Geometry from dynamics, classical and quantum

Moreover, thecurrent quantum descriptions of many physical systems are based on either aLagrangian or Hamiltonian description of a certain classical one.. We have found that such atask h

Classical and

Quantum

Jos é F Cariñena • Alberto Ibort

Geometry from Dynamics, Classical and Quantum

123

Departamento de Física Teórica

NapoliItalyGiuseppe MorandiINFN Sezione di BolognaUniversitá di BolognaBologna

Italy

ISBN 978-94-017-9219-6 ISBN 978-94-017-9220-2 (eBook)

DOI 10.1007/978-94-017-9220-2

Library of Congress Control Number: 2014948056

Springer Dordrecht Heidelberg New York London

© Springer Science+Business Media Dordrecht 2015

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The Birth and the Long Gestation of a Project

Starting a book is always a difficult task Starting a book with the characteristics ofthis one is, as we hope will become clear at the end of this introduction, evenharder It is difficult because the project underlying this book began almost 20 yearsago and, necessarily, during such a long period of time, has experienced ups anddowns, turning points where the project changed dramatically and moments wherethe success of the endeavor seemed dubious

However the authors are all very grateful that things have turned out as they did.The road followed during the elaboration of this book, the innumerable discussionsand arguments we had during preparation of the different sections, the puzzlinguncertainties we suffered when facing some of the questions raised by the problemstreated, has been a major part of our own scientific evolution and have madeconcrete contributions toward the shaping of our own thinking on the role ofgeometry in the description of dynamics In this sense we may say with the poet:

Caminante, son tus huellas1

el camino y nada m ás;

Caminante, no hay camino,

se hace camino al andar.

Al andar se hace el camino,

y al volver la vista atras

se ve la senda que nunca

se ha de volver a pisar.

Caminante no hay camino

sino estelas en la mar.

Antonio Machado, Proverbios y Cantares.

1 Wanderer, your footsteps are// the road, and no more;// wanderer, there is no road,// the road is made when we walk.// By walking the path is done,// and upon glancing back// one sees the path// that never will be trod again.// Wanderer, there is no road// only foam upon the sea.

Thus, contrary to what happens with other projects that represent the culmination

of previous work, in this case the road that we have traveled was not there beforethis enterprise was started We can see from where we are now that this work has to

be pursued further to try to uncover the unknowns surrounding some of thebeautiful ideas that we have tried to put together Thus the purpose of this book is toshare with the reader some of the ideas that have emerged during the process of

reflection on the geometrical foundations of mechanics that we have come up withduring the preparation of the book itself In this sense it would be convenient toexplain to the reader some of the major conceptual problems that were seeding themilestones marking the evolution of this intellectual adventure

The original idea of this book, back in the early 1990s, was to offer in anaccessible way to young Ph.D students some completely worked significantexamples of physical systems where geometrical and topological ideas play afundamental role The consolidation of geometrical and topological ideas andtechniques in Yang-Mills theories and other branches of Physics, not only theo-retical, such as in Condensed Matter Physics with the emergence of new collectivephenomena or the fractional quantum Hall effect or High Tc superconductivity,were making it important to have a rapid but well-founded access to geometry andtopology at a graduate level; this was rather difficult for the young student or theresearcher needing a fast briefing on the subject The timeliness of this idea wasconfirmed by the fact that a number of books describing the basics of geometry andtopology delved into the modern theories offields and other physical models thathad appeared during these years Attractive as this idea was, it was immediatelyclear to us that offering a comprehensive approach to the question of why somegeometrical structures played such an important role in describing a variety ofsignificant physical examples such as the electron-monopole system, relativisticspinning particles, or particles moving in a non-abelian Yang-Millsfield, required

us to present a set of common guiding principles and not just an enumeration ofresults, no matter how fashionable they were

Besides, the reader must be warned that because of the particular idiosyncrasies ofthe authors, we were prone to take such a road So we joyously jumped into theoceanic deepness of the foundations of the Science of Mechanics, trying to discussthe role that geometry plays in it, probably believing that the work that we hadalready done on the foundations of Lagrangian and Hamiltonian mechanics qualified

us to offer our own presentation of the subject Most probably it is unnecessary torecall here that, in the more than 20 years that had passed since publication of thebooks on the mathematical foundations of mechanics by V.I Arnold [Ar76],

R Abraham and J Marsden [Ab78] and J.M Souriau [So70], the use of geometry, orbetter, the geometrical approach to Mechanics, had gained a widespread acceptationamong many practitioners and the time was ripe for a second wave of the literature

on the subject Again our attitude was timely, as a number of books deepening andexploring complementary avenues in the realm of mechanics had started to appear.When trying to put together a good recollection of the ideas embracingGeometry and Mechanics, including our own contributions to the subject, a feeling

of uneasiness started to come over us as we realized that we were not completely

satisfied with the various ways that geometrical structures were currently introducedinto the description of a given dynamical system They run from the“axiomatic”way as in Abraham and Marsden’s book Foundations of Mechanics to the “con-structive” way as in Souriau’s book Structure des Systèmes Dynamiques where ageometrical structure, the Lagrange form, was introduced in the space of “move-ments” of the system, passing through the “indirect” justification by means ofHamilton’s principle, leading to a Lagrangian description in Arnold’s Méthodesmathématiques de la mécanique classique All these approaches to the geometry ofMechanics were solidly built upon ideas deeply rooted in the previous work ofLagrange, Hamilton, Jacobi, etc and the geometric structures that were brought tothe front row on them had been laboriously uncovered by some of the most brilliantthinkers of all times Thus, in this sense, there was very little to object in the variouspresentations of the subject commented above However, it was also beginning to

be clear at that time, that some of the geometrical structures that played such aprominent role in the description of the dynamical behavior of a physical systemwere not univocally determined For instance there are many alternative Lagrangiandescriptions for such a simple and fundamental system as the harmonic oscillator.Thus, which one is the preferred one, if there is one, and why? Moreover, thecurrent quantum descriptions of many physical systems are based on either aLagrangian or Hamiltonian description of a certain classical one Thus, if theLagrangian and/or the Hamiltonian description of a given classical system is notunique, which quantum description prevails? Even such a fundamental notion aslinearity was compromised at this level of analysis as it is easy to show the exis-tence of nonequivalent linear structures compatible with a given“linear” dynamics,for instance that of the harmonic oscillator again

It took some time, but soon it became obvious that from an operational point ofview, the geometrical structures introduced to describe a given dynamics were not apriori entities, but they accompanied the given dynamics in a natural way Thus,starting from raw observational data, a physical system will provide us with afamily of trajectories on some “configuration space” Q, like the trajectories pho-tographed on a fog chamber displayed below (see Fig.1) or the motion of celestialbodies during a given interval of time From these data we would like to build adifferential equation whose solution will include the family of the observed tra-jectories However we must point out here that a differential equation is not, ingeneral, univocally determined by experimental data The ingenuity of the theo-retician regarding experimental data will provide a handful of choices to startbuilding up the theory At this point we stand with A Einstein’s famous quote:Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations He will never be able to compare his picture with the real mech- anism and he cannot even imagine the possibility or the meaning of such a comparison But

he certainly believes that, as his knowledge increases, his picture of reality will become simpler and simpler and will explain a wider and wider range of his sensuous impressions.

He may also believe in the existence of the ideal limit of knowledge and that it is approached

by the human mind He may call this ideal limit the objective truth.

A Einstein, The Evolution of Physics (1938) (co-written with Leopold Infeld).For instance, the order of the differential equation will be postulated following

an educated guess of the theoretician Very often from differential equations weprefer to go to vectorfields on some (possibly) larger carrier space, so that evolution

is described in terms of one parameter groups (or semigroups) Thus afirst initialgeometrization of the theory is performed

At this point we decided to stop assuming additional structures for a givendescription of the dynamics and, again, following Einstein, we assumed that allgeometrical structures should be considered equally placed with respect to theproblem of describing the given physical system, provided that they were com-patible with the given dynamics, id est2with the data gathered from it Thus thisnotion of operational compatibility became the Occam’s razor in our analysis ofdynamical evolution, as geometrical structures should not be postulated butaccepted only on the basis of their consistency with the observed data The way totranslate such criteria into mathematical conditions will be discussed at lengththroughout the text; however, we should stress here that such emphasis on thesubsidiary character of geometrical structures with respect to a given set of data isalready present, albeit in a different form, in Einstein’s General Relativity, wherethe geometry of space–time is dynamically determined by the distribution of massand energy in the universe All solutions of Einstein’s equations for a given energy–Fig 1 Trajectories of particles on a fog chamber

2 ‘which is to say’ or ‘in other words’.

momentum tensor are acceptable geometrical descriptions of the universe Only ifthere exists a Cauchy surface (i.e., only if we are considering a globally hyperbolicspace–time) we may, after fixing some initial data, determine (locally) the particularsolution of equations compatible with a given energy–momentum tensor Fig.2.From this point on, we embarked on the systematic investigation of geometricalstructures compatible with a given dynamical system We have found that such atask has provided in return a novel view on some of the most conspicuous geo-metrical structures alreadyfilling the closet of mathematical tools used in the theory

of mechanical and also dynamical systems in general, such as linear structures,symmetries, Poisson and symplectic structures, Lagrangian structures, etc It isapparent that looking for structures compatible with a given dynamical systemconstitutes an“Inverse Problem” a description in terms of some additional struc-tures The inverse problem of the calculus of variations is a paradigmatic example ofthis The book that we present to your attention offers at the same time a reflection onthe geometrical structures that could be naturally attached to a given dynamicalsystem and the variety of them that could exist, creating in this way a hierarchy onthe family of physical systems according with their degree of compatibility withnatural geometrical structures, a system being more and more“geometrizable” asmore structures are compatible with it Integrable systems have played a key role inthe development of Mechanics as they have constituted the main building blocks forthe theory, both because of their simple appearance, centrality in the development ofthe theories, and their ubiquity in the description of the physical world The avenue

we follow here leads to such a class of systems in a natural way as the epitome ofextremely geometrizable systems in the previous sense

We may conclude this exposition of motives by saying that if any work has amotto, probably the one encapsulating the spirit of this book could be:

All geometrical structures used in the description of the dynamics of a given physical system should be dynamically determined.

Fig 2 The picture shows the movements of several planets over the course of several years The motion of the planets relative to the stars (represented as unmoving points) produces continuous streaks on the sky (Courtesy of the Museum of Science, Boston)

What you will Find and What you will not in This Book

This is a book that pursues an analysis of the geometrical structures compatible with

a given dynamical system, thus you will notfind in it a discussion on such crucialissues such as determination of the physical magnitudes relevant for description ofmechanical systems, be they classical or quantum, or an interpretation of theexperiments performed to gain information on it, that is on any theoreticaldescription of the measurement process Neither will we extend our enquiries to thedomain of Field Theory (Fig.3) (even though we included in the preparation of thisproject such key points but we had to discard them to keep the present volume at areasonable size) where new structures with respect to the ones described here areinvolved It is a work that focuses on a mathematical understanding of some fun-damental issues in the Theory of Dynamics, thus in this sense both the style and thescope will be heavily determined by these facts

Chapter 1 of the book will be devoted to a discussion of some elementaryexamples in finite and infinite dimensions where some of the standard ideas indealing with mechanical systems like constants of motion, symmetries, Lagrangian,and Hamiltonian formalisms, etc., are recalled In this way, we pretend to help thereader to have a strong foothold on what is probably known to him/her with respect

to the language and notions that are going to be developed in the main part of thetext The examples chosen are standard: The harmonic oscillator, an electronmoving on a constant magneticfield, the free particle on the finite-dimensional side,and the Klein–Gordon equation, Maxwell equations, and the Schrödinger equation

as prototypes of systems in infinite dimensions We have said that field theory willnot be addressed in this work, that is actually so because the examples in infinitedimensions are treated as evolution systems, i.e., time is a privileged variable and

Fig 3 Counter rotating vortex generated at the tip of a wing (American Physical Society ’s 2009

no covariant treatment of them are pursued Dealing with infinite-dimensionalsystems, already at the level of basic examples, shows that many of the geometricalideas that are going to appear are not restricted by the number of degrees offreedom Even though a rigorous mathematical treatment of them in the case of

infinite dimensions will be out of the scope of this book, the geometrical argumentsapply perfectly well to them as we will try to show throughout the book

Another interesting characteristic of the examples chosen in the first part ofChap 1 is that they are all linear systems Linear systems are going to play aninstrumental role in the development of our discourse because they provide aparticularly nice bridge between elementary algebraic ideas and geometricalthinking Thus we will show how a great deal of differential geometry can beconstructed from linear systems Finally, the third and last part of thefirst chapterwill be devoted to a discussion of a number of nonlinear systems that have managed

to gain their own relevant place in the gallery of dynamics, like the Calogero-Mosersystem, and that all share the common feature of being obtained from simpler affinesystems The general method of obtaining these systems out of simpler ones iscalled“reduction” and we will offer to the reader an account of such procedures byexample working out explicitly a number of interesting ones These systems willprovide also a source of interesting situations where the geometrical analysis isparamount because their configuration/phase spaces fail to be open domains on anEuclidean space The general theory of reduction together with the problem ofintegrability will be discussed again at the end of the book in Chap.7

Geometry plays a fundamental role in this book Geometry is so pervasive that ittends very quickly to occupy a central role in any theory where geometricalarguments become relevant Geometrical thinking is synthetic so it is natural toattach to it an a priori or relatively higher position among the ideas used to constructany theory This attitude spreads in many occasions to include also geometricalstructures relevant for analysis of a given problem We have deliberately subvertedthis approach here considering geometrical structures as subsidiaries to the givendynamics; however, geometrical thinking will be used always as a guide, almost as

a metalanguage, in analyses of the problems In Chap.2 we will present the basicgeometrical ideas needed to continue the discussion started here It would be almostimpossible to present all details of the foundations of geometry, in particular dif-ferential geometry, which would be necessary to make the book self-consistent.This would make the book hard to use However, we are well aware that manystudents who could be interested in the contents of this book do not possess thenecessary geometrical background to read it without introducing (with some care)some of the fundamental geometrical notions that are necessarily used in anydiscussion where differential geometrical ideas become relevant; just to name a few:manifolds, bundles, vector fields, Lie groups, etc We have decided to take apragmatic approach and try to offer a personal view of some of these fundamentalnotions in parallel with the development of the main stream of the book However,

we will refer to standard textbooks for more detailed descriptions of some of theideas sketched here

Linearity plays a fundamental role in the presentation of the ideas of this book.Because of that some care is devoted to the description of linearity from a geo-metrical perspective Some of the discourse in Chap.3is oriented toward this goaland a detailed description of the geometrical description of linear structures bymeans of Euler or dilation vectorfields is presented We will show how a smallgeneralization of this presentation leads naturally to the description of vectorbundles and to their characterization too Some care is also devoted to describe thefundamental concepts in a dual way, i.e., from the set-theoretical point of view andfrom the point of view of the algebras of functions on the corresponding carrierspaces The second approach is instrumental in any physical conceptualization ofthe mathematical structures appearing throughout the book; they are not usuallytreated from this point of view in standard textbooks.

After the preparation offered by the first two chapters we are ready to startexploring geometrical structures compatible with a given dynamics Chapter4will

be devoted to it Again we will use as paradigmatic dynamics the linear ones and wewill start by exploring systematically all geometrical structures compatible withthem: zero order, i.e., constants of motion, first order, that is symmetries, andimmediately after, second-order invariant structures The analysis of constants ofmotion and infinitesimal symmetries will lead us immediately to pose questionsrelated with the “integrability” of our dynamics, questions that will be answeredpartially there and that will be recast in full in Chap 8 The most significantcontribution of Chap 4 consists in showing how, just studying the compatibilitycondition for geometric structures of order two in the case of linear dynamics, wearrive immediately to the notion of Jacobi, Poisson, and Hamiltonian dynamics.Thus, in this sense, standard geometrical descriptions of classical mechanical sys-tems are determined from given dynamics and are obtained by solving the corre-sponding inverse problems All of them are analyzed with care, putting specialemphasis on Poisson dynamics as it embraces both the deep geometrical structurescoming from group theory and the fundamental notions of Hamiltonian dynamics.The elementary theory of Poisson manifolds is reviewed from this perspective andthe emerging structure of symplectic manifolds is discussed A number of examplesderived from group theory and harmonic analysis are discussed as well as appli-cations to some interesting physical systems like massless relativistic systems.The Lagrangian description of dynamical systems arises as a further step in theprocess of requiring additional properties to the system In this sense, the lastsection of Chap 5 can be considered as an extended exposition of the classicalFeynman’s problem together with the inverse problem of the calculus of variationsfor second-order differential equations The geometry of tangent bundles, which isreviewed with care, shows its usefulness as it allows us to greatly simplify expo-sition of the main results: necessary and sufficient conditions will be given for theexistence of a Lagrangian function that will describe a given dynamics and thepossible forms that such a Lagrangian function can take under simple physicalassumptions (Fig.4)

Once the classical geometrical pictures of dynamical systems have been obtained

as compatibility conditions forð2; 0Þ and ð0; 2Þ tensors on the corresponding carrier

space, it remains to explore a natural situation where there is also a complexstructure compatible with the given dynamics The fundamental instance of thissituation happens when there is an Hermitean structure admissible for ourdynamics Apart from the inherent interest of such a question, we should stress thatthis is exactly the situation for the dynamical evolution of quantum systems Let uspoint out that the approach developed here does not preclude their being an a priorigiven Hermitean structure But under what conditions there will exist an Hermiteanstructure compatible with the observed dynamics Chapter 6 will be devoted tosolving such a problem and connecting it with various fundamental ideas inQuantum Mechanics We must emphasize here that we do not pretend to offer aself-contained presentation of Quantum Mechanics but rather insist that evolution

of quantum systems can be dealt within the same geometrical spirit as otherdynamics, albeit the geometrical structures that emerge from such activity are ofdiverse nature Therefore no attempt has been made to provide an analysis of thevarious geometrical ideas that are described in this chapter regarding the physics ofquantum systems, even though a number of remarks and observations pertinent tothat are made and the interested reader will be referred to the appropriate literature

At this point we consider that our exploration of geometrical structures obtainedfrom dynamics has exhausted the most notorious ones However, not all geomet-rical structures that have been relevant in the discussion of dynamical systems arecovered here Notice that we have not analyzed, for instance, contact structures thatplay an important role in treatment of the Hamilton–Jacobi theory or Jacobistructures Neither have we considered relevant geometrical structures arising infield theories or the theory of integrable systems (or hierarchies to be precise) likeYang–Baxter equations, Hopf algebras, Chern-Simons structures, Frobenius man-ifolds, etc There is a double reason for that On one side it will take us far beyondthe purpose of this book and, more important, some of these structures are char-acteristic of a very restricted, although extremely significant, class of dynamics

Fig 4 Quantum stroboscope

based on a sequence of

iden-tical attosecond pulses that are

used to release electrons into a

strong infrared (IR) laser field

exactly once per laser cycle

However we have decided not tofinish this book without entering, once we are

in possession of a rich baggage of ideas, some domains in the vast land of the study

of dynamics, where geometrical structures have had a significant role In particular

we have chosen the analysis of symmetries by means of the so-called reductiontheory and the problem of the integrability of a given system These issues will becovered in Chap.7were the reduction theory of systems will be analyzed for themain geometrical structures described before

Once one of the authors was asked by E Witten,“how does it come that somesystems are integrable and others not?” The question was rather puzzling takinginto account the large amount of literature devoted to the subject of integrability andthe attitude shared by most people that integrability is a “non-generic” property,thus only possessed by a few systems However, without trying to interpret Witten,

it is clear that the emergence of systems in many different contexts (by that timeWitten had realized the appearance of Ramanujan’s τ-function in quantum 2Dgravity) was giving him a certain uneasiness on the true nature of“integrability” as

a supposedly well-established notion Without oscillating too much toward

V Arnold’s answer to a similar question raised by one of the authors: “An grable system is a system that can be integrated”, we may try to analyze theproblem of the integrability of systems following the spirit of these notes: given adynamics, what are the fundamental structures determined by the structural char-acteristics of theflow that are instrumental in the “integrability” problem?Chapter 8 will be devoted to a general perspective regarding the problem ofintegrability of dynamical systems Again we do not pretend to offer an inclusiveapproach to this problem, i.e., we are not trying to describe and much less to unify,the many theories and results on integrability that are available in the literature Thatwould be an ill-posed problem However, we will try to exhibit from an elementaryanalysis some properties shared by an important family of systems lying within theclass of integrable systems and that can be analyzed easily with the notionsdeveloped previously in this book We will close our excursion on the geometriesdetermined by dynamics by considering in detail a special class of them that exhibitmany of the properties described before, the so-called Lie–Scheffers systems whichprovide an excellent laboratory to pursue the search on thisfield

inte-Finally, we have to point out that the book is hardly uniform both in style andcontent There are wide differences among its different parts As we have tried toexplain before a substantial part of it is in a form designed to make it accessible to alarge audience, hence it can be read by assuming only a basic knowledge of linearalgebra and calculus However there are sections that try to bring the understanding

of the subject further and introduce more advanced material These sections aremarked with an asterisk and their style is less self-contained We have collected inthe form of appendices some background mathematical material that could behelpful for the reader

As mentioned in the introduction, we have been working on this project for over

20 years First we would like to thank our families for their infinite patience andsupport Thanks Gloria, Conchi, Patrizia and Maria Rosa

During this long period we discussed various aspects of the book with a lot ofpeople in different contexts and situations We should mention some particular oneswho have been regular through the years

All of us have been participating regularly in the “International Workshop onDifferential Geometric Methods in Theoretical Mechanics”; other regular partici-pants with whom we have interacted the most have been Frans Cantrjin, MikeCrampin, Janusz Grabowski, Franco Magri, Eduardo Martinez, Enrico Pagani,Willy Sarlet and Pawel Urbanski

A long association with the Erwin Schrödinger Institute has seen many of usmeeting there on several occasions and we have benefited greatly from the col-laboration with Peter Michor and other regular visitors

In Naples we held our group seminar each Tuesday and there we presented many

of the topics that are included in the book Senior participants of this seminar werePaolo Aniello, Giuseppe Bimonte, Giampiero Esposito, Fedele Lizzi and PatriziaVitale and of course, for even longer time, Alberto Simoni, Wlodedk Tulczyjew,Franco Ventriglia, Gaetano Vilasi and Franco Zaccaria

Our long association with A.P Balachandran, N Mukunda and G Sudarshanhas influenced many of us and contributed to most of our thoughts

In the last part of this long term project we were given the opportunity to meet inMadrid and Zaragoza quite often, in particular in Madrid, under the auspices of a

“Banco de Santander/UCIIIM Excellence Chair”, so that during the last 2 yearsmost of us have been able to visit there for an extended period

We have also had the befit of ongoing discussions with Manolo Asorey, ElisaErcolessi, Paolo Facchi, Volodya Man’ko and Saverio Pascazio of particular issuesconnected with quantum theory

During the fall workshop on Geometry and Physics, another activity that hasbeen holding us together for all these years, we have benefited from discussionswith Manuel de León, Miguel Muñoz-Lecanda, Narciso Román-Roy and XavierGracia.

1 Some Examples of Linear and Nonlinear Physical Systems

and Their Dynamical Equations 1

1.1 Introduction 1

1.2 Equations of Motion for Evolution Systems 2

1.2.1 Histories, Evolution and Differential Equations 2

1.2.2 The Isotropic Harmonic Oscillator 4

1.2.3 Inhomogeneous or Affine Equations 5

1.2.4 A Free Falling Body in a Constant Force Field 7

1.2.5 Charged Particles in Uniform and Stationary Electric and Magnetic Fields 8

1.2.6 Symmetries and Constants of Motion 12

1.2.7 The Non-isotropic Harmonic Oscillator 16

1.2.8 Lagrangian and Hamiltonian Descriptions of Evolution Equations 21

1.2.9 The Lagrangian Descriptions of the Harmonic Oscillator 27

1.2.10 Constructing Nonlinear Systems Out of Linear Ones 28

1.2.11 The Reparametrized Harmonic Oscillator 29

1.2.12 Reduction of Linear Systems 34

1.3 Linear Systems with Infinite Degrees of Freedom 41

1.3.1 The Klein-Gordon Equation and the Wave Equation 41

1.3.2 The Maxwell Equations 44

1.3.3 The Schrödinger Equation 50

1.3.4 Symmetries and Infinite-Dimensional Systems 53

1.3.5 Constants of Motion 55

References 61

2 The Language of Geometry and Dynamical Systems: The Linearity Paradigm 63

2.1 Introduction 63

2.2 Linear Dynamical Systems: The Algebraic Viewpoint 64

2.2.1 Linear Systems and Linear Spaces 64

2.2.2 Integrating Linear Systems: Linear Flows 66

2.2.3 Linear Systems and Complex Vector Spaces 73

2.2.4 Integrating Time-Dependent Linear Systems: Dyson’s Formula 79

2.2.5 From a Vector Space to Its Dual: Induced Evolution Equations 82

2.3 From Linear Dynamical Systems to Vector Fields 84

2.3.1 Flows in the Algebra of Smooth Functions 84

2.3.2 Transformations and Flows 86

2.3.3 The Dual Point of View of Dynamical Evolution 87

2.3.4 Differentials and Vector Fields: Locality 89

2.3.5 Vector Fields and Derivations on the Algebra of Smooth Functions 91

2.3.6 The‘Heisenberg’ Representation of Evolution 93

2.3.7 The Integration Problem for Vector Fields 95

2.4 Exterior Differential Calculus on Linear Spaces 100

2.4.1 Differential Forms 100

2.4.2 Exterior Differential Calculus: Cartan Calculus 102

2.4.3 The‘Easy’ Tensorialization Principle 108

2.4.4 Closed and Exact Forms 111

2.5 The General‘Integration’ Problem for Vector Fields 113

2.5.1 The Integration Problem for Vector Fields: Frobenius Theorem 113

2.5.2 Foliations and Distributions 115

2.6 The Integration Problem for Lie Algebras 118

2.6.1 Introduction to the Theory of Lie Groups: Matrix Lie Groups 119

2.6.2 The Integration Problem for Lie Algebras* 130

References 134

3 The Geometrization of Dynamical Systems 135

3.1 Introduction 135

3.2 Differentiable Spaces* 137

3.2.1 Ideals and Subsets 138

3.2.2 Algebras of Functions and Differentiable Algebras 141

3.2.3 Generating Sets 143

3.2.4 Infinitesimal Symmetries and Constants of Motion 145

3.2.5 Actions of Lie Groups and Cohomology 147

3.3 The Tensorial Characterization of Linear Structures and Vector Bundles 153

3.3.1 A Tensorial Characterization of Linear Structures 153

3.3.2 Partial Linear Structures 157

3.3.3 Vector Bundles 159

3.4 The Holonomic Tensorialization Principle* 163

3.4.1 The Natural Tensorialization of Algebraic Structures 163

3.4.2 The Holonomic Tensorialization Principle 165

3.4.3 Geometric Structures Associated to Algebras 169

3.5 Vector Fields and Linear Structures 171

3.5.1 Linearity and Evolution 171

3.5.2 Linearizable Vector Fields 172

3.5.3 Alternative Linear Structures: Some Examples 175

3.6 Normal Forms and Symmetries 180

3.6.1 The Conjugacy Problem 180

3.6.2 Separation of Vector Fields 184

3.6.3 Symmetries for Linear Vector Fields 186

3.6.4 Constants of Motion for Linear Dynamical Systems 188

References 192

4 Invariant Structures for Dynamical Systems: Poisson Dynamics 193

4.1 Introduction 193

4.2 The Factorization Problem for Vector Fields 194

4.2.1 The Geometry of Noether’s Theorem 194

4.2.2 Invariant 2-Tensors 195

4.2.3 Factorizing Linear Dynamics: Linear Poisson Factorization 200

4.3 Poisson Structures 210

4.3.1 Poisson Algebras and Poisson Tensors 210

4.3.2 The Canonical‘Distribution’ of a Poisson Structure 214

4.3.3 Poisson Structures and Lie Algebras 215

4.3.4 The Coadjoint Action and Coadjoint Orbits 219

4.3.5 The Heisenberg–Weyl, Rotation and Euclidean Groups 221

4.4 Hamiltonian Systems and Poisson Structures 227

4.4.1 Poisson Tensors Invariant Under Linear Dynamics 227

4.4.2 Poisson Maps 231

4.4.3 Symmetries and Constants of Motion 233

4.5 The Inverse Problem for Poisson Structures: Feynman’s Problem 243

4.5.1 Alternative Poisson Descriptions 244

4.5.2 Feynman’s Problem 247

4.5.3 Poisson Description of Internal Dynamics 249

4.5.4 Poisson Structures for Internal and External Dynamics 253

4.6 The Poincaré Group and Massless Systems 260

4.6.1 The Poincaré Group 260

4.6.2 A Classical Description for Free Massless Particles 267

References 269

5 The Classical Formulations of Dynamics of Hamilton

and Lagrange 271

5.1 Introduction 271

5.2 Linear Hamiltonian Systems 272

5.2.1 Symplectic Linear Spaces 273

5.2.2 The Geometry of Symplectic Linear Spaces 276

5.2.3 Generic Subspaces of Symplectic Linear Spaces 281

5.2.4 Transformations on a Symplectic Linear Space 282

5.2.5 On the Structure of the Group SpðωÞ 286

5.2.6 Invariant Symplectic Structures 288

5.2.7 Normal Forms for Hamiltonian Linear Systems 292

5.3 Symplectic Manifolds and Hamiltonian Systems 295

5.3.1 Symplectic Manifolds 295

5.3.2 Symplectic Potentials and Vector Bundles 300

5.3.3 Hamiltonian Systems of Mechanical Type 303

5.4 Symmetries and Constants of Motion for Hamiltonian Systems 305

5.4.1 Symmetries and Constants of Motion: The Linear Case 305

5.4.2 Symplectic Realizations of Poisson Structures 306

5.4.3 Dual Pairs and the Cotangent Group 308

5.4.4 An Illustrative Example: The Harmonic Oscillator 311

5.4.5 The 2-Dimensional Harmonic Oscillator 312

5.5 Lagrangian Systems 320

5.5.1 Second-Order Vector Fields 321

5.5.2 The Geometry of the Tangent Bundle 326

5.5.3 Lagrangian Dynamics 341

5.5.4 Symmetries, Constants of Motion and the Noether Theorem 351

5.5.5 A Relativistic Description for Massless Particles 358

5.6 Feynman’s Problem and the Inverse Problem for Lagrangian Systems 360

5.6.1 Feynman’s Problem Revisited 360

5.6.2 Poisson Dynamics on Bundles and the Inclusion of Internal Variables 366

5.6.3 The Inverse Problem for Lagrangian Dynamics 374

5.6.4 Feynman’s Problem and Lie Groups 383

References 404

6 The Geometry of Hermitean Spaces: Quantum Evolution 407

6.1 Summary 407

6.2 Introduction 407

6.3 Invariant Hermitean Structures 4096.3.1 Positive-Factorizable Dynamics 4096.3.2 Invariant Hermitean Metrics 4176.3.3 Hermitean Dynamics and Its Stability Properties 4206.3.4 Bihamiltonian Descriptions 4216.3.5 The Structure of Compatible Hermitean Forms 4246.4 Complex Structures and Complex Exterior Calculus 4306.4.1 The Ring of Functions of a Complex Space 4306.4.2 Complex Linear Systems 4336.4.3 Complex Differential Calculus and Kähler Manifolds 4356.4.4 Algebras Associated with Hermitean Structures 4376.5 The Geometry of Quantum Dynamical Evolution 4396.5.1 On the Meaning of Quantum Dynamical Evolution 4396.5.2 The Basic Geometry of the Space of Quantum States 4446.5.3 The Hermitean Structure on the Space of Rays 4486.5.4 Canonical Tensors on a Hilbert Space 4496.5.5 The Kähler Geometry of the Space of Pure

Quantum States 4536.5.6 The Momentum Map and the Jordan–Scwhinger Map 4566.5.7 A Simple Example: The Geometry of a Two-Level

System 4596.6 The Geometry of Quantum Mechanics and the GNS

Construction 4626.6.1 The Space of Density States 4636.6.2 The GNS Construction 4676.7 Alternative Hermitean Structures for Quantum Systems 4716.7.1 Equations of Motion on Density States

and Hermitean Operators 4716.7.2 The Inverse Problem in Various Formalisms 4716.7.3 Alternative Hermitean Structures for Quantum Systems:

The Infinite-Dimensional Case 481References 485

7 Folding and Unfolding Classical and Quantum Systems 4897.1 Introduction 4897.2 Relationships Between Linear and Nonlinear Dynamics 4897.2.1 Separable Dynamics 4907.2.2 The Riccati Equation 4917.2.3 Burgers Equation 4937.2.4 Reducing the Free System Again 4957.2.5 Reduction and Solutions of the Hamilton-Jacobi

Equation 499

7.3 The Geometrical Description of Reduction 5007.3.1 A Charged Non-relativistic Particle in a Magnetic

Monopole Field 5037.4 The Algebraic Description 5047.4.1 Additional Structures: Poisson Reduction 5067.4.2 Reparametrization of Linear Systems 5087.4.3 Regularization and Linearization of the Kepler

Problem 5147.5 Reduction in Quantum Mechanics 5207.5.1 The Reduction of Free Motion in the Quantum Case 5207.5.2 Reduction in Terms of Differential Operators 5227.5.3 The Kustaanheimo–Stiefel Fibration 5247.5.4 Reduction in the Heisenberg Picture 5277.5.5 Reduction in the Ehrenfest Formalism 532References 535

8 Integrable and Superintegrable Systems 5398.1 Introduction: What Is Integrability? 5398.2 A First Approach to the Notion of Integrability: Systems

with Bounded Trajectories 5418.2.1 Systems with Bounded Trajectories 5428.3 The Geometrization of the Notion of Integrability 5468.3.1 The Geometrical Notion of Integrability

and the Erlangen Programme 5488.4 A Normal Form for an Integrable System 5508.4.1 Integrability and Alternative Hamiltonian Descriptions 5508.4.2 Integrability and Normal Forms 5528.4.3 The Group of Diffeomorphisms of an Integrable

System 5558.4.4 Oscillators and Nonlinear Oscillators 5568.4.5 Obstructions to the Equivalence of Integrable Systems 5578.5 Lax Representation 5588.5.1 The Toda Model 5618.6 The Calogero System: Inverse Scattering 5638.6.1 The Integrability of the Calogero-Moser System 5638.6.2 Inverse Scattering: A Simple Example 5648.6.3 Scattering States for the Calogero System 565References 567

9 Lie–Scheffers Systems 5699.1 The Inhomogeneous Linear Equation Revisited 5699.2 Inhomogeneous Linear Systems 5719.3 Non-linear Superposition Rule 5789.4 Related Maps 581

9.5 Lie–Scheffers Systems on Lie Groups and Homogeneous

Spaces 5839.6 Some Examples of Lie–Scheffers Systems 5899.6.1 Riccati Equation 5899.6.2 Euler Equations 5959.6.3 SODE Lie–Scheffers Systems 5979.6.4 Schrödinger–Pauli Equation 5989.6.5 Smorodinsky–Winternitz Oscillator 5999.7 Hamiltonian Systems of Lie–Scheffers Type 6009.8 A Generalization of Lie–Scheffers Systems 605References 608

10 Appendices 611References 712

Index 715

Chapter 1

Some Examples of Linear and Nonlinear

Physical Systems and Their Dynamical

Equations

An instinctive, irreflective knowledge of the processes of nature will doubtless always precede the scientific, conscious apprehension, or investigation, of phenomena The former is the outcome of the relation in which the processes of nature stand to the satisfaction of our wants.

Ernst Mach, The Science of Mechanics (1883).

1.1 Introduction

This chapter is devoted to the discussion of a few simple examples of dynamics

by using elementary means The purpose of that is twofold, on one side after thediscussion of these examples we will have a catalogue of systems to test the ideas wewould be introducing later on; on the other hand this collection of simple systemswill help to illustrate how geometrical ideas actually are born from dynamics.The chosen examples are at the same time simple, however they are ubiquitous

in many branches of Physics, not just theoretical, and they constitute part of a cist’s wardrobe Most of them are linear systems, even though we will show how

physi-to construct non-trivial nonlinear systems out of them, and they are both finite andinfinite-dimensional

We have chosen to present this collection of examples by using just elementarynotions from calculus and the elementary theory of differential equations Moreadvanced notions will arise throughout that will be given a preliminary treatment;however proper references to the place in the book where the appropriate discussion

is presented will be given

Throughout the book we will refer back to these examples, even though new oneswill be introduced We will leave most of the more advanced discussions on theirstructure for later chapters, thus we must consider this presentation as a warmup andalso as an opportunity to think back on basic ideas

1.2 Equations of Motion for Evolution Systems

1.2.1 Histories, Evolution and Differential Equations

A physical system is primarily characterized by histories, histories told by observers:trajectories in a bubble chamber of an elementary particle, trajectories in the sky forcelestial bodies, or changes in the polarization of light The events composing thesehistories must be localized in some carrier space, for instance the events composingthe trajectories in a bubble chamber can be localized in space and time as well asthe motion of celestial bodies, but the histories of massless particles can be localizedonly in momentum space

In the Newtonian approach to the time evolution of a classical physical system,

a configuration space Q is associated with the system, that at this moment will be

assumed to be identified with a subset ofRN , and space-time is replaced by Q× Rthat will be the carrier space where trajectories can be localized Usually, temporal

evolution is determined by solving a system of ordinary differential equations on Q×

R which because of experimental reasons combined with the theoretician ingenuityfreedom, are chosen to be a system of second-order differential equations:

How differential equations are arrived at from families of ‘experimental trajectories’

is discussed in [Mm85] Assuming evolution is described by a second-order ential equation was the point of view adopted by Joseph-Louis Lagrange and it ledhim to find for the first time a symplectic structure on the space of motions [La16].The evolution of the system will be described by solving the system of Eq (1.1)

differ-for each one of the initial data posed by the experimentalist, i.e., at a given time t0,

both the positions and velocities q0andv0of the system must be determined The

solution q (t), that will be assumed to exist, of the initial value problem posed by

Eq (1.1) and q(t0) = q0, ˙q(t0) = v0, will be the trajectory described by the system

on the carrier space Q The role of the theoretician should be quite clear now We

started from a necessarily finite number of ‘known’ trajectories and we have found

a way to make previsions for an infinite number of them, for each initial condition

If we are able to solve the evolution Eq (1.1) for all possible initial data q0,v0,then we may alternatively think of the family of solutions obtained in this way asdefining a transformationϕ t mapping each pair(q0, v0) to (q(t), ˙q(t)) (for each t

such that the solution q (t) exists) The one-parameter family of transformations ϕ t

will be called the flow of the system and knowing it we can determine the state of

the system at each time t provided that we know the state of the system (described

in this case by a pair of points(q, v)) at a time t0

To turn the description of evolution into a one-parameter family of transformations

we prefer to work with an associated system of first-order differential equations Inthis way there will be a one-to-one correspondence between solutions and initial

1.2 Equations of Motion for Evolution Systems 3

data A canonical way to do that is to replace our Eq (1.1) by the system of 2Nequations

dq i

dt = v i , d v i

dt = F i (q, v) , i = 1, , N , (1.2)where additional variablesv i, the velocities of the system, have been introduced

If we start with a system in the form given by Eq (1.1) we can consider asequivalent any other description that gives us the possibility to recover the trajectories

of our starting system This extension however has some ambiguities The one we aredescribing is a ‘natural one’ However other possibilities exist as has been pointed out

in [Mm85] In particular we can think of, for instance, a coordinate transformationthat would transform our starting system into a linear one where such a transformationwould exist We will consider this problem in depth in relation with the existence ofnormal forms for dynamical systems at various places throughout the text A largefamily of examples fitting in this scheme is provided by the theory of completelyintegrable systems

A completely integrable system is characterized by the existence of variables,called action-angle variables denoted by(J i , φ i ), such that when written in this new

set of variables our evolution Eq (1.2) look like:

d φ i

dt = ν i (J) , d J i

dt = 0, i = 1, , N (1.3)The general solution of such a system is given by:

φ i = φ i

0+ ν i

0t , J i = J i (t0).

whereν i

0= ν i (J(t0)) and J i (t0) is the value taken by each one of the variables J iat

a given initial time t0

If det(∂ν i/∂ J j ) = 0, this system can be given an equivalent description as follows:

d dt

sys-d x

where A is an n × n real matrix and x ∈ R n Here and hereafter use is made ofEinstein’s convention of summation over repeated indices The Eq (1.5) is then thesame as:

d x i

Then the solution of Eq (1.5) for a given Cauchy datum x(0) = x0is given by:

x (t) = exp (t A) x0, (1.7)where the exponential function is defined as the power series:

(see Sect.2.2.4for a detailed discussion on the definition and properties of exp A).

1.2.2 The Isotropic Harmonic Oscillator

As a first example let us consider an m-dimensional isotropic harmonic oscillator

of unit mass and proper frequency ω Harmonic oscillators are ubiquitous in the

description of physical systems For instance the electromagnetic field in a cavity may

be described by an infinite number of oscillators An LC oscillator circuit consisting

of an inductor (L) and a capacitor (C) connected together is described by the harmonicoscillator equation In classical mechanics, any system described by kinetic energy

plus potential energy, say V (q), assumed to have a minimum at point q0, may be

approximated by an oscillator in the following manner We Taylor expand V (q)

around the equilibrium point q0, provided that V is regular enough, and on taking only the first two non-vanishing terms in the expansion we have: V (q) ≈ V (q0) +

1.2 Equations of Motion for Evolution Systems 5

by introducing the vectors q , v ∈ R m , x = (q, v) T ∈ R2m and the 2m × 2m matrix:

exp(t A) = cos(ωt) I N+ω1 sin(ωt) A , (1.11)

as well as the standard solution for (1.9):

x (t) = e t A x0.

given explicitly by:

q (t) = q0cos(ωt) + v0

ω sin(ωt), v(t) = v0cos(ωt) − ωq0sin(ωt) (1.12)

1.2.3 Inhomogeneous or Affine Equations

Because of external interactions very often systems exhibit inhomogeneous terms inthe evolution equations Let us show now how we can deal with them in the samesetting as the homogeneous linear ones Thus we will consider an inhomogeneousfirst-order differential equation:

d x

First of all we can split off b in terms of its components in the range of A, and a

supplementary space, i.e., we can write:

where b2= A · c form some c ∈ R n Then Eq (1.13) becomes

with ˜x = x + c Note that the splitting of b is not unique but depends on the choice

of a supplementary space to the range of A If b1 = 0, we are back to the previoushomogeneous case If not, we can define a related dynamical system on Rn+1by

and the solutions of Eq (1.15) will correspond to those of Eq (1.16) with a(0) = 1

(and vice versa) The latter will be obtained again by exponentiating ˜A Note that we

giving raise to Eq (1.14), in such case A· b1= 0, and then,

exp(t ˜A) = exp(t ˜A1) exp(t ˜A2) (1.21)where, explicitly

exp(t ˜A1) = exp t

1.2 Equations of Motion for Evolution Systems 7

x (t) = e t A (x0+ c + tb1) − c , (1.23)

with initial value x (0) = x0, and only the explicit exponentiation of A is required.

Very often this particular situation is referred to as the ‘composition of independent

motions’ Note that the fact that b can be decomposed into a part that is in the range and a part that is in the kernel of A is only guaranteed when ker A ⊕ ran A = R n.Let us consider now two examples that illustrate this situation

1.2.4 A Free Falling Body in a Constant Force Field

We start by considering the case of the motion of a particle in constant force field,

a simple example being free fall in a constant gravitational field As for any choice

of the initial conditions the motion takes place in a plane, we can consider directly

Q= R2with the acceleration g pointing along the negative y-axis direction Then, again we take x = (q, v) T ∈ R4, q , v ∈ R2and Newton’s equations ofmotion are:

If the initial velocity is not parallel to g, i.e., v1

0= 0, then the solutions of Eq.(1.24)will be the family of parabolas:

v1 0

⎞

⎟

A free particle would have been described by the previous equation with g = 0 Insuch a case the solutions would have been a family of straight lines.

Now, ker A = ran A and it consists of vectors of the form (x, 0) T Hence b is neither in ker A nor in ran A, and in order to obtain the solution the simple method

discussed in the previous section cannot be used We would use the general procedureoutlined above and would be forced to exponentiate a 5× 5 matrix In this specificcase, using the decomposition Eq (1.14), one might observe that, due to the fact that

A is nilpotent of order 2 (A2 = 0), both ˜A1 and ˜A2commute with[ ˜A1, ˜A2], andthis simplifies greatly the procedure of exponentiating ˜A However, that is specific

to this case, that can be solved, as we did, by direct and elementary means, so wewill not insist on this point

1.2.5 Charged Particles in Uniform and Stationary Electric

and Magnetic Fields

Let us consider now the motion of a charged particle in an electromagnetic field in

R3 Denoting the electric and magnetic fields respectively by E and B, and by q, and v the position and velocity of the particle (all vectors inR3), we have that theequations of motion of the particle are given by Lorentz equations of motion:

consider the second equation in (1.26) as an autonomous inhomogeneous equation

onR3 Let us work then on the latter

We begin by defining a matrixB by setting B · u = u × B for any u ∈ R3, i.e.,

B i j =  i j k B k The matrixB is a 3 × 3 skew–symmetric matrix, hence degenerate Its

kernel, kerB, is spanned by B (we are assuming that B is not identically zero), and

ranB is spanned by the vectors that are orthogonal to B Hence, R3= ker B ⊕ ran B

and we are under the circumstances described after Eq (1.20) We can decompose Ealong kerB and ran B as follows:

1.2 Equations of Motion for Evolution Systems 9

The equations of motion (1.15) become:

m B2(E · B) B.

If S is the matrix sending B into its normal form, i.e., S is the matrix defining a

change of basis in which the new basis is made up by an orthonormal set with two

orthogonal vectors to B such that e1, e2, B/ B is an oriented orthonormal set,

and we find that e et B/m is a rotation around the axis defined by B with angular

velocity given by the cyclotron (or Larmor) frequency = eB/m (recasting the

light velocity c in its proper place we would find = eB/mc).

Proceeding further, the first of Eq (1.26) can be rewritten as

By applying e et B/mto this decomposition we get

We can examine now various limiting cases:

1 When E× B = 0, E is in ker B and (exp (etB/m) − 1)E = 0 So,

and the motion consists of a rotation around the B axis composed with a uniformly

accelerated motion along the direction of B itself.

2 As a subcase, if E = 0, we have a rotation around B plus uniform motion along B:



et B m

2

+ O(B3).

Now,

B(B × v0− E) = B2v0− ((v0· B)B + E × B)

1.2 Equations of Motion for Evolution Systems 11

So, if E· B = 0 and E < B (actually under normal experimental conditions

E B ) there is a frame in which the electric field can be boosted away.

1.2.5.1 Classical Hall Effect

If we have a sample with n charged particles per unit volume, the total electric current

will be j= nev(t), with v(t) given by Eq (1.32) Averaging over times of order −1,

the second term in Eq (1.32) will average to zero, and the average current J= j(t)

1.2.6 Symmetries and Constants of Motion

A symmetry of the inhomogeneous equation (1.13) will be, for the moment, any

smooth and invertible transformation: x → x = F(x) that takes solutions into

solutions Limiting ourselves to affine transformations, it can be easily shown that

the transformation x= M · x + d, with M and d constant, will satisfy

d x

dt = A · x+ b

iff[M, A] = 0 and A · d = (M − I ) · b.

Using Eq (1.6) we can compute, for any smooth function f inRn(the set of such

functions will be denoted henceforth as C∞(R n ), F (R n ), or simply as F if there is

Then a constant of motion will be any (at least C1) function f (x) such that d f/dt = 0.

Limiting ourselves to functions that are at most quadratic in x, i.e., having the form:

f N (x) = x t N x + a t x = N i j x i x j + a i x i (1.40)

where N is a constant symmetric matrix, we find that

1.2 Equations of Motion for Evolution Systems 13

x=



q v



, b =



0 b

dv = ˆβb, Cd v = (ˆδ − I)b.

There are no conditions on dq, and this corresponds to the fact that arbitrary

trans-lations of q alone are trivial symmetries of Eq (1.26) Let us consider in particular

the case in which M is a rotation matrix Then the condition M T M = I leads to the

additional constraints:

α Tα = δ T δ+  β T β = I,  α T β = 0  (1.43)But then, as ˆα is an orthogonal matrix, ˆβ = 0, and we are left with



γ =  α,  α t = α−1, d v = 0,  α t

b= b, δ, C

= 0 (1.44)

As C itself is proportional to the infinitesimal generator of rotations about the

direc-tion of B, this implies thatˆα must represent a rotation about the same axis, and that b

must be an eigenvector of ˆα with eigenvalue one As b is parallel to E, this implies,

of course, that E × B = 0 It is again pretty obvious that, if E and B are parallel, then

rotations about their common direction are symmetries

More generally, a simple counting shows that because of Eq (1.43) the matricesˆα,

ˆβ, ˆδ, generate in general a six–parameter family of symmetries Special relationships

between E and B (or vanishing of some of them) may enlarge the family.

The transformations determined by Eq (1.44) (whether it is a symmetry or not)

is an example of a point transformation, i.e., a transformation q → q = q(q), of

the coordinates together with the transformation: v → v = dq/dt, that preserve

the relation between the position and the velocity Such transformations are calledalso Newtonian

For a given system of second-order differential equations one can permit alsotransformations (in particular, symmetries) that preserve the relationship between

q and v without being point transformations (also-called ‘Newtonoid’

1.2.6.1 Non-point Transformation Symmetries for Linear Systems

We describe now briefly a way of obtaining symmetries that are non-point mations starting from a system of differential equations written in the form Eq (1.5).Let us first remark that if we start from a homogeneous linear system

transfor-1.2 Equations of Motion for Evolution Systems 15

d ξ

i.e., Eq (1.50) is a symmetry As long as taking powers generates independent ces, this procedure will generate new symmetries at each step Notice howeverthat only a finite number of them, will be independent, because of the celebratedHamilton–Cayley theorem [Ga59], according to which any matrix satisfies its char-acteristic equation

matri-In the particular case we are considering here,

Let us discuss now briefly the constants of motion A general symmetric matrix

N can be written, in terms of 3× 3 blocks, as

can spell out explicitly the condition for f Nto be a constant of motion as

Let us examine the case in which

1 ξ = 0 leads to

f N = Etot= 1

2m [v]

where Etotis the total energy

2 Forξ = 0 we find then that

f ξ = e

m



(E · B) (ξ · q) − (ξ · B) (E · q)− (E × v) · ξ (1.59)

is another constant of motion

We have discussed at length this example to show that the subsequent physicalinterpretation of the solutions of a given system contains much more than the generalsolution provided by Eq (1.7), which from the mathematical point of view is ratherexhaustive

1.2.7 The Non-isotropic Harmonic Oscillator

Let us consider now in more detail the m-dimensional anysotropic harmonic oscillator

If the oscillator is not isotropic, the system of equations (1.8) generalizes to:

d x

dt = A · x+ b

iff[M, A] = and A · d = (M − I ) · b.... symmetries that are non-point mations starting from a system of differential equations written in the form Eq (1.5).Let us first remark that if we start from a homogeneous linear system