown in the case of KdV, the simple knowledge of the asymptotic behavior of the operator B is enough to determine the scattering data, namely
s = {xn, cn(xn,t), R ( k , t )vlc E 8) of the Lax operator L, associated with the solution u ( z , t ) of Eq. (12.15), corresponding to the given initial value uo. The knowledge of scattering data of the operator L allows us, by means of the Gel'fand-Levitan-Marchenko formula (GLM), to explicitly write L and then u(z,t ) . We can synthesize the described procedure with the following picture:
I'
The Kd V case For KdV, the Lax pair is given by
L = a,,
1 + -u(z, 6 t) ,
KdV Equation
324
1 2
B = -.laxxx- U ( Z , t)a, - - u ~ ( zt ,) . Let us consider the eigenvalue problem
L$ = k2$ , which will have as solutions
f i e states, corresponding to k2 > 0, represented by waves +O(x, k , t ) , whose asymptotic behavior is given by
where
are the unknown t r a ~ s m ~ s s coeficient ~on and the reflection coefficient, respectivefy ; bound states, corresponding to k2 < 0, represented by eigenfunctions $ : ( x , k n ) , with ki = -x$ (or better, kn = iXn, xn > 0), whose asymptotic behavior is given by N
$n(Z,
ixn,t)x4--00 exPlxn4
$3.1
ixn, t ) zf;bo
Cn(Xn7
1
t>.XP[-Xn.I
1
where the coefficients c i ( x n ,t ) are the unknown normalization constants. The set
s= { x n , cn(xn,t.),R(k,t )
1
V k E Ill}
is the set of scattering data which can be determined by the asymptotic behavior
of the operator B
Conservation Laws
325
Indeed, the equation
$=B$, given in Eq. (12.19), will be true also ~ymptotically
400 = B&, J
9
so that we can write e
for bound states &(xnrt) @XP[-Xn4 = 4X%n(Xn, t )exP[-Xn4
r
which is trivially integrated to 4 X n , t ) = &xn> exp[4x3ntI ;
for fTee states
C(k,t)e-+kc = -4ik3c(k,
t)e-ik”,
C-(k,t)e-ik” + &+{ktt)eikx= -4ik3fC- (k, t)e-ikx - C+(k, t)e’ik”2] , which gives
C(k,t ) = -4ik3C(k, t), C-(k, t) = -4ik3C-(k, t) ,
C + ( k ,t) = 4ik3C+(k,t) , and then,
T ( k ,t ) = TO(k), R(k,t) = Ro(k) =p(8ik3t]
.
At this point, we know the time evolution of scattering data when the “potential” changes according to the KdV equation. We also know that the number of bound states does not change in time and is determined by the initial condition UO(Z). In our time-dependent case, the GLM formula (see Appendix G) must be written in the form
A ( z ,y,t )
+ F ( z + y, t ) +
1
CQ
X
+
A ( z , z, t ) F ( z y, t)dz = 0,
326
Kd V Equation
and the solution of KdV will be given by
d dx
U(Z,t ) = - ~ - A ( x , Z,
t ).
The o ~ e - s o s~ ~ o~ o~~ Let ~ us~ consider ~ o a ~t r ~~ n ~ p ~~o ~t ennt t ~i.e. u ~an initial condition UO(Z),such that the direct scattering problem gives Ro(k)= 0. Suppose, moreover, that there exists only one bound state, with eigenvalue -x2 and normalization constant CO. The kernel F of GLM will be given by ~ ( 2 t ),= c;e4x3t-xz
,
so that GLM gives
or
Since this kernel is separable, we can try to find a solution of the following form:
so that the equation simplifies to
and algebraically, we obtain
Conseruation Laws
327
Thus, we can write 2 4x%-x(xSy) 2 co 4 x 3 t - 2 x x
COe
A ( x , y , t )= -
7
1+-e 2%
and since A(%,y , t ) is continuous and differentiable at y = z, C$e4X9t - 2 x x
A ( z , ~ , t=) -
2
1 + Cge4x3t-2xx
2X Finally, after simple calculations, we can write
d dx
U(Z,t ) = -2--A(z,
Z, t ) =
-
2X2 cosh2[x(x - 4x2t) - 61 '
The N-solitons solution. Let us again consider a transparent potential, but this time, with N eigenvalues -$ and normalization constant c;. Then, the kernel of GLM equation is given by F(Z,
t)=
C ci(xjlt)e-xjz , j
and we can try to find a solution of the following form: hj(x, t)e-xjY .
A ( z ,y , t ) = j
By replacing the above formula in the GLM equation, we obtain
The vanishing of the above sum implies the vanishing of the factor in the square brackets, so that performing the integral where contained, we obtain
The above equation can be rewritten as follows:
Kd V Equation
328
or by introducing the matrices
M = (mi$), K
= (e-xj"),
H =( ~ ~ ( ~ , t ) ) ,
with
in the following compact form:
MH=--K. Since det M # 0, we have
H=M-lK. The reader can perform remaining calculations to have the following simple final solution:
u(x,t ) = -2aX. In 1 det MI. 12.4 KdV as a Hamiltonian Dynamics
It was observed by Gardner that KdV is a Hamiltonian dynamics with infinitely many degrees of freedom. Indeed, Eq. (12.1) can be written in the following form. ut = -ax-
6K3 du '
where the functional K3[21]is given by
Thus,the "time derivative" of any Fkechet differentiable functional F[u] is given by
where GF I= SF/Gu and G3 3 6K3/6u are the gradients of F and K J ,respectively, and .) is the Lz scalar product. (0,
KdV
ap
a Hamiltonian Dynamics
329
It is easy to check that, for any two Frechet differentiable functional F and F', the bracket
on the chosen class of functions, is antisymmetric. Moreover, it satisfies the Jacobi identity and is a derivation on the associative product of functionals. Thus, we can conclude that KdV is a Poisson dynamics whose Poisson bivector field A has components, in the formal basis e(z) = 6/6u(z), given by the operator 8,. The operator ax has a kernel given by constants c E IR, so that it does not have an inverse, and KdV is not strictly a Hamiltonian dynamics. However, on the quotient manifold, namely M , KdV is a Hamiltonian dynamics; indeed, the manifold M can be endowed the symplectic structure q u ]=
s'" -"
dx
[; dY[6+)
A
WY)]
9
which having constant coefficients, is trivially closed. Moreover, it is easy to show that functionals Kn, whose gradients are constructed by means of Lenard's sequence (12.11),
a
-Gn+1 dX
= EkGn 1
n = 1,2,3,
are pairwise in involution. Indeed,
Thus, if n < m, an index k can be found such that
with A, one of the two antisymmetric operators 8, or Ek, so that
A further important step on the study of KdV, is represented by the following result by Faddeev and Zakharo~.'~
Kd V EquatQopt,
330
The sympIe~ticform w and the ~ ~ m i l t o n i a&, n once expressed in terms of the scattering data S = { R ( k ) , xz,cn(k,)) via the GLM transformation,
U 3 S , become +w
G J ( k ) A S@(k)dk+
C 6Ja A 6
~ ;
i
and
respectively, where
k J ( k ) = --ln(1 - l R ( h ) ( 2 ) @ ( k=) argC+(k,t), 7r
Ji = xf , pi = Zlnbi, with
I
Ik=iXj
Thus, ( J ( k ) ,@ ( k ) ,Ji,cpi) are Darboux’ coordinates for the symplectic form w. The expression of the Hamiltonian K3 suggests that they play the same role of usual action-angle coordinates for Hamiltonian systems with finitely many degrees of freedom.
12.5
KdV as a Completely Integrable Hamiltonian Dynamics
A crucial step in the study of KdV was finaliy performed by F. Magri137~138 who observed that KdV is a Poisson dynamics also with respect to the Poisson bivector field A k whose components are given, in the formal basis e(z) = b/Su(z),by Lenard’s operator &. Indeed, KdV can be written in the form
KdV as a Completely Integrable Hamiltonian Dynamics
331
and the bracket
{F,F'} 3 ( G F E , ~GF~) satisfies the Jacobi identity, so that the above bracket, being Ek an antisymmetric operator, is a Poisson bracket. The Jacobi identity can be verified directly or, indirectly, by inverting the operator Ek on the phase manifold quotiented by its null space. Thus, on the quotient manifold N,we can define a nondegenerate functional 2-form by
wju](X[UI,YI.1)
Jw(4)
= (X('LL),
1
(12.20)
where X ( u ) and Y ( u ) ,shortly denoted by X and Y , are C"" numerical functions defined on N and representing the components of vector fields X [ U ] , Y [ u ]E TUNin the basis e(z) = 6/6u(z); i.e. +m
d
X(U)-dz, W X ) +m
Y[UI=
s_,
s Y(u)--dx. a+>
The exterior derivative 6w' of the functional 2-form w' is given by 3
C
awjUi(Xi[~],Xz[~],X3[~]) = ~ i j r c ( X i( ,E i l ) u ( X j , X k ) ) , i,j , k = l
where
+
d) - ' ( u X X i ) ) X j l ~ = o . ( E ~ l ) u ( X ~ ,= X j-(Ek dX
Since EkEi'X = X , we have
(Ek)u(xi) EF'Xj) + Ek(Zr')u(Xai x j ) = 0
7
so that
( E ; ' ) u ( x i , X j ) = -E;l(E&(Xi, EL'Xj). By using the expression of Ek, given by Eq. (12.12), we obtain 2
(Ek)u(xi,E,-'Xj) = -Xi(&E;lXj) 3
+ -31( E ; l X j ) ( a z x i ) ,
(12.21)
KdV Equation
332
and then
so that swj,, = 0 .
Thus, the wiU1is a symplectic structure on N and the interior product is given by
~AW'
that is
so that KdV is, on N,a Hamiltonian dynamics also with respect to w' We can define an operator i" such that
gk
=Z
&p,
which allows us to write the Lenard recursion of gradients of conserved functionals K,(u] in the following form:
On the quotient manifold M , the operator ax can be inverted to give
D-'P(x) and we have
=
L[(
cp(z:)dx-
1'" X
~ ( x ) d x ) , Vtp E M ,
KdV as a Completely Integrable Hamiltonian Rpamics
333
Between the operator rf' and the Lax operator L = exists the following remarkabte reiation:
L$ = A$
===$- T$2
ax,+ (1/6)u, there
= 4Xzf12,
(12.24)
that is, if is an eigenstate of the Schrodinger operator belonging to the eigenvalue A, then $a is an eigenstate of i" belonging to the eigenvalue 4X. Indeed, if L$ = A$, we have $xx
1 + -gu$ = A$
and
= 29: = 2$Z
1 1 + 2X$2 - -u$J' + p$2 + 4X 3
+ 4X$2 - 2$:
L:
$$)ydy - 4 J x
$v$l.'%ratdy
--oo
= 4Xzf12.
Let us observe that the relation, expressed by Eq. (12.24), does not depend on the fact that u satisfies the KdV equation. Moreover, by supposing that the @'s are normalized, we have
so that
and this implies that
Kd V Equation
334
We can conclude that the eigenvalues of L are eigenvalues of and that the associated eigenstates of T are the gradients of the eigenvalues
6X 6X = 4X-. 6% 6U Let us now consider the Poisson bracket of two functionals K and K', "
T-
(k,k'}
= fG,axG')
(12.25)
>
and suppose that the function~sare in invo~uti~n; that is, their Poisson bracket is vanishing:
(G, O,,G') = 0 .
If G,, and GLu denote the derivatives at the point u of G and G', the derivative, with respect to u, of the above equation gives (G*,(~%),8,G') The operator G, : T M product, so that
+ fG, OXG~,(6a))= 0 , -+ T M
6% E T,M
.
is symmetric with respect to the La scalar
(&a,G*O,G') - (G:,aZG, 6 ~ z=) (621,G,,O=G'
- G',,a,G)
=0 ,
or equivalently G,,O,G'
= G',,azG.
(12.26)
If G' is the gradient of &, the left-hand side of Eq. (12.26) is -G,where the dot denotes the derivatives of G along integral curves given by the solutions of KdV, while the right hand side is given by FG, where F = a,,, ua,. Thus, we may write
+
G=-A~G,
-
(12.27)
where A = -a,,, - ua, u,, the adjoint of the operator F , is the derivative of dynamics. The "time derivative" of the sequence expressed by Eq. (12.22) gives G,+I = +G, -tpGn, so that, by using Eq. (12.27), we have
TGn = [T,At]G,,
Kd V as a Completely Integrable Hamiltonian Dynamics
335
where the bracket [. , . f denotes the usual commutator. The reader can easily check that
?' = [-At, f']. The analogy between L and
(12.28)
can be resumed as folfows: "
6X
T-
6U
6X
=: 4X-
624
L = [B,L],
?'= [ - A t , F ] ,
.rl=BlCI,
G = -AtG.
~ q ~ t i (12.28) o n can dso be written in the fo~lowingform:
rf, = [ A , T ],
(12.29)
where the operator f' is the adjoint, with respect to the scalar product Lz,of f' = D-'Ef$
T*= a,,
2
. f-u. 3
1 3
+-u,L?-1*
.
(12.30)
Chapter 13
General Structures
In spite of its success as an integration algorithm, a compact a priori criterion of integrability in terms of Lax pairs is, to date, lacking. On the other hand, the inverse scattering method being a transformation from generic coordinates (potentials) to action-angle variables,g2 makes it only natural for us to state an integrability criterion, for soliton equations, by looking at them as dynamical system on an infinite-dimensional phase m&fold. 137,138,103,179,168,81,82,144,78,80,147,73~100~166 This point of view is also suggested by the occurrence in these remarkable systems of a peculiar Opera~or174,175,137,132,138,103,179,104,81,82,l44,78,80,147,73,100,20,162,98,99,ll7,67 relevant for the effectiveness of the methods, which naturally fits in this geometrical setting as a mixed tensor field on the phase manifold M . 13.1
Notation and Generalities
Many geometrical concepts, introduced in Part 11, can be extended to infinite dimensional manifolds, whose local model is an infinite-dimensional topological vector space, if the necessary care, connected to the passage from finitedimensional case to the infinite-dimensionalone, is taken. Many properties of the finite-dimensional case, still hold, in the infinite dimension, only if the considered manifold is a Banach manifold; that is, a manifold locally homeomorphic to a Banach space. The reason is that the 337
General Stmctwes
338
implicit functions theorem does not hold in an arbitrary topological vector space. Given a nonlinear operator A; i.e. a function of u and its space derivatives, the generic evolution equation
8th
- = A(u) at
(13.1)
will be considered as a dynamical equation on the functional space M of field functions u ( x ,t ) regarded as functions of the space coordinates only, defined on the whole real axis, and satisfying suitable boundary conditions. The rate of change, along the solutions of Eq. (13.1),of any functional F[u] will be given by
d dt
-F[u]
=
1,
+"6F
Lm $00
8u - -dx du at
=
6F dxA(u) * 6u(x) l
(13.2)
and then, by the action of the operator
s_,
+m
A[u] =
6 6u(x)
~ x A ( u* )
which will be called the dynamical vector field. In the functional space M , first order differential operators S/6u(z) constitute a basis for vector fields. The dual basis is given by the variations 6u(x) and, as it is usual
where 6 is the Dirac delta function. Then, any vector field X [ u ]can be written in the following form:
6 dxX(u)6u(x) l and dual vectors or covectors a[u]as follows:
L m
a[.] =
dya(u)Su(y).
The contraction ( a ,X ) between vectors and covectors will be given by 00
Notation and Generalities
339
The rate of change, along the solutions of Eq. (13.1), of a vector field X [ u ] is given by +m
6
[Xu A(u) - A, X ( U ) ]6u(z) *
1
dt
(13.3)
where the operators Xu and A,, defined by
d
X,C~:= -X(U d&
+
d
E(P)[~=o,
A,(P := -A(u d&
+
E(P)[~=o,
are the week derivatives, or Gateaux derivatives of X ( u ) and A(u), respectively. Let ua observe that Eqs. (13.2) and (13.3) correspond to the usual Lie derivatives, with respect to A[.], of F[u]and X [ u ] respectively. , So such time derivatives will be denoted* by L AF and L A X ; LA just being the Lie derivative operator with respect to A. We notice that Eq. (13.3) can be written in the form
where the bracket denotes the usual commutator between differential o p erators. The tangent space and the cotangent space of M in u, will be denoted by 7,M and C M , respectively. In the continuous (formal) frame 6/6u(z)and coframe 6u(z),the evolution equation can be regarded &s an ordinary differential equation, [a,
a]
du = A[u] dt In order to simplify notations and formulae, in the following a vector field
X [ u ]will be identified with its components X ( u ) and a mixed tensor field T with its associated endomorphisms T , or defined by T(a,X ) = (a, PX)= (Pa,X ) . These endomorphisms will be, in general, represented as operators acting on vector fields or their dual. *Henceforth, to avoid confusion with the Lax operator L,the Lie derivative with respect to a vector field X will be denoted by Cx.
General Structures
340
Thus, with abuse of saying, the Lie derivative L A X of a vector field X with respect to A will be identified with X u . A(u)- A, X ( u ) , and the symmetries X of a given dynamics At will be given by the solutions of the following linear differential equation:
-
X u . A(u) - A, . X ( U )= 0 . The Lie derivative, with respect to A , of an operator T ; that is, of an endomorphism on vector fields associated with a mixed tensor field, will be given by the operator or endomorphism C A T given by
-
,CATV = t ( A , 'P) -
[&,T]'P,
where (13.4)
Therefore, the invariance with respect to the dynamics of such a tensor field will be expressed as*
13.1.1
Backward to KdV
In the case of the KdV equation, M is the manifold of C" field functions u, considered as functions only of x , and vanishing at the infinity together with its space derivatives. The dynamics is given by the vector field
so that the solutions of KdV correspond to the integral curves of A. Let us observe that Eq. (12.29) is simply the expression, in local coordinates, of the invariance, under the KdV flow, of the mixed tensor field T , defined by t A very general and fundamental approach to the analysis of symmetries of nonlinear partial differential equations, is described in Refs. 182 and 30. tEquation (13.4),in spite of its form, does not correspond, generally, to the Lax representation. A possible tensorial version of this has been given by several authors, some of them in the context of alternative L a g r a n g i a n ~ ~ ~ 'or , ' ~in reading it has the vanishing, along the dynamics, of the covariant derivative of a section of an M-based bundle.81
Notation and Generalities
341
T ( a , X )= ( P X , a ) = (X,?Q), Q E T*M, X Indeed,
E
TM.
(13.5)
&. (12.29) can be written, in geometrical terms, as follows: CAT=O,
(13.6)
where &A is the Lie derivative with respect to A. The tensor T , which in local coordinates can be written in the form (13.7)
satisfies the condition
TU(PX,Y )- PU(PX,Y )= P[PU(X,Y )- TU(X,Y)I ,
(13.8)
which is the analogue, in this i n f i n i t ~ d ~ e n s i o nsetting, al of Eq. (6.483. The above condition can also be written as follows:
(LpxT)"Y = T(CxT)"Y , X , Y E 7 , M .
(13.9)
We recall that Eq. (13.9), or Eq. (13.8), is called the Nijenhuis condition or the ~ ~ j e ~ ~~ rU~i c$and ~ ethat ~ ,the tensor field NT[U](a,x, y )=
(a, (L.fxT)"y- P(LXT)^y>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<'
Wij
=
0
.
(16.16)
I f p ( w ) = 1, then dim S = (m,m) and there exist local coordinates such that
w =duiAdc,
wij =
(
':)
. -1, By having a Poisson structure, we can deal with Hamilton equations. From Eq. (16.12), if H is the Hamiltonian, the corresponding equations are (16.17)
Now we would like to maintain the possibility of explicitly constructing the flow of Eq. (16.17). This requires that the dynamical evolution be an even vector field. In turn this implies that the Poisson structure and the Hamiltonian function should have the same parity so that in particular, with an odd Poisson structure, we need an odd Hamiltonian function. Before we analyze the graded Nijenhuis condition, we will study a few examples.
Integrability of Fermionic Dynamics
396
Example 43 (Bosonic -Fermionic oscillator) The mized bosonic-fermionic harmonic oscillator in (2,2) dimensions is described with coordinates ( q , p ,q, () and has the following equations of motion:
(16.18)
Equations (16.18) can be given two Hamiltonian descriptions, The Hamiltonians are the usual even one
H = ;@2+42)+i
(16.19)
and an odd one
K
(16.20)
= PE -k 477 7
while the two Poisson structures am respectively 0 A H = ( - ;
o
1
0
0
:a
I)
o
i
o
i
and
We can construct a mixed invariant tensor field T by
T
=: W H o AK =
0
0
1
0
0
0
0
1
0
4
0
0
i
0
0
0
(16.23)
Powson Supermanafold
397
However, this odd tensor field ( p ( T ) = 1) is not a recursion operator. Indeed, it is easy to check that
so that T d K = dH
, d(TdH)# 0 .
(16.24)
If we evaluate the Poisson brackets of the coordinate variables, by using the two symplectic structure given by Eqs. (16.21) and (16.22), we find that {q,p)H = 1 >
{P,q}H = -1
9
{V,q}H = i
{<,<}H = i 7
(16.25)
{7),P}K = 1 >
(16.26)
and {Q,<}K = 1 ,
{ < , Q } K = -1
9
{P,V)K = -1
>
the remaining ones being identically zero. W e see that the sum {., .}+ of the two structures is itself a Poisson structure with the property
{F,G}+ = -(-l)P(F)J’(G)
{G,F}+ >
but it has not definite parity. Moreover the bracket
(
0
,
(16.27)
.}+ is degenerate.
Example 44 (Witten dynamics) Interesting examples come f r o m superthat the dynamics of Witten’s symmetric dynamics. It has been Hamilton systems184 can be given a bi-Hamiltonian description with an even Poisson bracket and Grassmann even Hamiltonian, or with an odd bracket and Grassmann odd Hamiltonians. Instead of considering the general case we shall study a supersymmetric Toda chain with coordinates (q,p , q, <). The even Hamiltonian is given by
H
1
= z(p2
With the even Poisson structure
+ eq) + Ji57/e$ . 2
(16.28)
Integrability of Fermaonic Dynamics
398
the equations of motion read
(16.30)
Then the following functions are constants of the motion
(16.31)
W e can use K ( o r L ) in Eq. (16.31) as a n alternative Hamiltonian function. The corresponding symplectic structure is given by WK
= dq A d<
+ dp A dq(e-4q) + dp A dq(-2e-4) + df A d H ,
(16.32)
where f ( q , p , q , [ ) is a function explicitly given by
f
= A[
+ Bq
with
(16.33)
The symplectic structure W K can also be written in the following form: WK
= d{dq(-t)
+ dp(2e-$7) + f d H } .
If r is the dynamical vector field of the Toda system, as given by Eq. (16.30), then, the function f will satisfy the relation irdf = e-qI2q and this, in turn,
Poisson Supemanajold
ensures that
+WK
399
= dK.
It take some algebra to check that the (1,1)-tensor
field
T =WX 0AH,
(16.34)
is such that
TdH = d K , d(T2dH)f: 0 .
(16.35)
Again, the tensor field T in Eq. (16.34) is not a recursion operator 16.3.1
Super N'jenhuis torsion
Let us recall that one of the most relevant consequences deriving from a (not graded) (1 - 1)-tensor field T,with a vanishing Nijenhuis torsion, is the possibility to generate sequences of exact differential l-forms according to
NT = 0 , d(TdF) = 0 ===+ d(TkddF)= 0 .
(16.36)
The above relation is a consequence of the identity
X AY Id(T2a)= { X ATY
+ T X A Y } Id(Ta)- { T X A T Y } Ida
- 3CT(X,Y ) Ia: *
(16.37)
in which a is any differential l-form. Indeed, by assuming that both a and Ta:are closed, Eq. (16.37) implies that T2ais closed if and only if 311. = 0, namely if and only if the Nijenhuis torsion of T vanishes. Let us analyze now the graded situation. Suppose T is a graded (1,l)tensor field that is homogeneous of parity ~ ( 7 ' )Then, . if a is any differential l-form, by using the definition Eq. (16.5), after some (graded) algebra, the analogue of Eq. (16.37) reads
x A y 1_ d(T2a)= {(-l)p(T)p(Y)XA TY + (-l)p(T)[p(X)+p(Y)lTX Ay} Id(Ta)- (-1) P(T)[P(X)+P(T)lTX A TY Id a - ( - l ) p ( T ) G 3 C ~ ( x Y, ) i Q
+ ( - l ~ p ( ~ ) p ( x-) ~( -l l ) p ( T ) ] ~ ~ xI~a), TY where G 7 . 1 ~is defined in Eq. (16.9).
(16.38)
400
Integrability of Fermionic Dynamics
It is clear then, that for a (1, 1) odd tensor field a (1, P)-tensor field corresponding to its super Nijenhuis torsion can be defined only whenp(T) = 0. The same result is attained with the use of the general approach dT o dT = 0. Summing up, we have shown that there are examples of dynamical systems whose dynamical vector field I? admits two Hamiltonian descriptions, odd and even, respectively, and that the tensor field T , constructed out of the corresponding Poisson structures is not a recursion operator since it cannot generate new integrals of motion after the first two ones. We have also shown that this fact is general and that for a generic-graded (1, 1)-tensor field T , a graded Nijenhuis torsion cannot be defined unless T is even. F’rom the nature of the proof it seems plausible that a similar theorem should hold true also in infinite dimensions. The no go theorem we have proved in our paper does not exhausts, obviously, the analysis of complete integrability for graded Hamiltonian systems. Much more attention must be paid, however, in generalizing to the graded case geometrical structures that play a relevant and natural role in the nongraded situation.
Further Readings 0
0
0
0
M. Batchelor, “Graded Manifols and Supermanifolds,” in Mathematical Aspects of Superspace, Clarke, C. J. S., Rosenblum, A., Seifert, H.-J. eds. (Dordrecht, Reidel, 1984). M. DuboisViolette, “DQrivationet calcul differbntiel non commutatif,” C. R. Acad. Sci. Paris 307 SJrie I 4 0 3 (Paris, 1988). B. A. Dubrovin, Geometry of Hamiltonian Ewolvtionary S y s t e m (Bibliopolis, Naples, 1991). Faddeev and V. E. Korepin, L‘QuantumTheory of Solitons,” Physics Reports 42C, 1 (1978).
0
S. P. Novikov, S. V. Manakov, L. P. Pitaevskii, and V. E. Zakharov, Theory of Solitons: The Inverse Scattering Method (Consultants Bureau, New York, 1984).
Appendix A
Lagrange: A Short Biography
Lagrange is considered one of the greatest mathematicians of the modern age and it is jmpossibIe, in a few pages, to quote his enormous contribution to mathe~aticsand physics. Thus, we shajl limit ourselves to a short b i o g r ~ p h i ~ note. Giuseppe Luigi Lagrangia was born in Torino on January 25, 1736, and died in Paris on April 10, 1813. At the age of 19 he aIready w w Professor of Mathematics at Artillery’s School in Torino and soon after associate founder of Sciences of Torino. The first fruit of Lagrange’s works here of the A c ~ ~ m y wits his letter, wrjtten when he was still only 19, to Euler, in which he solved the isoperimetrical problem, which for more than half a century, had been ;t subject of discussion. “To effect the solution he enunciated the principles of the calculua of variations. Euler recognized the generality of the method adopted, and its superiority to that used by h i m s e ~and ~ with rare courtesy he withheld a paper he had previously written, which covered some of the same ground, in order that the young Italian might have time to complete his work, and claimed the undisputed invention of the new calculus”.46 Most of Lagrange’s early writings are to be found in the five voIumes of sacti ti#^ of Turin Academy, usually known as the ~ ~ s c e ~ l ~u a~ e~ ur ~ n e n s ~ ~ . The first volume contains a memoir on the theory of sound propag~tion. In this he indicates a mistake made by Newton, obtains the general differential equation for the motion and integrates it for motion in a straight line. In 401
402
Lagrange: A Short Biography
this volume is to be found a complete solutions of the problem of a string vibrating transversely. In particular, the article points out a lack of generality in the solutions previously given by Taylor, D’Alembert, and Euler; arrives at the conclusion that the form of the curve at any time t is given by y = a s i n m x s i n n t , and concludes with a masterly discussion of echoes, beats, and compound sounds. In this volume, other articles concern recurring series, probabilities, and calculus of variations. The second volume includes remarks on the theory and notation of the calculus of variations, already discussed in the first volume, the derivation of Least Action Principle as an illustration of the method, and solutions of various dynamical problems. The third volume, besides the solutions of additional dynamical problems by means of the calculus of variations, and some articles on the integral calculus, includes the general differential equations of motion for three bodies moving under their mutual attractions. In a word, in 1761 Lagrange stood without a rival as the foremost mathematician living. In his paper in 1764, on the libration of the moon, he explains, with the aid of the Principle of the Virtual Work, why the moon always turns to the earth the same face. Here there was already, in germ, the future generalized equations of the motion. “In 1766 Euler left Berlin, and Frederick the Great immediately wrote expressing the wish that ‘the greatest King in Europe’ t o have ‘the greatest mathematician in Europe’ resident at his court. Lagrange accepted the offer and spent the next twenty years in Prussia, where he produces, not only the long series of memoirs published in the Berlin and Torino transactions, but his monumental work, the Me‘canique Analytique” .46 Indeed, during these 20 years, Lagrange contributed one memoir per month, on the average, to the Academies of Berlin, Torino, and Paris. All his memoirs are of high scientific level. Moreover, some of them are actually treatises. Among the ones sent to Paris it is worth to mention the memoir on the Jovian system (1766), the essay on the three body problem.(1772), the article on the secular equation of the moon (1773), and the treatise on cometary perturbation (1778). For all these memoirs, the Acadkmie the fiance, who had proposed the subjects, awarded a prize to Lagrange. In 1787, after the death of Frederick, Lagrange “who had found the climate of Berlin trying, gladly accepted the offer of Louis XVI to migrate to Paris. He received similar invitations from Spain and Naples”.46 The decree
Lagrange: A Short Biography
403
of October 1793, which ordered all foreigners to leave F’rance, specially exempted him by name. He was offered the presidency of the commission for the reform of weights and measures and the different revolutionary governments loaded him with honors and distinctions. In 1795, Lagrange was appointed to a mathematical chair at the newly-established dcole ~ o ~ u lwhich e , enjoyed only a brief existence of four months, and in 1797, he was made professor at the l$cole Polytechnique. In appearance Lagrange was of medium height, and slightly formed, with pale blue eyes and a colorless complexion. In character, he was nervous and timid, he detested controversy, and to avoid it willingly allowed others to take the credit for what he had himself done. Indeed, no inconsiderable part of the discoveries of his great contemporary, Laplace, consists of the application of the Lagrangian formulae to the facts of nature. Even the introduction of the mornentu and of the P o ~ s s obracket ~ occur in the writings of Lagrange as well as the theory of ~ d ~of the c ~t r o~ b of ~ ne ~bodzes. ~
Appendix B
Concerning the Lie Derivative
Let us observe that from
it follows that
or equivalently,
Then, we may write
404
Concerning the Lie Derivative
405
h=l
axk
= --
0x6
‘
In this way, multiplying by 8 x i / 8 x k and summing over k, we obtain
and then
Therefore,
Appendix C
Concerning the Kepler Action Variables
The two closed curves of integration in the integrals
are fixed by requiring the vanishing of the corresponding velocities or, better, of the corresponding momenta p8 and pr expressed, of course, in terms of variables re and rv. In this way, the integration limits are fixed by
Therefore, the “6” integration must be performed between the limits and 192 given by the solutions of
$1
where Eq. (4.25) has been used. Since 19 itself always lies between 0 and r, where sin8 > 0, we have sin291 = sin& = COSLY. Thus, the integration goes 406
407
Concerning the Kepler Action Variables
from 81 = n/2 - a to 7r/2 to 8 2 = n / 2 + a and again back to 61; the sin6 goes from cosa to 1, then to COSQ. In this way, we obtain a - cos28 d 8 2x8
--
IT
sin2a
*/2
1
cos2r dr, 1 - sin2 a sin2 r
with r defined by cos 8 = sin a sin r
Therefore, with x
5
t a n r , we have
Jo=y+m[,,+ 1 1+x2
7r
=
27r# 7r
cos2a
1
2 d2 xcos2 a
(z--cosa ) 7r
7r
2
= 7rfi(l - cos a ) ,
and then, by using again Eq. (4.25),
The "r" integration requires the application of the method of residues. The roots r1 and r 2 of the equation
7ri
2mk --o 2 m E + -- - -
r
T2
are positive if E < 0 and correspond to the radii of turning points. In the complex z plane, the function
has two branch points at
Concerning the KepEer Action Variables
408
and a simple pole at z = 0, so that J, = i(R(2= 0) + R(z = + 0 3 ) ) .
Since R(z = 0 ) =
a
and R(z = +m) = r n k / e , we finally obtain mlc Jr --Z$
+ 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
:
are given by
d
E G + ( G ( g ) = z @ e t t ( g ) l t = o E TgG
v< E 6
and
ET*G
d * (ag)lt=o E : Og E T*G + f T * G ( a g )= -ae-t€
dt
Tag(T*G),
is a commutative diagram too. Of course, by using the commutative diagram (E.2), we have
d dt
T*cz,,( < T * G ( Q h ) )= -T(@:-t€
d
(%))lt=O
= -@€?C(T(ah))(t=O
dt
On the Canonical Differential I-Form
412
d - -&a& (h)lt=o =kdh) =~
G ( T ( ~ ~ ) ) .
From Eq. (10.24), we have J€(ag) = ( i € p G @ ) ( % ) = @ag ( b * C ( a g ) ) = a g ( T * a g ( < T * G ( a g ) ) ) = ag(tG(g))* Let us go back to the general case. It is not difficult to prove that
p*s = 8 for every differential l-form on T * M ,
p : q E M -+
p ( q ) = pq E
T;M.
(E.3)
We start by observing that the derivative of Eq. fE.3) defines a map
p* : T M + T ( T * M ), so that, if d
v, = -q(t)lt=o dt i.e.
7
V, E TqM with an integral curve q ( t ) ,where q(0) = q, then ( ~ s ) q (=~s ) ~ 9 ( ~ * q (= ~p ) q ) ( T * ~ 9 ( p * q ( *~ ) ) )
On the other hand, d
T*P,(P*q(l/P))
= g'("q(t)))lt=o
d
= -q(t)It=o dt
=
v,
7
so that (P*%(Vq) =P*(V*)r
vvq
E TqM.
If w = -d0 is the canonical symplectic form on T * M ,we have
P*w
= -p*&
= -dp*s = -@
for every differentia^ f-form P on T * M .
Appendix
~ o n c e r n Rigid i~~
F
ody Equations
Let us observe that given the map
Adra : g E G -+ ( A ~ a ) ( g=) A d i a E g;*
CY
E g;* ,
its deri~tive,at identity, gives the map
d ( d d : a ) * e ( ~=) -Ad:tcblt=O = QdgaI
E 0*
dt Similarly, the derivative of the map
Ad:AdiCY : h E G -+ Ad~Ad;TCYE 8% is given by
d ~ u~ ~a; (l A~d~~.bo ) (Ad:Adla)..(() == - A d ~ ~ ~ A =z dt On the other hand, we also have d d ( A d ~ A d i b ) * ~= ( e--Ad$cAd;Ttu)t=o ) = -dd&t
that
-
If g = ~
(Ad:a)*g((Lg)*e(t))= ad; ( A ~ i a ~ ( tis)a curve in G and a(t)=I A ~ is a curve ~ in Q",~ the above ~
relation is equivalent to Eq. (10.49). 413
~
Appendix G
The Gelfand-Levitan-Marchenko Equation
Let us consider the stationary Schrodinger equation (with fi = 1,m = 1/2), on the real line R, d2 -9 dx2
+ (k2- U(z))(P= 0 ,
(G.1)
where the potential U ( x ) is assumed to be a fast-decreasing function at f o o : lim U ( z )= 0 . X+fCQ
If 4(z, Ic) is a solution of Eq. ( G . l ) with the following asymptotic behavior:
4(z,k)
G o 0
exp[ikz]
(G.2)
then, by parity, 4(z, -k) is a solution of Eq. ( G . l ) whose asymptotic behavior is given by $(z, -k)
,,”, exp[-ikz]
.
Moreover, it is not difficult to provel4?l6that the solution +(z,k)can be expressed in the following form:
To this end let us observe that 414
The Gelfan+LevitarrMarchenko Equation 0
415
the following theorem holds:
Theorem 45 (Titchmarch) A necessary and suficient condition f o r a real function F ( q ) E Lz(-03, +CO) be the real limit F(q) =
6 i F(q + ib) ,
vq E 9
> 0)
of a function F ( t ) holomorphic in the upper complex plane ( b and satisfying the condition
L
+a
IF(q
+ ib)ldq = O(exp[-2abl)
9
is that
p(t) 0
for b > 0, k such that
2T
Jtm
F(q)exp[-iqtldq = 0 ,
V t
-a
# 0 and M ( x )
1," /U(y)(dy<
03,
a constant C exists
l d x , k) - 112< c ,
(G.4)
where
g ( x , k) = r#(x, k) exp[-ikx]
.
Indeed, by multiplying Eq. (G.l), written for p 3 #(y, k), by sink(y x), we obtain
U(Y)+(Y, k) sink-(y -
k) sink(y - x) - k+(y, k) cos k(y which, integrated between x and pressed by Eq. (G.l), gives
or
00
1
- x)
,
with the boundary condition ex-
The Gelfan&~evata~~archenko Equation
416
The above equation, expressed in terms of g(z, k) = #(x, k) exp[--ikx], reads
A solution of the above equation is expressed by the expansion gn@, k)
d z , k) =
1
nEN
with m
SnCl(X7
k) =
f,
exp'-2ik'z 2ik
-
U(Y)g"(Y, k)dy , go(%,k) = 1,
whose uniform convergence can be easily checked. Indeed, since y 3 x and k = q ib with b > 0,we have
+
so that, with ~ ( x =} f,"
~U~Y)~~Y,
Moreover,
so that from the induction hypothesis
we have
Thus,
417
The Ceuand-Levitan-Marhenko Equation
It follows that, for b > 0, the function
h(z,k) = #(s, k) - expfikz] = exp[ikz][g(s,k) - 11, k = q
+ ib
)
is square integrable with respect p E 8, since
Ih(z, k)l = I eXP[ik4fs(z*k) - 111 = exp [ - - b ~19 ] (z, k) - 11,
so that
lh(z,k)I2 = exp[-2bz]Ig(e, k) - 112 < ~exp[-2bz]o( o(exp[-2bz]). Therefore,
L
+m
I&,
W124= O(eXP[-2W)
9
and we can apply Titchmarch' theorem, to write
A(z,y)
2n
s'" I+"
h(z,k) exp[-iqyjdq = 0 , Vy < z ,
--M
whose inversion, for y > z, gives
h(z,k) =
A(z,y) exp[iky]dy, y > z .
F'rom the expression of h(z,k) and the above equation, we have Eq. (G.3), Thus, it has been shown that the solution +(z, k) = exp[ikz]
+
I"
A @ , y) exp[iky]dy
(G.5)
is analytic in the complex open upper plane defined by Qmk > 0. Moreover, can be expressed as a linear combination of #(z, k) and every solution of (G.l) #(z, -k), the last being linearly independent. Therefore, two solutions $(z, k) and $(s, -k) of (G.l), having the following asymptotic behavior: N
+(t, k) I-+--oo
-k)
N
a+--M
exPI-ik.zl
Y
exp[W,
The Gelfanct-LevitawMarchenko Equation
418
The coefficients a ( k ) , P ( k ) , d ( k ) and p ( k ) can be easily expressed in terms of the functions $(x, k) and $(x, k). Indeed, by using Eq. (G.6), the Wronskian W of 4(x7k) and $(x, k),
W[4(x7k),$(x, k)l
= +(XI
is related to the Wronskian W of
d k)-&(X’k)
d
- $(x, k ) Z 4 ( Z j k) >
4 ( x ,k) and +(x, -k) by
W[4(., k),$(x, k>l = a ( ~ ) W [ 4 ( xk), , 4(x, 4 1 1 * Since the Wronskian of two solutions of the Schrodinger equation does not depend on x, we have
The Gelfan&Levitan-Marchenko Equation
419
so that
b ( k ) = a ( k ) = - Wr6(z,k),442, k>l 2ik Similarly, from
we also obtain
so that
14V12= 1 + I P ( W
*
(G.9)
F’rom the first of Eq. (G.6), we can introduce the scattering function defined by (G.lO)
which well describes the following physical process:
A wave +(z, -k) exp[-ikz], coming from +oo o n the obstacle represented by the potential U ( z ) , is partially reflected at +oo as ( P ( k ) / a ( k ) ) $ ( z , k ) N
-
N
( P ( k ) / a ( k ) exp[-ikz], ) and partially transmitted at --oo as ($(qk ) / a ( k ) ) (l/a(k)) exp[-ikz]. The ratios
are called the reflection coeficient and the transmission coeficient, respectively. They, owing to Eq. (G.9), satisfy the relation
+
IR(k)I2 IT(k)I2= 1 Moreover, R(k), a ( k ) and P ( k ) are analytic in the complex open upper plane Smk > 0.
The Gelfan&Levitan-Marchenko Equation
420
By muItip~yingboth sides of Eq. (G.10) by exp[iky] and integrating over k from -m to $00, we write
(G.11)
Therefore,
where
Fc(x)= 2;; 1
L,
+O0
~ ~ k ) e ~ ~ x ~ k
is the Fourier transform of the refiection coefficient. +Weobserve that A(x, g) is defined only for y > x and that in this case we have 6(zfgf z fi/zz) J_*,-ctkexp[iB(zrt y)j = 0.
The Gelfond-Levitan-Marchenko Equation
421
In order for evaluate the integral on the left-hand side of Eq. (G.ll),
let us observe that a ( k ) is an analytic function in the complex open upper plane Smk > 0, where it has simple zeroes corresponding to bound states. Indeed, if a ( k ) vanishes at the point ko, then
W[4(.,
ko), $ ( x ,
w1
= 01
so that + ( x , ko) and $(z, Ico) are linearly dependent. Therefore, we obtain
$ ( x , ko) = P ( k o ) d ( z ,ko) '
(G.12)
On the other hand, the solution 4(z, k ) decreases exponentially for z 00 as well w $(z,k ) when 2 -+ -00, since Smk > 0, so that we can conclude that 4 ( x , Icg) and $(z,ko) are the wave functions of a bound states if Smko > 0. Since, k: is real, ko will be purely imaginary ko = ixo. In order to show that ko is a simple zero, let us introduce, for the sake of simplicity, the notation
and let us consider the Schrodinger equations for
4" + k2$ = U4,
$'I
4 and $:
+ k2$ = U$ .
By taking the derivative, with respect to k , of the second equation, we have
8" + k 2 d = Ud - 2 k $ . The difference between the first Schrodinger equation multiplied by above equation multiplied by 4, gives
or equivalently,
= 2k4$,
11and the
The Gelfan&LevituPrMarchenlco Equation
422
so that (G.13)
where 1 is an arbitrary parameter. With the same procedure we also have
(G.14)
On the other hand, by using Eq, (G.81, we can write d
-(2ika) dk
= 2Za(k)
+ Zik&{k) = -W[$, @] - W [ # 41. ,
((2.15)
Let us now observe that 0
e 0
the above equation does not depend on x; for k = ko, both 4s and $ vanish; both W[$,$] and W [ # , q ]vanish at z = il -+ f m .
Therefore, in such limits, by adding Eqs. (G.13) and (G.14), Eq, (G.15) gives
L
+a
flZikoCy(k0)= 2k.O
#(x,kaFa)$(s,ko)ds = 2koP(ko)
f
+m
4s2(s,k.o)dg
# 0,
-m
(G.16) where Eq. (G.12) has been used. The above equation shows that ka is a simple zero of a ( k ) . Let us continue the calculation of the integral D,by applying the ~ s z ~ ~ e s ~ e according ~ to ~ which o ~ 7t
for every domain d’D completely belonging to the field of the analyticity of the function # and containing it finite number of singular isolated points Zk. For our purpose, let us choose the half-circle, with an infinite radius, contained in the upper plane, csink > 0. For an infinite radius of the ha~f-circle,the factors of the type e x p ~ ~ k . ~ ~ ~ with y > z, give rise to a vanishing contribution from the integral along the
b o ~ d ~ r yTherefore3 , the only c o n ~ r ~ b ~ to~the ~ ointegral n comes from the integral dong the real k axis; that is, from D. Thus, n
j=l
where the x’s correspond to the bound states; i.e. a ( i x j ) = 0. On the other hand,
where Eq. (G.16) has been used and where the c’s are the normalization constants of the #s, spec~fi~d below, The waves functions of bound states ((z,i x j ) are given by
where Eq. (6.12) has been used. The c’s are the normalization constants of the 4’sdefined by
424
The G e ~ f ~ n ~ ~ ~ ~Equation 5 ~ ~ a ~
and appear in the asymptotic behavior
of the 5’s. Thus, we have shown that ~ ( i x j= ) ~ c ~ e - X 3 ~ #i(x% j f, .
Therefore, by using for # the expression given by Eq. (G.5),we finally have
By setting
and
we can write Eq. (G.11); i.e. B = D,in the following form:
The above equation, which is an integro-differentialequation of Volterra type, is known as the ~ a r c ~ e en q~~ua t ~ ofor n x E R and as the G ~ ~ ~ a ~ ~ - L e ~ equation for 3: E R+ u (0). The Gel’fand-Levitan-Marchenlco equation allows us to recover the interaction potential U once the scattering data S E {xJ,c j , R ( k ) }are given, since
The Gelfand-Levitan-Marchenko Equation
425
the following relation can be provent: d
v(z)= -2-A(x, dx
z),
In order to prove that the above relation holds, let us compu~e,from Eq. (G.5), the derivative of $ ( x , k) with respect to x . We have
so that
- [Az(x,y)Jy=s + A,(%,gflv=xfeik" - ikA(z,~ ) h = ikx ~ e. On the other hand, performing the integral in Eq, (G.5) by part twice, with the assumption lim A ( z ,y) = 0 ,
5-00
we obtain
tIt is worth recalling that A(z,y) is defined only for y > z. Thus,
Of course, A(z,y) can be defined by continuity at zf = x, and if it is differentiable at the point, the potential will be given by
426
By replac~ngthe previous expressions of $ and Schrodinger’sequation, we obtain
s,”[ A z x ( z , - A,,(x,
-2A,(z, z)eikxi .
y)
4” on the eft-hand side of
y>]eik”dy= U+(z,k) ,
and, by using once again Eq. (G.5) on the right-hand side, also
which finally gives
Bibliography
Books 1. R. Abraham and J. E. Marsden, Foundations of Mechanics (Benjamin/Cummings, Reading, Mass., 1978). 2. V. I. Arnold, Mathematical Methods of Classical Mechanics (Mir, Moscow 1976). 3. V. I. Arnold, V. V. Kozlov, and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Dynamical Systems 111 (Springer-Verlag, Berlin, 1988). 4. D. V. Alekseevskij, A. M. Vinogradov, and V. V. Lychagin, “Basic ideas and concepts of differential geometry,” ~ n c y c l o p ~ e d iof a ~ a ~ ~ e m Sciences, a ~ ~ c ~ l Vol. 28, ed. R. V. Gamkrelidz, (Springer-Verlag, BerIin, 1992). 5, F. A. Berezin, Introduction to Supemnalysis, ed. Kirillov, (D. Reidel, Dordrecht, 1987). 6. L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953). 7. I. Choquet-Bruhat, C. Dewitt-Morette, and Dillard-Bleik, Analysis, Manifold and Physics (North-Holland P. C., 1977), 8. F.Calogero and A. De Gasperis, “Spectral transform and solitons 1,” Studies in Mathematics and its Applications, Vol. 13 (Amsterdam, 1982). 9. C. Cercignani, Spazio, tempo e movimento (Zanichelli, Bologna). 10. R. Chernoff and J. Marsden, Lect. Notes in Math (Springer-Verlag, Berlin, 19741, p. 425. 11. C. Chevdley, heo ow of Lie Gmups (Princeton University Press, Princeton, 1946). 427
12. A. Connes, on-Commutative Differential Geometry, Pub. I.H.E.S. 62,257 (Paris, 1986). 13. R. Cushman and L. Bates, Qmsicat Integrable Systems (1993). g Amsterdam, 14. V. de Alfaro and T. Regge, Potent~al~ c a t t e r i ~(North-Holl~d, 1965). 15. B. De Witt, Supermanifolds (Cambridge University Press, London, 1984). 16. A. di Giacomo, Ledoni d i Fisica Teorica (Edizioni ETS, Pisa, 1992). 17. P. A. M. Dirac, Lectures in Quantum Mechanics, Belfer Graduate School of Sciences (Yeshiva University, New York, 1964). 18. P. A. M. Dirac, Lectures on Quantum Faeld Theory, Belfer Graduate School of Science (Yeshiva University, New York, 1966). 19. B, Dubrovin, S. Novikov, A. Fomenko, GBomBtric Contemporaine, MIR (Moscow 1979, 1982). 20. J. J. Duistermaat, “Global action-angle coordinate,” Comm. Pure Appl. Math., Vol. XXXIII (1980), p. 687. 21. G. Eilemberger, Solitons (Springer-Verlag, Berlin, 1981). 22. A. Fasano and S. Marmi, Meccanzca A n ~ ~ t z (Boringhieri, ca Thrino, 1994). 23. L. D. Faddeev and L. A. Takhtadjan, Hamiltonian Method in the Theory of Solitons (Springer-Verlag, Berlin, 1986). 24. G. Gallavotti, Meccanica Elementam (Boringhieri, Torino, 1986). 25. F. Gantmacher, Lectures on Ana~yticalMechanzcs (Mir, Moscow, 75). 26. E. Goursat, Lecons sur I’ z n t e g ~ ~ ~des o nequations aux derivees partielles du premier ordre (Hermann, Paris, 1921). 27. A. Hanson, T. Regge, and C. Teitelboim, Constrained ~ a ~ ~ l t o Systems n~an (Accademia Nazionale dei Lincei, Roma, 1976). 28. A. Kirillov, El&m~nts de la thgoorie des reprksentat~ons(Mir, Moscow, 1974). 29. A, Kolmogorov and S. Fomine, &&tents de la thkorie des fonctions et de 1’ analyse fonctionelle (Mir, Moscow, 1977). 30. S. Kobaiashi and K. Nomizu, ~ o u n d a t i oof~ D a f f e ~ n t ~Geome~ry, a~ I, 11 (Interscience Publishers, New York, 1963). 31. I. S. Krasil’shchik, V. V. Lichagin, and A. M. Vinogradov, Geometry of Jet Spaces and Nonlznear Partial Differential Equations (Gordon and Breach, 1986). 32, B. Kupershmidt, ~ l e m e n t sof Superin~egmb~e s y s ~ (D. e ~ Riedel, Dordrecht, 1987). 33. C. Lanczos, The Vanatzonal Principles of Mechanics (Dover, 1986). 34. G. Landi, “An introduction to noncommutative spaces and their geometries,” Lectzires Notes in Physics, New Series m: M o n o g ~ pspringer-Verl~, ~ Berlin, 1997). 35. S. L a g , Linear Algebm (Addison-Wesley, Reading, Mass., 1966). 36. J. W. Leech, Classical Mechanics, Science Paperbacks (Chapman and Hall, London, 1965). 37. T. Levi-Civita and W. Amaldi, Lezioni di Meccanica Razionale (Zanicheili, Bologna, 1929).
~ablio~ph~
429
38. S. Lie, Theorie der t ~ ~ f o ~ a ~ a o Zweiter ~ ~ p ~Abschnitt, e n , unter mitwirkung von. Prof. Dr. Frederich Engel (Teubner, Uipzig, 1890). 39. L.Landau and €3, Lifchitz, M e c h a n ~(Mir, ~ Moscow, 1966). 40. S , D.L~ndenbaum,Ana~ytica~ D ~ n a m i c s(World Scientific, 1994). h i s t o ~ g c h - ~ t ~ sdcah~ e s ~ e l i t 41. E. Mach, ‘{Die Mechanik,” in ihrer En~wic~elung (1883). 42. G. Marmo, E.J. Saletan, A. Simoni, and B. Vitale, Dynamical Systems (John Wiley, 1985). 43. J. Moser, “Various aspects of integrable Hamiltonian systems,“ in P q r . Math., Vol. 8 ~ ~ i r k ~ ~ uBoston, s e r , 1980). 44. E, Nelson, Teasop.Andpsis (Princeton University Press, Princeton, 1967). ~ l e of ~ ~ s i c Mec~anzcs al and Lie A ~ g e b ~ 45. A. M.Pereiornov, ~ n t e g ~ Systems (Birkhauser Verlag Basel, 1990). 46. M. Rasetti, Modern Methods in E ~ ~ l i b ~Statistical um Mechanics (World Scientific, 1986). 47. W. W.Rouse Ball, A Short Account of the ~ i s of~~ ao ~ ~e m a t i (Dover, cs New York, 1960). 48. E.J. Saletan and A. H. Cromer, Theoretical Mechanics (John Wiley, New York, 1971) 49, J. A. Schouten, Ricci Calculus, 2nd edition (Springer, Berlin, 1954). 50. R, Schmid, Infinite Dimensional Hamiltonian Systems (Bibliopolis, Naples, 1987). 51, €3.Schutz, G e o m e t M ~ ~e~ ~ ofoM ~a ~ h e m a tPhpsics i~ ( C ~ b r i d g University e Press,1980). 52. M. Spivak, A CompPiensive Intmduc~zonto Dz~erentaalGeometry (Publish or Perish, Boston, 1970). 53. J. M. Souriau, Stwctare des s ~ s t ~ m dpnamiques es (Dunod, Paris, 1970). 54. L.Tonelli, F o ~ ~ a m e dni tCalcolo ~ delle V a ~ a ~ (Zaniche~i, ~ o n ~ Bologna, 1921). f o ~ ~ Verjag 55. 1. Vaisman, Lectures on the G e o m e ~of ~Po~sson~ a n ~{Birkhauser Basel, 1990). 56. F. W. Wwner, Foundations of Da;treen?ntiable Manifolds and Lie Gmups (Springer-Verlag, Berlin, 1983), 57. H. Weyl, Space, Time and Matter, 4th edition (Dover, New York, 1952). 58, E,T.Whittaker, A Reatise on the Analptical D ~ n a ~ i of c sParticles and Rigid Bodies (Cambridge University Press, London, 1904,4th edition, 1959). 59. A. Win~ner,The A n a ~ p ~t o~ u~n~d ~ tofi oC ~e ~ e s ~ e~c ah a~n z c sprincet ton University Press, Princeton, 1941). a
Articles 60. M,J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, Phps. Rev. Lett. 30, 1262 (1973);Stud. Appl. Math. 53,249 (1974). 61. M. Adams, T.Ratiu, and R. Schmid, Math. Ann. 273,529 (1986);275 (1986).
62. M. Adler, “On the trace functional for formal p~udodifferentialoperators and the symplectic structure for Korteweg-de Vries type equations,” Inu. Math. 50, 219 (1979). 63. V. I. Arnold, Ann. Inst. Fourier Grenoble 16(1), 319 (1966). 64. 0. Babelon and C. M. Viallet, SISSA 54, EP (1989). 65. C. Bartocci and U. Bruzzo, “Some differential-geometric approach to supermanifolds,” J. Geom. and Phys. 4, 391 (1987). 66. S. Benenti, Lect. Notes in Math. No. 836, 512 (1980). 67. A. M. Bloch, Phys. Lett. A116, 353 (1986). 68. A. Bogoyavlenskij, Comm. Math. Phys. 180, 529 (1996). 69. F. Cantrijn and A. Ibort, ~ n t ~ d u c t ~too Poisson n S ~ p e ~ a n i f opreprint, l~, 1989. 70. G. Cassinelli, E. De Vito, and A. Levrero, The ~ y n a ~ Euolution ~ c a ~ of Quantum ~ y s t in e the ~ Adiabatic A p p ~ ~ m apreprint ~ ~ ~ n(Universitii , di Genova, 1996). 71. D, V. Choodnovsky and G. V. Choodnovsky, I1 Nuovo Cimento 40,339F (1977). 72. R. Cirelli and L. Pizzocchero, “On the integrability of quantum mechanics as an infinite dimensional Hamiltonian system,” Nonlinearity 3 (1990). 73. J. D. Cole, Q. Appl. Math. 9, 225 (1951). 74. M. Crampin, G. Marmo, and C. Rubano, Phys. Lett. A37, 88 (1983). 75. A. Das and S. Roy, J. Math. Phys. 31, 2145 (1990). 76. A. Das, When-Jui Huang, and S. Roy, J. Math. Phys. 32, 2733 (1991). 77. S. De Filippo, G. Landi, G. Marmo, and G. Vilasi, “Tensor field defining a tangent bundle structure,” Ann. Inst. N. Poinc. 53, 205 (1989); “An algebraic description of the electron-monopole dynamics,” Phys. Lett. B220,576 (1989); G. Landi and G. Marmo, “Algebraic Lagrangian formalism,” SISS‘A, Tkieste (SISSA 90/9O/FM, 1990). 78. S. De Filippo, G. Landi, G. Marmo, and G. Vilasi, “An algebraic characterization of complete integrability for Hamiltonian systems,” in Diflerential Geometric Methods in Theoretical Physics, Rapallo (June, 1990), Lect. Notes in Physics 375 (Springer-Verlag, 1991). 79. S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi, On the Phase Manifold Geometry of Integrable Nonlinear Field Theory, preprint IFUSA (Salerno, 1982), unpublished. 80. S.De Filippo, G. Marmo, M. Salerno, and G. Vilasi, “Phase manifold geometry of Burgers hierarchy,” Lett. Nuovo Cimento 37(3), 105 (1983). 81. S. De Filippo, G. Marmo, M. Salerno, and G. Vilasi, “A new c~aracterjz~tion t ~ 97 (1984). of complete integrable systems,” It N . C ~ m e nB83, 82. S. De Filippo, G. Marmo, and G. Vilasi, “A geometrical setting for the Lax representation,” Phys. Lett. S117(6), 418 (1982). 83. S. De Filippo and G. Vilasi, L‘Geometricalmethods for infinite dimensional dynamical systems,” Proceedings of the Second World Conference on Mathematics (LasPalmas, Spain, 1982), p. 236. 84. S. De Filippo, M. Salerno, and G. Vilasi, Lett. Math. Phys. 9, 85 (1985).
Bibliogmphy
431
85. L. A, Dikii, finkt. Anal. Pril. 6(4), 83 (1972). 86. P.A, M.Dirac, “Generalized Hamiltonian dynamics,” Canad. J. Math. 2,129 (1950). 87. P.Di Stasio and G. Vilasi, Lett. ~ a t h Phys. . 11, 299 (1986~. 88. I. Ya Dorfman, ~ e f o ~ a t of ~ offam~lton~an n S t ~ c t u r e sand I ~ t e g m b ~System e (1983). 89. V. G. DrinfeId and V. V. Sokolov, Lie Algebras and Korteweg-de V?ies Lzspe Equations, VINITI, Contemporary Problems of Mathematics. Recent Deveboprnenta (Moscow, 1984) (in Russian), pp. 81-180. 90. B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, Russ. Math. Survey 31(1), 59 (1976). 91. A. Einstein, “Die Grundlage der dgemeinen Relativitatstheorie,” Annalen der Physik 49,769 (1916). 92. L. Faddeev, “Symplecticstructures and quantization of the Einstein gravitation theory,” Actes du Congrds Internat, Math. 3,35 (1970). 93. L.D. Faddeev and V. E. Zakharov, finct. Anal. Appl. 5,18 (1971). 94. L.D,Faddeev and L. A. T ~ h t a j Theor. ~ , Math. Phys, 21, 160 (1974). 95. A. S. Fokas and 8.~ ~ s t e i n e“On r , the structure of symp~ecticoperators and Hereditary symmetries,” Lett. Nuovo Cimento 28(8), 299 (1980); A. S. Fokas and B. Fuchsteiner, Nuntinear Anal. TMA 5, 423 (1981); A. S . Fokas and R.L. Andersen, “On the use of isospectral eigenvalue problem for obtaining hereditary symmetries for Hamiltonian systems,” J. Math. Phys. 23(6), 1066
(1982). 96. A. P.Fordy and P. P. Kulish, “Nonlinear Schrodinger equation and simple Lie algebras,” Comm. Math. Phya. 89(4), 423 (1983). 97. A. Rolicher and A. Nijenhuis, Indag. Math. 23,338 (1956). 98. A. Rolicher and A. Nijenhuis, “Theory of vector valued differential forms 1-11,’’ Neded. Acad. Westinch. Pmc. A59, 338 (1965). 99. B. Fbchsteiner, ‘“rhe Lie algebra structure of the NEE ~ ~ t t i infinite n g dimensional abelian s y ~ e t r group,’) y Pmg. Z’heor. Phys. 65(3), 861 (1981). 100. B.Fbchsteiner, “The Lie algebra structure of the degenerate Hamiltonian and bi-Hamiltonian system,” Prog. Theor. Phys. 68(4), 1082 (1982). 101. B. Florko and A. Yanovski, On Magn’ls theorem for complete integmbility, Communication JINR (Joint Institute for Nuclear Research), Dubna, E5-83-831 ( 1983). 102. C.S.Gardner, S. M. Greene, M. D. Kruskal, and R. M. Miura, Phys. Rev. Lett. 19, 1095 (1967). 103. I. M.Gel’fand and 3.M. Levitan, “On the determination of a differential equation from its spectral function,’’ Isv. Akad. Nauk SSSR Ser. Mat. 15 (1951); Arne?-. Math. SOC. Band. 1,253 (1955). 104. 1. M. Gelfand and I. Ya. Dorfman, “The Schouten bracket and Hamiltonian operators,” finct. Anal. Awl. 14,71 (1980). 105. V. S. Gerdjikov and P. P. Kulish, “The generating operator for n x n linear system,” Physica D D3(3), 549 (1981).
106. V. S. Gerdjikov and A. B. Yanovski, “ G a u g ~ c o y ~formulation i~t of the generating operator” (1. The Zakharov-Shabat system; 2. Systems on homogeneous spaces), Phys. Lett. A103(5), 232 (1984); A110(2), 53 (1985); Comm. Math. Phys. 103, 549 (1986). 107. R. Giachetti, “Hamiltonian systems with symmetry: An introduction,” La rivista del Nuovo Cimento 4(12) (1981). 108. J. Grabowski, G. Landi, and G. Marmo, “Generalized reduction procedure: Symplectic and Poisson formalism,” Forthscritte der Physik 42(5) (1994). 109. J. Gotay, J, Nester, and G. Hinds, “Presymplectic manifolds and the DiracBergman theory of constraints,” J. Math. Phys. 19, 2388 (1978). 110. F. Guerra and L. Morato, Phys. Rev. D27, 1774 (1983); F. Guerra and R. Marra, Phys. Rev. D28,1916 (1983); D29,1647 (1984). 111. J. Haantjes, “On X,-forming sets of eigenvectors,~’Nederl. Acad. ~ e s t ~ n c h . Pmc. A58(2),158 (1955). 112. W. R. Hamilton, h n s . R. Irish Acad. 15, 69 (1828). 113. W. R. Hamilton, Phil. h n s . Roy. SOC.Part I247 (1834); Phil. ‘Ihns. Roy. SOC. Part II 95 (1835). In The Mathematical Papers of Sir W. R. Hamilton (Cambridge University Press, 1940). 114. J. Wietarinta, Quantum Integrability and Classical Integrability (Turku University, 1984). 115. E. Hopf, Comm. Pure Appl. Math. 3, 201 (1950). 116. C.G. J. Jacobi, Crelle’s J. 17, 97 (1837); Liouville’s J. 3, 60, 161 (1837). 117. R. Jackiw and V. E. Zakharov, Phys. Rev. I, 133 (1979). 118. D. J. Kaup, J. Math. Anal. Appt. 54, 849 (1976). 119. B. Kay and H.E. Moses, II Nuovo C ~ m e n 3, ~ o276 (1956). 120. 3. G. Konopelchenko and V. G. Dubrovski, “General N t h order differential spectral problem: general structure of the integrable equations, nonuniqueness of the recursion operator and gauge invariance,” Ann. Phys. li56(2), 256 (1984). 121. D. J. Korteweg and G. de Vries, Philos. Mag. Ser. 5 39, 422 (1895). 122. Y. Kosmann-Schwarzbach, “Geom6trie des systemes bihomiltoniennes” , Seminaire de Mathemetiques Superieurs aur les systemes dynamiques nonlineaires, Pub IRMA Lille Vol. 2, I (1986). 123. B. Kostant, “Graded manifolds, graded Lie theory and prequantization,” Differential Geometric Methods in Mathematical Physics,LNM 570 (Springer-Verlag, Berlin, 1977), pp. 177-306. 124. P. P. Kulish and A. G. Reymann, “Hierarchy of s y m p l ~ t i cforms for the S c h r ~ i n g e requation and for the Dirac equation on the line,” Zap. LOMI (Notes of the L e n i n g r ~Branch of the M a t ~ e m a t i c a~nstitute) ~ 77 (1978) (in Russian), pp. 134-147. 125. P. P. Kulish and E. K. Sklyanin, “About the Yang-Baxter equation solutions,” Zap. LOMI (Notes of the Leningrad Branch of the Mathematical Institute) 95 (1980) (in Russian), pp. 129-160, 126. B. A. Kupershmidt, Phys. Lett. A109, 136 (1985).
127. G.Landi, “An algebraic Betting for gauge theories,” Doctor ~ h i l o ~ p Thehi~ sis, SISSA (Trieste, 1988);G. Landi and G. Masmo, “Lie algebra extensions and Abelian monopoles,” Phys. Lett. B196,429 (1987);Extensions of Lie superalgebres and supersymmetric Abelian gauge fields, Phys. Lett. B193,61 (1987); Graded Chern-Simons terms, Phys. Lett. B192,61(1987);Algebraic reduction of the ’t Hooft-Polyakov monopole to the Dirac monopole, Phys. Lett. B201, 101 (1988);Einstein algebras and the algebraic Kaluza-Klein monopole, Phys. Lett. B210,68(1988);Algebraic instantons, Phys. Lett. B216,338 (1988);Algebraic differential calculus for gauge theories, Nuclear Physics B (Proc. Suppl.) A18,171 (1990). 128. G. Landi, G . Marno, and G. Vilasi, “An algebraic approach to ~ n t e ~ a b i ~ t y , , ’ Journal of Gmup in Physics 2, 1 (1994). 129. G . Landi and G. Marmo, “Algebraic Lagrangian F o ~ s m , ”SISSA, Qieste (SISSA 90/90/FM,1990). 130. 0.Landi, G.Marmo, G. Sparano, and G.Vilasi, “A general reduction procedure for dynamical systems,” Mod. Phys. Lett. A6(37), 3445 (1991). 131. G, Landi, G.Marmo, and G. Vilasi, “Remarks on the complete integrability of dynamical systems with fermionic variables,” J. Phys. A26,4413 (1992). 132. G. Landi, G. Marmo, and G. Vilasi, “Recursion operators: Meaning and existence for completely integrable systems,” J. Math. Phys. 36(2), 808 (1994). 133. P.D. Lax, Comm. Pure Math. 21,467 (1968);28, 141 (1975);SIAM Rev. 18, 351 (1975). 134. D. L.Lebedev and J. I. Manin, “Gel’fand-Dikii, H ~ i l t o n i a noperator and the ~ . Appl. 13(4), 40 adjoint r e p r ~ n t a t i o nof the Volterra group,” ~ n c An. (1979). 135. D. A, LGtes, “New Lie superalgebras and mechanics,” Soviet Math. Bokl. 18, 1277 (1977). 136. S. Lie, “Theorie der partiallen gleichungen erster Ordnung,” Ann. Math. Bd IX,1875. 137. J. Liouvjlle, Acta Math. 20,239 (1897). 138. L. Lusanna, Phys. Rep. 186,1 (1990);Riv. Nuovo Cimento 14(3), 1 (1991); Int. J. Mod. Phys. At44193 (1993). 139. F. Magri, “A simple model of the integrable Hamiltonian equations,” J. Math. Phys. 19,1156 (1978). 140. F, Magri, “A geometrical approach to the nonlinear solvable equations,” in Lectuw Notes in Phys~cs(Springer-Verlag, New York, 1980), p. 120. 141. F. Magri and C. Morosi, “A geometrical c h ~ ~ t e r i z a t i oofn integrable Hamiltonian systems through the theory of Pois~n-Nijenhuis manifolds~Quaderni del Dipartimento di Matematica (UniversitB di Milano, 1984). 142. F. Magri, C. Morosi, and 0. Ragnisco, “Reduction techniques for infinitedimensional Hamiltonian systems. Some ideas and applications,” Comm. Math. Phys. 99,115 (1985). 143. S. V. Manakov, finkt. Anal. Pril. 10,93 (1976). ~~~
434
Bibliography
144. Y. Manin and A. 0. Radul, Comm. Math. Phys. 98, 65 (1985). 145. V. A. Marchenko, ‘‘On the reconstruction of the potential energy from phases of the scattered waves,’’ Dokl. Akad. Nauk. SSSR 104,695 (1955). 146. G. Marmo, “A geometrical characterization of completely integrable systems,” Proceedings on Geometry and Physics (Florence, 1982), p. 257. 147. G. Marmo, “Nijenhuis operators in classical dynamics,” Proceedings of Seminar on Group Theoretical Methods in Physics (USSR Academy of Sciences, Yurmala, Latvian SSR, 1985). 148. G. Marmo, N. Mukunda, and J. Samuel, “Dynamics and symmetry for constrained systems: a geometrical analysis,” La rivista de I1 Nuovo Cimento 6,2 (1983). 149. G. Marmo and C. Rubano, I1 N . Cimento B78,70 (1983). 150. G. Marmo and E. J. Saletan, I1 Nuovo Cirnento B40,67 (1977). 151. G. Marmo, G. Morandi, and G. Mukunda, “A geometrical approach to the Hamilton-Jacobi form of dynamics and its generalizations,” La rivista del Nuovo Cimento, No. 8 (1990). 152. G. Marmo and G. Vilasi, “When do recursion operators generate new conservation laws?” Phys. Lett. B277,137 (1992). 153. G. Marmo and G. Vilasi, “Symplectic structures and quantum mechanics,” Mod. Phys. Lett. B10(12), 545 (1996). 154. G. Marmo, A . M. Vinogradov, and G. Vilasi, “The local structure of n-Jacobi and n-Poisson manifold,” J . Geom. Phys. 25(1/2) (1997). 155. J. E. Marsden and T. Ratiu, “Reduction of Poisson manifolds,” Lett. Math. Phys. 11, 161 (1986). 156. A. S. Mischenko, Funkt. Anal. Pril. 4(3), 73 (1970). 157. R. M. Miura, C. S. Gardner, and M. D. Kruskal, J. Math. Phys. 9, 1204 (1968). 158. R. M. Miura, SIAM Rev. 3,412 (1976); C.Cercignani, Riu. Nuovo Cimento 7, 429 (1977); M. J. Ablowitz, Lect. Appl. Math. 58,17 (1978); L.D. Faddeev and V. E. Korepin, Phys. Rep. C42,1 (1978); V. E. Zakharov and S. V. Manakov, Phys. Rev. 1, 133 (1979). 159. G. Morandi, C . Ferrario, G. Lo Vecchio, G. Marmo, and C. Rubano, “The inverse problem in the calculus of variations and the geometry of the tangent bundle,” Phys. Rep. 188,147 (1990). 160. N. N. NehoroSev, “Action-angle variables and their generalizations,” Zhns. Moscow Math. SOC.26,180. 161. A. Nijenhuis, Indag. Math. 13,200 (1951); 15,233 (1953); 16, 17 (1954). 162. A. Nijenhuis, Indag. Math. 49, 2 (1987). 163. M. A. Olshanetsky and A. M. Perelomov, Phys. Rep. 71,313 (1981). 164. J. P. Olver, J . Math. Phys. 18,1212 (1977). 165. R. S. Palais, “The symmetries of solitons,” Bulletin of the American Mathematical Society, 34(4), 339 (1997). 166. H. Poincark, Acta Math. 13,1 (1899). 167. T. Ratiu, Indiana Univ. Math. J. 29(4),609 (1980).
168. H. J. Resheti~inand L. D. Faddeev, ‘‘Harn~Iton~an structures of the integrable fie€d theory models,” T h e o ~ and t ~ M~ a~~ e ~ ~ t ~s hc yas~~ 66(3), cs 323 (1983). 169. A. Rogers, “A global theory of supe~anifolds,” J . ma^. Phys, 21, 1352 (1980);“Graded manifolds, supermanifolds and in~ite-dimensionalGrassmann algebras,” Comm. Ma& Phys. 106,375 (1986). 170. A. G.Reiman and M. A. Semenov Tyan-Shanskii, Invent. Math. 64,81(1979). A. G. Reymann and M. A. Semenov-Tian Shauski, “Families of Hamiltonian structures, hierarchies of Hamiltonian functions and reductions for first order matrix-valued differential operators,” finct. Anal. Appl. 14(2), 77 (1980). 171. M,Rothstein, “The axioms of supermanifoIds and a new structure arising from them,” rr(t-9w. AMS 297, 159 (1986). 172. A. Scott, F. Chu, and D. Mc Laughlin, Pmc. IEEE61, 1443 (1973). 173. M.A. Semenov-Ti~-Shanski, “What is the classical r-matrix,” fine. Ana~. Appl. 17(4), 17 (1973). 174. V. A. Soroka, Lett. mat^. P h ~ s .17,201 (1989). 175. R,Sorkin, J. Phys- A10, 717 (1977). 176. A. P. Stone, “Higher order conservation laws,” J. BiB. G e o m e t ~3, 447 (1969). 177. A. P,Stone, “Some remarks on the Nijenhuis tensor,” Canad. J. Math. 25(5), 903 (1973). 178. F.Strocchi, Rev. Mod. Phys., Vol. 4 (1966). 179. K. Sundermeyer, “Constrained dynamics,” Lecture Notes in Physics, Vol. 169 (1982). 180. W. W, Symes, Physica 131,339 (1980). 181. G.Vila& “On the H ~ l t o u ~ structure au of KdV and sineGordon theories,” Phys. Lett. B94(2), 195 (1980). 182. G.Vilasi, “Invariant endomorphism and conserved functiouds of the Liouville equation,” Phys. Lett. B174203 (1986). 183. G. Vilasi, ‘%ecursion operator and I”-scheme for Kepler Dynamics,” Conference P~eedzngs,Vol. 48, N a t ~ ~o o~T~~ h on o pNon~anear~ynamzcs,SocietA Italiana di Fisica (Bologna, 1995). 184. A. M,Vinogradov, “Local symmetries and conservation laws,” Acta Aplicandae Mathematicae, No. 2 (1984),pp. 21-78. 185. A. M. Vinogroolov and B. A. Kupershmit, “The Structures of Hamiltonian Mechanics”, Rwsian Math. Surveys 32:4, 177-243 (1977). 186. D,V. Volkov, A. I. Pashnev, V. A. Soroka, and V. I. Tkach, JETP Lett, 4470 (1986). 187. E.Witten, NucI. Phys. B188,513 (1981). 188. V, E. Zakharov and B. G.K o n o p ~ ~ e n k o “On , the theory of recursion operator,” ~ o m ~ u ma^. n , Phys. 94,483(1984). 189. V. E.Zakharov and S. V. Manakov, “About the complete ~ntegrabilityof nonlineax S ~ r ~ i nequation,” ~ e r T h ~ ~ t and ~ cMa ~~ t h e ~Physics ~ t ~ 19(3), c ~ 332 (1974); Sou. Scient, Rev. 1, 133 (1979).
436
B a b i ~ ~ p h ~
190. V.E. Zakharov and A. B. Shabat, Sov. Phys. JEPT34,62(1972); F. Lund and T. Regge, Phys. Rev. 114,1524 (1976); A. S. Bugadov and L. A. Tak~tajan, Sou. Php. ~ o 22,~ 428. (1977); M. Chaichain and P. Kulish, Phys. Lett. B78,413 (1978); V. E. Zakharov and A. V. ~ i n ~ i l o Sou. v , Phys. JETP 47, 1017 (1978); V. E. Zakharov and A. B. Shabat, finct. Anal. AppL 13, 166 (1979); R. D’Auria, T. Regge, and S, Sciuto, Phys. Lett. B89,363 (1980). 191. V. E. Zakharov and L. A. Takhtadjan, “Equivalence between nonlinear Schrodinger equation and Heisenberg ferromagnet equation,’’ Theowtical apad ~ a t h e ~ u ~ Physics z c a ~ 38(1), 26 (1979).
Index
Gauss, 296 Hamilton, 14 Hertz, 18 Jacobi, 51 Lagrange, 401 Laplace, 83 Legendre, 40 Levi-Civita, 55 Lie, 66 Ljapunov, 356 Maxwell, 295 Newton, 6 Planck, 92 PoincarL-, 121 Poisson, 50 Riccati, 27 Ricci, 189 Riemann, 188 Schrodinger, 94 Stokes, 204 Bohr formula, 95 Brackets commutator, 52 Lagrange, 76, 244 Lie, 156, 219 Nijenhuis, 192 Poisson, 50, 237, 244 Bundle base, 145
Algebra Deformation of a Lie algebra, 386 Grammann, 166 Lie algebra, 59, 219, 236 Lie algebra ideal, 214 Ampere Zform, 299 Atlas, 129 symplectic, 240 Basis anholonornic, 143 holonomic, 143, 181 tangent space, 134 vector space, 137 Biography Arnphre, 296 Beltrami, 209 Bernoulli, 26 Betti, 208 Bianchi, 189 Bohr, 95 Coulomb, 104 Darboux, 240 Dirac, 100 Einstein, 190 Euier, 26 Faraday, 296 Galilei, 5 437
438
cotangent, 146, 239 fiber, 145 fiber bundle, 145 frame bundle, 147 Mobius, 147 principal fiber bundle, 147 structure group, 145 tangent, 143, 146 typical fiber, 145 Calculus of Variations, 24 Canonical d i ~ e r e n t i a1-form, ~ 239 symplectic structure, 239 Canonical system, 43 Chaotic behavior, 122 Constraints anholonomic, 18 holonomic, 18 Coordinates action-angle coordinates, 115, 250 action-angle type coordinates, 250 Lagrangian, 22 Covector, 137 Curve brachjstochrone, 25 integral curve, 86, 148 Degrees of freedom, 22 Derivative covariant, 185 exterior, 138, 172 exterior Nijenhuis, 387 Frechet, 29 Gateaux, 30, 339 Gradient, 31 Lie, 86,151, 168 L i ~ N i j e n h u387 ~, Distribution, 155 integral manifold, 156 involutive, 156 local basis, 155 maximal integral manifold, 156 Dual
Index
Hodge, 200 Dynamics General Hamiltonian, 57 Jacobi-Poisson, 59 ternary Jacobi-Poisson, 61 Equation Einstein equations, 190 Euler equation, 292 Gel’fand-Levitan-Marchenko, 325 Hamilton-Jacobi equation, 96 Korteweg-de Vries, 311 Liouvi~~e, 345 modified Korteweg-de Vries, 317 nonlinear Schrijdinger, 380 Schrijdinger, 377 sine-Gordon, 343 Yang-Baxter, 388 Faraday 2-form, 299 Flow Hamiltonian, 86 isospectral, 322 of the vector field, 151 Foliation, 105, 157 leaf, 105, 157 Form canonical 1-form, 410 closed, 173 differential k-form, 171 exact, 173 harmonic, 209 twisted, 197 Group adjoint representation, 220 coadjoint representation, 225 Lie group, 211 local Lie group, 213 group, 225 Hamiltonian function, 233
Index
Hamiltonian vector field, 51 Hessian, 45 Identity Bianchi, 189 Cartan, 177 contracted Bianchi, 189 Jacobi, 51 Inertial frame, 6, 8, 11 1ocalIy inertial frame, 9 Integral absolute integral invariant, 246 complete integrd, 95 first integral, 237 integral invariant, 82 relative integral invariant, 247 Integration H ~ t o n - J a c o b imethod, 95 Involution, 54, 106 Jacobi Jacobi identity, 51 Jacobi multip~er,84 ternaxy Jacobi-Poisson bracket, 61 Lagrange bracket, 76, 244 intrinsic equation, 241 Lagrangian, 13 function, 13 LaX Lax pair for KdV, 323 Lax Representation, 320 Lorentz force, 14 Manifold level, 248 maximally symmetric, 184 orientable, 130, 195 submanifold, 154 sympiectic, 231 Map diff~morphism,132 embedding, 154
439
homeomorphism, 129 immersion, 154, 251 momentum, 273 PoincaxB, 122 Maxwell differential equations, 297 geometrical equations, 300 phenomenological equations, 295 Operator codifferential, 208 contraction, 174 endomorphism, 190 Laplace-Beltrami, 209 recursion operator, 259, 359 strong recursion operator, 368 weak recursion operator, 367 operators, 359 Poisson Poisson bracket, 50 Poissonian action, 274 ternary Jacobi-Poisson bracket, 61 Principle Einstein principle of equivalence, 9 Einstein principle of relativity, 9 Galilei principle of relativity, 5 Hamilton principle, 24, 34 Mac~-Einsteinprinciple of relativity, 12 of general covariance, 10 Product exterior, 166 interior, 176 tensor, 161 projection stereographic, 130 Scattering inverse scattering method, 322 reflection coefficient, 322 transmission coefficient, 322 Space cotangent, 138
440
dual vector space, 136 phase space, 48, 241 reduced phase space, 277 state space, 47, 241 tangent, 134 vector space, 135 Symplectic strongly structure, 342 structure, 231 weakly structure, 342, 343 Tensor, 161 field, 167 metric tensor, 166 metric tensor field, 181 Nijenhuis torsion, 192 Ricci, 189 Riemann, 188 scalar curvature, 189 super Nijenhuis torsion, 399 torsion, 187 Theorem Ado, 224 Cartan, 224 Daxboux, 240 Frobenius, 156 Hodge, 210 Jacobi-Poisson, 54 KAM, 121 Lee Hua-Chung, 68 Liouville, 107, 249 Liouville-Arnold, 362 rnonodromy, 224 Nother, 242, 245 Stokes, 204 virial, 46 Torus, 115, 116 Tkansformation ~ea-prese~ing 73, Biicklund, 315, 316 canonical, 66 completely canonical, 71,244 generating function, 70 Hopf-Cole, 315
Index
Legendre, 40, 242 pull-back, 140 push-forward, 140 symplectic, 73, 244 volume preserving, 75 Vector complete vector field, 151 globally Hamiltonian vector field, 233 Killing vector field, 182 Laplace-Runge-Len2 vector, 90 left invariant vector field, 214 locally H ~ i l t o n i a nvector field, 233 Pauli, 93 tangent, 133, 134 vector field, 84, 140, 245
