Geometry from dynamics, classical and quantum

José F. Cariñena
Alberto Ibort
Giuseppe Marmo
Giuseppe Morandi

Geometry from
Dynamics,
Classical and
Quantum


Geometry from Dynamics, Classical and Quantum

www.pdfgrip.com


José F. Cariđena Alberto Ibort
Giuseppe Marmo Giuseppe Morandi
•

•

Geometry from Dynamics,
Classical and Quantum

123
www.pdfgrip.com


José F. Cariđena
Departamento de Física Trica

Universidad de Zaragoza
Zaragoza
Spain

Giuseppe Marmo
Dipartimento di Fisiche
Universita di Napoli “Federico II”
Napoli
Italy

Alberto Ibort
Departamento de Matemáticas
Universidad Carlos III de Madrid
Madrid
Spain

Giuseppe Morandi
INFN Sezione di Bologna
Universitá di Bologna
Bologna
Italy

ISBN 978-94-017-9219-6
DOI 10.1007/978-94-017-9220-2

ISBN 978-94-017-9220-2

(eBook)

Library of Congress Control Number: 2014948056

Springer Dordrecht Heidelberg New York London
© Springer Science+Business Media Dordrecht 2015
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or
information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed. Exempted from this legal reservation are brief
excerpts in connection with reviews or scholarly analysis or material supplied specifically for the
purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the
work. Duplication of this publication or parts thereof is permitted only under the provisions of
the Copyright Law of the Publisher’s location, in its current version, and permission for use must
always be obtained from Springer. Permissions for use may be obtained through RightsLink at the
Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of
publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for
any errors or omissions that may be made. The publisher makes no warranty, express or implied, with
respect to the material contained herein.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)

www.pdfgrip.com


Foreword

The Birth and the Long Gestation of a Project
Starting a book is always a difficult task. Starting a book with the characteristics of

this one is, as we hope will become clear at the end of this introduction, even
harder. It is difficult because the project underlying this book began almost 20 years
ago and, necessarily, during such a long period of time, has experienced ups and
downs, turning points where the project changed dramatically and moments where
the 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 discussions
and arguments we had during preparation of the different sections, the puzzling
uncertainties we suffered when facing some of the questions raised by the problems
treated, has been a major part of our own scientific evolution and have made
concrete contributions toward the shaping of our own thinking on the role of
geometry in the description of dynamics. In this sense we may say with the poet:
Caminante, son tus huellas1
el camino y nada más;
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.

v

www.pdfgrip.com


vi

Foreword

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 before
this 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 the
beautiful ideas that we have tried to put together. Thus the purpose of this book is to
share 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 with
during the preparation of the book itself. In this sense it would be convenient to
explain to the reader some of the major conceptual problems that were seeding the
milestones marking the evolution of this intellectual adventure.
The original idea of this book, back in the early 1990s, was to offer in an
accessible way to young Ph.D. students some completely worked significant
examples of physical systems where geometrical and topological ideas play a
fundamental role. The consolidation of geometrical and topological ideas and
techniques in Yang-Mills theories and other branches of Physics, not only theoretical, such as in Condensed Matter Physics with the emergence of new collective
phenomena or the fractional quantum Hall effect or High Tc superconductivity,
were making it important to have a rapid but well-founded access to geometry and
topology at a graduate level; this was rather difficult for the young student or the
researcher needing a fast briefing on the subject. The timeliness of this idea was
confirmed by the fact that a number of books describing the basics of geometry and
topology delved into the modern theories of fields and other physical models that

had appeared during these years. Attractive as this idea was, it was immediately
clear to us that offering a comprehensive approach to the question of why some
geometrical structures played such an important role in describing a variety of
significant physical examples such as the electron-monopole system, relativistic
spinning particles, or particles moving in a non-abelian Yang-Mills field, required
us to present a set of common guiding principles and not just an enumeration of
results, no matter how fashionable they were.
Besides, the reader must be warned that because of the particular idiosyncrasies of
the authors, we were prone to take such a road. So we joyously jumped into the
oceanic deepness of the foundations of the Science of Mechanics, trying to discuss
the role that geometry plays in it, probably believing that the work that we had
already done on the foundations of Lagrangian and Hamiltonian mechanics qualified
us to offer our own presentation of the subject. Most probably it is unnecessary to
recall here that, in the more than 20 years that had passed since publication of the
books 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, or
better, the geometrical approach to Mechanics, had gained a widespread acceptation
among 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 and
exploring complementary avenues in the realm of mechanics had started to appear.
When trying to put together a good recollection of the ideas embracing
Geometry 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

www.pdfgrip.com


Foreword

vii

satisfied with the various ways that geometrical structures were currently introduced
into 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 “constructive” way as in Souriau’s book Structure des Systèmes Dynamiques where a
geometrical structure, the Lagrange form, was introduced in the space of “movements” of the system, passing through the “indirect” justification by means of
Hamilton’s principle, leading to a Lagrangian description in Arnold’s Méthodes
mathématiques de la mécanique classique. All these approaches to the geometry of
Mechanics were solidly built upon ideas deeply rooted in the previous work of
Lagrange, Hamilton, Jacobi, etc. and the geometric structures that were brought to
the front row on them had been laboriously uncovered by some of the most brilliant
thinkers of all times. Thus, in this sense, there was very little to object in the various
presentations 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 a
prominent role in the description of the dynamical behavior of a physical system
were not univocally determined. For instance there are many alternative Lagrangian
descriptions 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, the
current quantum descriptions of many physical systems are based on either a
Lagrangian or Hamiltonian description of a certain classical one. Thus, if the
Lagrangian and/or the Hamiltonian description of a given classical system is not
unique, which quantum description prevails? Even such a fundamental notion as
linearity was compromised at this level of analysis as it is easy to show the existence 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 of
view, the geometrical structures introduced to describe a given dynamics were not a
priori entities, but they accompanied the given dynamics in a natural way. Thus,
starting from raw observational data, a physical system will provide us with a
family of trajectories on some “configuration space” Q, like the trajectories photographed on a fog chamber displayed below (see Fig. 1) or the motion of celestial
bodies during a given interval of time. From these data we would like to build a
differential equation whose solution will include the family of the observed trajectories. However we must point out here that a differential equation is not, in

general, univocally determined by experimental data. The ingenuity of the theoretician regarding experimental data will provide a handful of choices to start
building 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 mechanism and he cannot even imagine the possibility or the meaning of such a comparison. But

www.pdfgrip.com


viii

Foreword

Fig. 1 Trajectories of particles on a fog chamber
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 we
prefer to go to vector fields on some (possibly) larger carrier space, so that evolution
is described in terms of one parameter groups (or semigroups). Thus a first initial
geometrization of the theory is performed.
At this point we decided to stop assuming additional structures for a given

description of the dynamics and, again, following Einstein, we assumed that all
geometrical structures should be considered equally placed with respect to the
problem of describing the given physical system, provided that they were compatible with the given dynamics, id est2 with the data gathered from it. Thus this
notion of operational compatibility became the Occam’s razor in our analysis of
dynamical evolution, as geometrical structures should not be postulated but
accepted only on the basis of their consistency with the observed data. The way to
translate such criteria into mathematical conditions will be discussed at length
throughout the text; however, we should stress here that such emphasis on the
subsidiary character of geometrical structures with respect to a given set of data is
already present, albeit in a different form, in Einstein’s General Relativity, where
the geometry of space–time is dynamically determined by the distribution of mass
and energy in the universe. All solutions of Einstein’s equations for a given energy–

2

i.e., ‘which is to say’ or ‘in other words’.

www.pdfgrip.com


Foreword

ix

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)

momentum tensor are acceptable geometrical descriptions of the universe. Only if
there exists a Cauchy surface (i.e., only if we are considering a globally hyperbolic

space–time) we may, after fixing some initial data, determine (locally) the particular
solution of equations compatible with a given energy–momentum tensor Fig. 2.
From this point on, we embarked on the systematic investigation of geometrical
structures compatible with a given dynamical system. We have found that such a
task has provided in return a novel view on some of the most conspicuous geometrical structures already filling 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 is
apparent that looking for structures compatible with a given dynamical system
constitutes an “Inverse Problem” a description in terms of some additional structures. The inverse problem of the calculus of variations is a paradigmatic example of
this. The book that we present to your attention offers at the same time a reflection on
the geometrical structures that could be naturally attached to a given dynamical
system and the variety of them that could exist, creating in this way a hierarchy on
the family of physical systems according with their degree of compatibility with
natural geometrical structures, a system being more and more “geometrizable” as
more structures are compatible with it. Integrable systems have played a key role in
the development of Mechanics as they have constituted the main building blocks for
the theory, both because of their simple appearance, centrality in the development of
the 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 of
extremely geometrizable systems in the previous sense.
We may conclude this exposition of motives by saying that if any work has a
motto, 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.

www.pdfgrip.com


x

Foreword

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 not find in it a discussion on such crucial
issues such as determination of the physical magnitudes relevant for description of
mechanical systems, be they classical or quantum, or an interpretation of the
experiments performed to gain information on it, that is on any theoretical
description of the measurement process. Neither will we extend our enquiries to the
domain of Field Theory (Fig. 3) (even though we included in the preparation of this
project such key points but we had to discard them to keep the present volume at a
reasonable size) where new structures with respect to the ones described here are
involved. It is a work that focuses on a mathematical understanding of some fundamental issues in the Theory of Dynamics, thus in this sense both the style and the
scope will be heavily determined by these facts.
Chapter 1 of the book will be devoted to a discussion of some elementary
examples in finite and infinite dimensions where some of the standard ideas in
dealing with mechanical systems like constants of motion, symmetries, Lagrangian,
and Hamiltonian formalisms, etc., are recalled. In this way, we pretend to help the
reader 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 the
text. The examples chosen are standard: The harmonic oscillator, an electron
moving on a constant magnetic field, 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 will
not be addressed in this work, that is actually so because the examples in infinite
dimensions 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
Gallery of Fluid Motion)

www.pdfgrip.com


Foreword

xi

no covariant treatment of them are pursued. Dealing with infinite-dimensional
systems, already at the level of basic examples, shows that many of the geometrical
ideas that are going to appear are not restricted by the number of degrees of
freedom. 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 arguments
apply perfectly well to them as we will try to show throughout the book.
Another interesting characteristic of the examples chosen in the first part of
Chap. 1 is that they are all linear systems. Linear systems are going to play an
instrumental role in the development of our discourse because they provide a
particularly nice bridge between elementary algebraic ideas and geometrical
thinking. Thus we will show how a great deal of differential geometry can be
constructed from linear systems. Finally, the third and last part of the first chapter
will 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-Moser
system, and that all share the common feature of being obtained from simpler affine
systems. The general method of obtaining these systems out of simpler ones is
called “reduction” and we will offer to the reader an account of such procedures by
example working out explicitly a number of interesting ones. These systems will
provide also a source of interesting situations where the geometrical analysis is
paramount because their configuration/phase spaces fail to be open domains on an
Euclidean space. The general theory of reduction together with the problem of
integrability 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 it

tends very quickly to occupy a central role in any theory where geometrical
arguments become relevant. Geometrical thinking is synthetic so it is natural to
attach to it an a priori or relatively higher position among the ideas used to construct
any theory. This attitude spreads in many occasions to include also geometrical
structures relevant for analysis of a given problem. We have deliberately subverted
this approach here considering geometrical structures as subsidiaries to the given
dynamics; 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 basic
geometrical ideas needed to continue the discussion started here. It would be almost
impossible to present all details of the foundations of geometry, in particular differential 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 many
students who could be interested in the contents of this book do not possess the
necessary geometrical background to read it without introducing (with some care)
some of the fundamental geometrical notions that are necessarily used in any
discussion where differential geometrical ideas become relevant; just to name a few:
manifolds, bundles, vector fields, Lie groups, etc. We have decided to take a
pragmatic approach and try to offer a personal view of some of these fundamental
notions 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 the
ideas sketched here.

www.pdfgrip.com


xii

Foreword

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 geometrical perspective. Some of the discourse in Chap. 3 is oriented toward this goal

and a detailed description of the geometrical description of linear structures by
means of Euler or dilation vector fields is presented. We will show how a small
generalization of this presentation leads naturally to the description of vector
bundles and to their characterization too. Some care is also devoted to describe the
fundamental concepts in a dual way, i.e., from the set-theoretical point of view and
from the point of view of the algebras of functions on the corresponding carrier
spaces. The second approach is instrumental in any physical conceptualization of
the mathematical structures appearing throughout the book; they are not usually
treated from this point of view in standard textbooks.
After the preparation offered by the first two chapters we are ready to start
exploring geometrical structures compatible with a given dynamics. Chapter 4 will
be devoted to it. Again we will use as paradigmatic dynamics the linear ones and we
will start by exploring systematically all geometrical structures compatible with
them: zero order, i.e., constants of motion, first order, that is symmetries, and
immediately after, second-order invariant structures. The analysis of constants of
motion and infinitesimal symmetries will lead us immediately to pose questions
related with the “integrability” of our dynamics, questions that will be answered
partially there and that will be recast in full in Chap. 8. The most significant
contribution of Chap. 4 consists in showing how, just studying the compatibility
condition for geometric structures of order two in the case of linear dynamics, we
arrive immediately to the notion of Jacobi, Poisson, and Hamiltonian dynamics.
Thus, in this sense, standard geometrical descriptions of classical mechanical systems are determined from given dynamics and are obtained by solving the corresponding inverse problems. All of them are analyzed with care, putting special
emphasis on Poisson dynamics as it embraces both the deep geometrical structures
coming from group theory and the fundamental notions of Hamiltonian dynamics.
The elementary theory of Poisson manifolds is reviewed from this perspective and
the emerging structure of symplectic manifolds is discussed. A number of examples
derived from group theory and harmonic analysis are discussed as well as applications to some interesting physical systems like massless relativistic systems.
The Lagrangian description of dynamical systems arises as a further step in the
process of requiring additional properties to the system. In this sense, the last
section of Chap. 5 can be considered as an extended exposition of the classical

Feynman’s problem together with the inverse problem of the calculus of variations
for second-order differential equations. The geometry of tangent bundles, which is
reviewed with care, shows its usefulness as it allows us to greatly simplify exposition of the main results: necessary and sufficient conditions will be given for the
existence of a Lagrangian function that will describe a given dynamics and the
possible forms that such a Lagrangian function can take under simple physical
assumptions (Fig. 4).
Once the classical geometrical pictures of dynamical systems have been obtained
as compatibility conditions for ð2; 0Þ and ð0; 2Þ tensors on the corresponding carrier

www.pdfgrip.com


Foreword

xiii

Fig. 4 Quantum stroboscope
based on a sequence of identical attosecond pulses that are
used to release electrons into a
strong infrared (IR) laser field
exactly once per laser cycle

space, it remains to explore a natural situation where there is also a complex
structure compatible with the given dynamics. The fundamental instance of this
situation happens when there is an Hermitean structure admissible for our
dynamics. Apart from the inherent interest of such a question, we should stress that
this is exactly the situation for the dynamical evolution of quantum systems. Let us
point out that the approach developed here does not preclude their being an a priori
given Hermitean structure. But under what conditions there will exist an Hermitean
structure compatible with the observed dynamics. Chapter 6 will be devoted to

solving such a problem and connecting it with various fundamental ideas in
Quantum Mechanics. We must emphasize here that we do not pretend to offer a
self-contained presentation of Quantum Mechanics but rather insist that evolution
of quantum systems can be dealt within the same geometrical spirit as other
dynamics, albeit the geometrical structures that emerge from such activity are of
diverse nature. Therefore no attempt has been made to provide an analysis of the
various geometrical ideas that are described in this chapter regarding the physics of
quantum systems, even though a number of remarks and observations pertinent to
that are made and the interested reader will be referred to the appropriate literature.
At this point we consider that our exploration of geometrical structures obtained
from dynamics has exhausted the most notorious ones. However, not all geometrical structures that have been relevant in the discussion of dynamical systems are
covered here. Notice that we have not analyzed, for instance, contact structures that
play an important role in treatment of the Hamilton–Jacobi theory or Jacobi
structures. Neither have we considered relevant geometrical structures arising in
field theories or the theory of integrable systems (or hierarchies to be precise) like
Yang–Baxter equations, Hopf algebras, Chern-Simons structures, Frobenius manifolds, etc. There is a double reason for that. On one side it will take us far beyond
the purpose of this book and, more important, some of these structures are characteristic of a very restricted, although extremely significant, class of dynamics.

www.pdfgrip.com


xiv

Foreword

However we have decided not to finish 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 reduction
theory and the problem of the integrability of a given system. These issues will be

covered in Chap. 7 were the reduction theory of systems will be analyzed for the
main geometrical structures described before.
Once one of the authors was asked by E. Witten, “how does it come that some
systems are integrable and others not?” The question was rather puzzling taking
into account the large amount of literature devoted to the subject of integrability and
the 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 time
Witten had realized the appearance of Ramanujan’s τ-function in quantum 2D
gravity) 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 integrable system is a system that can be integrated”, we may try to analyze the
problem of the integrability of systems following the spirit of these notes: given a
dynamics, what are the fundamental structures determined by the structural characteristics of the flow that are instrumental in the “integrability” problem?
Chapter 8 will be devoted to a general perspective regarding the problem of
integrability of dynamical systems. Again we do not pretend to offer an inclusive
approach 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. That
would be an ill-posed problem. However, we will try to exhibit from an elementary
analysis some properties shared by an important family of systems lying within the
class of integrable systems and that can be analyzed easily with the notions
developed previously in this book. We will close our excursion on the geometries
determined by dynamics by considering in detail a special class of them that exhibit
many of the properties described before, the so-called Lie–Scheffers systems which
provide an excellent laboratory to pursue the search on this field.
Finally, we have to point out that the book is hardly uniform both in style and
content. There are wide differences among its different parts. As we have tried to
explain before a substantial part of it is in a form designed to make it accessible to a
large audience, hence it can be read by assuming only a basic knowledge of linear
algebra and calculus. However there are sections that try to bring the understanding

of the subject further and introduce more advanced material. These sections are
marked with an asterisk and their style is less self-contained. We have collected in
the form of appendices some background mathematical material that could be
helpful for the reader.

www.pdfgrip.com


Foreword

xv

References
Abraham, R, Marsden, J.E.: Foundations of mechanics, (2nd ed.). Benjamin, Massachussetts
(1978)
Arnol’d, V.I.: Méthodes mathématiques de la mécanique classique (Edition Mir), 1976.
Mathematical Methods of Classical Mechanics. Springer, New York (1989)
Souriau, J.-M.: Structure des systemes dynamiques. Dunod, Paris (1970)

www.pdfgrip.com


Acknowledgments

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 and
support. Thanks Gloria, Conchi, Patrizia and Maria Rosa.
During this long period we discussed various aspects of the book with a lot of
people in different contexts and situations. We should mention some particular ones
who have been regular through the years.

All of us have been participating regularly in the “International Workshop on
Differential Geometric Methods in Theoretical Mechanics”; other regular participants with whom we have interacted the most have been Frans Cantrjin, Mike
Crampin, 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 us
meeting there on several occasions and we have benefited greatly from the collaboration 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 were
Paolo Aniello, Giuseppe Bimonte, Giampiero Esposito, Fedele Lizzi and Patrizia
Vitale 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. Sudarshan
has 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 in
Madrid 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 years
most of us have been able to visit there for an extended period.
We have also had the befit of ongoing discussions with Manolo Asorey, Elisa
Ercolessi, Paolo Facchi, Volodya Man’ko and Saverio Pascazio of particular issues
connected with quantum theory.

xvii

www.pdfgrip.com


xviii

Acknowledgments

During the fall workshop on Geometry and Physics, another activity that has
been holding us together for all these years, we have benefited from discussions
with Manuel de Ln, Miguel Moz-Lecanda, Narciso Román-Roy and Xavier
Gracia.

www.pdfgrip.com


Contents

1

2

Some Examples of Linear and Nonlinear Physical Systems
and Their Dynamical Equations . . . . . . . . . . . . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Equations of Motion for Evolution Systems . . . . . . . . . . . .
1.2.1 Histories, Evolution and Differential Equations . . . . .
1.2.2 The Isotropic Harmonic Oscillator . . . . . . . . . . . . . .
1.2.3 Inhomogeneous or Affine Equations . . . . . . . . . . . .
1.2.4 A Free Falling Body in a Constant Force Field . . . . .
1.2.5 Charged Particles in Uniform and Stationary Electric
and Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . .
1.2.6 Symmetries and Constants of Motion. . . . . . . . . . . .
1.2.7 The Non-isotropic Harmonic Oscillator . . . . . . . . . .
1.2.8 Lagrangian and Hamiltonian Descriptions
of Evolution Equations. . . . . . . . . . . . . . . . . . . . . .
1.2.9 The Lagrangian Descriptions of the Harmonic
Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2.10 Constructing Nonlinear Systems Out of Linear Ones .
1.2.11 The Reparametrized Harmonic Oscillator . . . . . . . . .
1.2.12 Reduction of Linear Systems . . . . . . . . . . . . . . . . .
1.3 Linear Systems with Infinite Degrees of Freedom . . . . . . . .
1.3.1 The Klein-Gordon Equation and the Wave Equation .
1.3.2 The Maxwell Equations . . . . . . . . . . . . . . . . . . . . .
1.3.3 The Schrödinger Equation . . . . . . . . . . . . . . . . . . .
1.3.4 Symmetries and Infinite-Dimensional Systems . . . . .
1.3.5 Constants of Motion . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The
The
2.1
2.2

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

1
1
2
2
4
5
7

...
...
...

8
12
16

...

21

.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

27
28
29
34
41
41
44

50
53
55
61

Language of Geometry and Dynamical Systems:
Linearity Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Linear Dynamical Systems: The Algebraic Viewpoint . . . . . . . .

63
63
64

.
.
.
.
.
.
.
.
.
.
.

xix

www.pdfgrip.com

xx

Contents

2.2.1
2.2.2
2.2.3
2.2.4

Linear Systems and Linear Spaces. . . . . . . . . . . . . .
Integrating Linear Systems: Linear Flows . . . . . . . . .
Linear Systems and Complex Vector Spaces. . . . . . .
Integrating Time-Dependent Linear Systems:
Dyson’s Formula. . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 From a Vector Space to Its Dual: Induced Evolution
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 From Linear Dynamical Systems to Vector Fields . . . . . . . .
2.3.1 Flows in the Algebra of Smooth Functions . . . . . . . .
2.3.2 Transformations and Flows. . . . . . . . . . . . . . . . . . .
2.3.3 The Dual Point of View of Dynamical Evolution . . .
2.3.4 Differentials and Vector Fields: Locality . . . . . . . . .
2.3.5 Vector Fields and Derivations on the Algebra
of Smooth Functions . . . . . . . . . . . . . . . . . . . . . . .
2.3.6 The ‘Heisenberg’ Representation of Evolution. . . . . .
2.3.7 The Integration Problem for Vector Fields . . . . . . . .
2.4 Exterior Differential Calculus on Linear Spaces . . . . . . . . . .
2.4.1 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Exterior Differential Calculus: Cartan Calculus . . . . .
2.4.3 The ‘Easy’ Tensorialization Principle . . . . . . . . . . . .

2.4.4 Closed and Exact Forms . . . . . . . . . . . . . . . . . . . .
2.5 The General ‘Integration’ Problem for Vector Fields . . . . . .
2.5.1 The Integration Problem for Vector Fields:
Frobenius Theorem . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 Foliations and Distributions . . . . . . . . . . . . . . . . . .
2.6 The Integration Problem for Lie Algebras . . . . . . . . . . . . . .
2.6.1 Introduction to the Theory of Lie Groups:
Matrix Lie Groups. . . . . . . . . . . . . . . . . . . . . . . . .
2.6.2 The Integration Problem for Lie Algebras* . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3

The Geometrization of Dynamical Systems . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Differentiable Spaces* . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Ideals and Subsets . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Algebras of Functions and Differentiable Algebras
3.2.3 Generating Sets . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4 Infinitesimal Symmetries and Constants of Motion
3.2.5 Actions of Lie Groups and Cohomology . . . . . . .
3.3 The Tensorial Characterization of Linear Structures
and Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 A Tensorial Characterization of Linear Structures .
3.3.2 Partial Linear Structures . . . . . . . . . . . . . . . . . . .
3.3.3 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . .

www.pdfgrip.com

...
...

...

64
66
73

...

79

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

82
84
84
86
87
89

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.

91
93
95
100
100
102
108
111
113

...
...
...

113
115
118

...
...
...

119
130
134

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

135
135
137
138
141
143
145
147

.
.

.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

153
153
157
159


Contents

xxi

3.4

The Holonomic Tensorialization Principle* . . . . . . . . . . . .
3.4.1 The Natural Tensorialization of Algebraic Structures
3.4.2 The Holonomic Tensorialization Principle . . . . . . .
3.4.3 Geometric Structures Associated to Algebras . . . . .
3.5 Vector Fields and Linear Structures . . . . . . . . . . . . . . . . .
3.5.1 Linearity and Evolution . . . . . . . . . . . . . . . . . . . .
3.5.2 Linearizable Vector Fields . . . . . . . . . . . . . . . . . .
3.5.3 Alternative Linear Structures: Some Examples . . . .
3.6 Normal Forms and Symmetries . . . . . . . . . . . . . . . . . . . .
3.6.1 The Conjugacy Problem. . . . . . . . . . . . . . . . . . . .
3.6.2 Separation of Vector Fields . . . . . . . . . . . . . . . . .
3.6.3 Symmetries for Linear Vector Fields . . . . . . . . . . .
3.6.4 Constants of Motion for Linear Dynamical Systems.
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4

.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.

163
163
165
169
171
171
172
175
180
180
184
186
188
192

Invariant Structures for Dynamical Systems: Poisson Dynamics .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Factorization Problem for Vector Fields . . . . . . . . . . . . .
4.2.1 The Geometry of Noether’s Theorem. . . . . . . . . . . . .
4.2.2 Invariant 2-Tensors . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Factorizing Linear Dynamics: Linear Poisson
Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Poisson Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Poisson Algebras and Poisson Tensors . . . . . . . . . . . .

4.3.2 The Canonical ‘Distribution’ of a Poisson Structure. . .
4.3.3 Poisson Structures and Lie Algebras . . . . . . . . . . . . .
4.3.4 The Coadjoint Action and Coadjoint Orbits . . . . . . . .
4.3.5 The Heisenberg–Weyl, Rotation and Euclidean
Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Hamiltonian Systems and Poisson Structures . . . . . . . . . . . . .
4.4.1 Poisson Tensors Invariant Under Linear Dynamics . . .
4.4.2 Poisson Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.3 Symmetries and Constants of Motion. . . . . . . . . . . . .
4.5 The Inverse Problem for Poisson Structures:
Feynman’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Alternative Poisson Descriptions . . . . . . . . . . . . . . . .
4.5.2 Feynman’s Problem . . . . . . . . . . . . . . . . . . . . . . . . .
4.5.3 Poisson Description of Internal Dynamics. . . . . . . . . .
4.5.4 Poisson Structures for Internal and External Dynamics .
4.6 The Poincaré Group and Massless Systems . . . . . . . . . . . . . .
4.6.1 The Poincaré Group. . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 A Classical Description for Free Massless Particles . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.

.
.
.
.

.

193
193
194
194
195

.
.
.
.
.
.

.
.
.
.
.
.

200
210
210
214
215
219

.

.
.
.
.

.
.
.
.
.

221
227
227
231
233

.
.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.
.

243
244
247
249
253
260
260
267
269

www.pdfgrip.com

.
.
.
.
.
.
.
.
.
.
.

.
.
.


xxii

5

6

Contents

The Classical Formulations of Dynamics of Hamilton
and Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Linear Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . .
5.2.1 Symplectic Linear Spaces . . . . . . . . . . . . . . . . . .
5.2.2 The Geometry of Symplectic Linear Spaces . . . . .
5.2.3 Generic Subspaces of Symplectic Linear Spaces . .
5.2.4 Transformations on a Symplectic Linear Space . . .
5.2.5 On the Structure of the Group SpðωÞ . . . . . . . . . .
5.2.6 Invariant Symplectic Structures . . . . . . . . . . . . . .
5.2.7 Normal Forms for Hamiltonian Linear Systems . . .
5.3 Symplectic Manifolds and Hamiltonian Systems . . . . . . .
5.3.1 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . .
5.3.2 Symplectic Potentials and Vector Bundles . . . . . .
5.3.3 Hamiltonian Systems of Mechanical Type . . . . . .
5.4 Symmetries and Constants of Motion
for Hamiltonian Systems. . . . . . . . . . . . . . . . . . . . . . . .

5.4.1 Symmetries and Constants of Motion:
The Linear Case . . . . . . . . . . . . . . . . . . . . . . . .
5.4.2 Symplectic Realizations of Poisson Structures . . . .
5.4.3 Dual Pairs and the Cotangent Group . . . . . . . . . .
5.4.4 An Illustrative Example: The Harmonic Oscillator .
5.4.5 The 2-Dimensional Harmonic Oscillator . . . . . . . .
5.5 Lagrangian Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Second-Order Vector Fields . . . . . . . . . . . . . . . .
5.5.2 The Geometry of the Tangent Bundle . . . . . . . . .
5.5.3 Lagrangian Dynamics . . . . . . . . . . . . . . . . . . . .
5.5.4 Symmetries, Constants of Motion
and the Noether Theorem . . . . . . . . . . . . . . . . . .
5.5.5 A Relativistic Description for Massless Particles . .
5.6 Feynman’s Problem and the Inverse Problem
for Lagrangian Systems . . . . . . . . . . . . . . . . . . . . . . . .
5.6.1 Feynman’s Problem Revisited . . . . . . . . . . . . . . .
5.6.2 Poisson Dynamics on Bundles and the Inclusion
of Internal Variables . . . . . . . . . . . . . . . . . . . . .
5.6.3 The Inverse Problem for Lagrangian Dynamics . . .
5.6.4 Feynman’s Problem and Lie Groups . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.

.
.
.
.
.
.

271
271
272
273
276
281
282
286
288
292
295
295
300
303

.....

305

.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.

305
306
308
311
312
320
321
326
341

.....
.....

351

358

.....
.....

360
360

.
.
.
.

.
.
.
.

366
374
383
404

The Geometry of Hermitean Spaces: Quantum Evolution. . . . . . . .
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407
407
407

www.pdfgrip.com

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.
.

.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.
.

.
.
.
.


Contents

xxiii

6.3

Invariant Hermitean Structures. . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Positive-Factorizable Dynamics . . . . . . . . . . . . . . . . . .
6.3.2 Invariant Hermitean Metrics . . . . . . . . . . . . . . . . . . . .
6.3.3 Hermitean Dynamics and Its Stability Properties . . . . . .
6.3.4 Bihamiltonian Descriptions . . . . . . . . . . . . . . . . . . . . .

6.3.5 The Structure of Compatible Hermitean Forms . . . . . . .
6.4 Complex Structures and Complex Exterior Calculus. . . . . . . . .
6.4.1 The Ring of Functions of a Complex Space . . . . . . . . .
6.4.2 Complex Linear Systems . . . . . . . . . . . . . . . . . . . . . .
6.4.3 Complex Differential Calculus and Kähler Manifolds. . .
6.4.4 Algebras Associated with Hermitean Structures . . . . . . .
6.5 The Geometry of Quantum Dynamical Evolution. . . . . . . . . . .
6.5.1 On the Meaning of Quantum Dynamical Evolution . . . .
6.5.2 The Basic Geometry of the Space of Quantum States . .
6.5.3 The Hermitean Structure on the Space of Rays . . . . . . .
6.5.4 Canonical Tensors on a Hilbert Space . . . . . . . . . . . . .
6.5.5 The Kähler Geometry of the Space of Pure
Quantum States. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.6 The Momentum Map and the Jordan–Scwhinger Map . .
6.5.7 A Simple Example: The Geometry of a Two-Level
System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 The Geometry of Quantum Mechanics and the GNS
Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6.1 The Space of Density States . . . . . . . . . . . . . . . . . . . .
6.6.2 The GNS Construction. . . . . . . . . . . . . . . . . . . . . . . .
6.7 Alternative Hermitean Structures for Quantum Systems . . . . . .
6.7.1 Equations of Motion on Density States
and Hermitean Operators . . . . . . . . . . . . . . . . . . . . . .
6.7.2 The Inverse Problem in Various Formalisms. . . . . . . . .
6.7.3 Alternative Hermitean Structures for Quantum Systems:
The Infinite-Dimensional Case . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7

Folding and Unfolding Classical and Quantum Systems . .

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Relationships Between Linear and Nonlinear Dynamics
7.2.1 Separable Dynamics . . . . . . . . . . . . . . . . . . .
7.2.2 The Riccati Equation . . . . . . . . . . . . . . . . . . .
7.2.3 Burgers Equation. . . . . . . . . . . . . . . . . . . . . .
7.2.4 Reducing the Free System Again. . . . . . . . . . .
7.2.5 Reduction and Solutions of the Hamilton-Jacobi
Equation . . . . . . . . . . . . . . . . . . . . . . . . . . .

www.pdfgrip.com

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

409
409

417
420
421
424
430
430
433
435
437
439
439
444
448
449

.
.

453
456

.

459

.
.
.
.

462
463
467
471

.
.

471
471

.
.

481
485

.
.
.
.
.
.
.

489
489
489
490
491

493
495

.......

499

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.


xxiv

Contents

7.3

The Geometrical Description of Reduction . . . . . . . . . . . . .
7.3.1 A Charged Non-relativistic Particle in a Magnetic
Monopole Field. . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 The Algebraic Description . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Additional Structures: Poisson Reduction . . . . . . . . .
7.4.2 Reparametrization of Linear Systems . . . . . . . . . . . .
7.4.3 Regularization and Linearization of the Kepler
Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.5 Reduction in Quantum Mechanics . . . . . . . . . . . . . . . . . . .
7.5.1 The Reduction of Free Motion in the Quantum Case .
7.5.2 Reduction in Terms of Differential Operators . . . . . .
7.5.3 The Kustaanheimo–Stiefel Fibration. . . . . . . . . . . . .
7.5.4 Reduction in the Heisenberg Picture . . . . . . . . . . . .
7.5.5 Reduction in the Ehrenfest Formalism . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8

9

...

500

.
.
.
.

.
.
.
.

.
.
.
.

503
504
506
508

.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.

514
520
520
522
524
527
532
535

..
..

539
539

..
..
..

541
542
546

Integrable and Superintegrable Systems . . . . . . . . . . . . . . . . . .
8.1 Introduction: What Is Integrability? . . . . . . . . . . . . . . . . . . .
8.2 A First Approach to the Notion of Integrability: Systems
with Bounded Trajectories . . . . . . . . . . . . . . . . . . . . . . . . .
8.2.1 Systems with Bounded Trajectories . . . . . . . . . . . . . .
8.3 The Geometrization of the Notion of Integrability . . . . . . . . .
8.3.1 The Geometrical Notion of Integrability
and the Erlangen Programme . . . . . . . . . . . . . . . . . .
8.4 A Normal Form for an Integrable System . . . . . . . . . . . . . . .
8.4.1 Integrability and Alternative Hamiltonian Descriptions .
8.4.2 Integrability and Normal Forms. . . . . . . . . . . . . . . . .
8.4.3 The Group of Diffeomorphisms of an Integrable
System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.4.4 Oscillators and Nonlinear Oscillators . . . . . . . . . . . . .
8.4.5 Obstructions to the Equivalence of Integrable Systems .
8.5 Lax Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.5.1 The Toda Model . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.6 The Calogero System: Inverse Scattering . . . . . . . . . . . . . . .
8.6.1 The Integrability of the Calogero-Moser System . . . . .
8.6.2 Inverse Scattering: A Simple Example . . . . . . . . . . . .
8.6.3 Scattering States for the Calogero System. . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.

.
.

.
.
.
.

548
550
550
552

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.

555
556
557
558
561
563
563
564
565
567

Lie–Scheffers Systems. . . . . . . . . . . . . .
9.1 The Inhomogeneous Linear Equation
9.2 Inhomogeneous Linear Systems . . . .
9.3 Non-linear Superposition Rule . . . . .
9.4 Related Maps . . . . . . . . . . . . . . . .

.
.
.
.
.

.
.

.
.
.

569
569
571
578
581

.......
Revisited
.......
.......
.......

www.pdfgrip.com

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.


Contents

xxv

9.5

Lie–Scheffers Systems on Lie Groups and Homogeneous
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.6 Some Examples of Lie–Scheffers Systems . . . . . . . . . . .
9.6.1 Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . .
9.6.2 Euler Equations. . . . . . . . . . . . . . . . . . . . . . . . .
9.6.3 SODE Lie–Scheffers Systems . . . . . . . . . . . . . . .
9.6.4 Schrödinger–Pauli Equation . . . . . . . . . . . . . . . .
9.6.5 Smorodinsky–Winternitz Oscillator . . . . . . . . . . .
9.7 Hamiltonian Systems of Lie–Scheffers Type . . . . . . . . . .
9.8 A Generalization of Lie–Scheffers Systems . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.

583
589
589
595
597
598

599
600
605
608

10 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

611
712

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

715

www.pdfgrip.com

.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

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 the
discussion of these examples we will have a catalogue of systems to test the ideas we
would be introducing later on; on the other hand this collection of simple systems
will 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 physicist’s wardrobe. Most of them are linear systems, even though we will show how
to construct non-trivial nonlinear systems out of them, and they are both finite and
infinite-dimensional.
We have chosen to present this collection of examples by using just elementary
notions from calculus and the elementary theory of differential equations. More
advanced 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 ones
will be introduced. We will leave most of the more advanced discussions on their
structure for later chapters, thus we must consider this presentation as a warmup and
also as an opportunity to think back on basic ideas.

© Springer Science+Business Media Dordrecht 2015
J.F. Cariñena et al., Geometry from Dynamics, Classical and Quantum,
DOI 10.1007/978-94-017-9220-2_1

www.pdfgrip.com

1