Zehnder E. Lectures on Dynamical Systems: Hamiltonian Vector Fields and Symplectic Capacities

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Eduard Zehnder

Lectures on

Dynamical Systems

Hamiltonian Vector Fields and Symplectic Capacities

Author:

Eduard Zehnder

Departement Mathematik

ETH Zürich, LEO D 3

Leonhardstrasse 27

8092 Zürich

Switzerland

E-mail: zehnder@math.ethz.ch

2010 Mathematical Subject Classification (primary; secondary): 37-01; 37A25, 37B25, 37C10,

37C20, 37C27, 37C29,37C50, 37C57, 37D05, 37J05, 37J35, 37J45, 37N05, 34C25, 34C28,

34C37, 34C45, 34D20, 70H15,70H05, 70H25, 53D35, 53C24, 53D05, 58E05, 58E30

Key words: Dynamical systems, ergodicity, transitivity, structural stability, stable and unstable mani-

folds, hyperbolic sets, shadowing lemma, homoclinic and heteroclinic orbits, Bernoulli systems, chaos,

limit sets, Lyapunov function, gradient systems, Morse inequalities, symplectic maps, Hamiltonian vec-

tor fields, Hamiltonian formalism, integrable systems, symplectic capacities, Hofer–Zehnder capacity,

action functional, mini-max principles, periodic motions on prescribed energy surfaces

ISBN 978-3-03719-081-4

The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography,

and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch.

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Contact address:

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Typeset using the author’s TEX files: I. Zimmermann, Freiburg

Printed in Germany

9 8 7 6 5 4 3 2 1

Preface

The ﬁeld of dynamical systems originated in the difﬁcult mathematical questions

related to movements of the planets and the moon, questions like: Are there periodic

orbits? Will the solar system keep its present beautiful form, also in the distant

future, or could it happen that one of the planets, Jupiter for instance, leaves the

system? Or could it come to a collision between planets, leading to a dramatic

change of the solar system?

The mathematical theory of dynamical systems provides concepts, ideas and

tools, in order to analyze and model dynamical processes in all ﬁelds of natural

sciences, making use of nearly all branches of mathematics. On the other hand,

already in the past, questions of dynamical systems in the real world have triggered

new mathematical developments and led to whole new branches of mathematics.

Here, typical questions would be: Knowing its present state, how will a dynamical

system develop in the long run? Will it, for example, tend to an equilibrium state

or will it come back to itself? What will happen to the long-time behavior if we

change the initial conditions a little bit? And what will happen to the whole orbit

structure of a system if we perturb the system itself?

The book addresses readers familiar with standard undergraduate mathematics.

It is not a systematic monograph, but rather the lecture notes of an introductory

course in the ﬁeld of dynamical systems given in the academic year 2004/2005 at

the ETH in Zürich for third year students in mathematics and physics. I selected

relatively few topics, tried to keep the requirements of mathematical techniques

minimal and provided detailed (sometimes excruciatingly detailed) proofs.

The introductory chapter discusses simple models of discrete dynamical sys-

tems, in which the dynamics is determined by the iteration of a map. There are

examples for minimal, transitive, structurally stable and ergodic systems. Map-

pings that preserve the measure of a ﬁnite measure space have strong recurrence

properties in view of a classical result due to H. Poincaré. In order to describe the

statistical distribution of their orbits, the ergodic theorem of G. Birkhoff is proved.

Chapters II and III are devoted to unstable phenomena caused by a hyperbolic

ﬁxed point of a diffeomorphism. Such a point gives rise to two global invariant

sets, the so-called stable, respectively unstable, invariant manifolds issuing from

ﬁxed point. These consist of points which tend to the ﬁxed point under the iteration

of the map and under the iteration of the inverse map, respectively. The transver-

sal intersection of the stable and unstable manifolds in the so-called homoclinic

points is one of the roads to chaos. The existence of homoclinic points, discovered

by H. Poincaré in the 3-body problem of celestial mechanics, complicates the orbit

structure considerably and gives rise to invariant hyperbolic sets. The chaotic struc-

ture of the orbits near such sets is analyzed by means of the Shadowing Lemma,

vi Preface

which is also used to demonstrate S. Smale’s theorem about the embeddings of

Bernoulli systems near homoclinic orbits. The interpretation of unpredictable or-

bits, which are determined by random sequences, will be demonstrated in the simple

system of a periodically perturbed mathematical pendulum.

In Chapter IV we deal with smooth ﬂows generated by vector ﬁelds and with

continuous ﬂows on metric spaces. The concepts of limit set, attractor and Lyapunov

function are introduced. The bounded solutions of a gradient-like ﬂow tend to rest

points in forward and in backward time. The rest points are found by mini-max

principles. The intimate and fruitful relation between the dynamics of gradient

ﬂows and the topology of the underlying compact manifold is described by the

Morse inequalities. The Morse theory is sketched at the end of the chapter.

ChapterV introduces the special class of Hamiltonian vector ﬁelds that are deter-

mined by a single function and deﬁned on symplectic manifolds. These manifolds

are even-dimensional and carry a symplectic structure. A symplectic structure is

a 2-form that is closed and nondegenerate. In contrast to Riemannian structures

which do exist on every manifold, not every even-dimensional manifold admits

a symplectic structure. Symplectic manifolds of the same dimension are locally

indistinguishable (Darboux). There are no local symplectic invariants. The Hamil-

tonian formalism is developed in the convenient language of the exterior calculus

which will be brieﬂy introduced. The very special integrable Hamiltonian systems

are characterized by the property that they possess sufﬁciently many integrals of

motion, so that the task of solving the Hamiltonian equations for all time becomes

almost trivial. This will follow from the existence of action- and angle-variables

established by V. Arnold and R. Jost.

Chapter VI motivates the study of global symplectic invariants different from

the volume which will be introduced in Chapter VII. They are called symplectic

capacities and go back to I. Ekeland and H. Hofer. In view of their monotonicity

properties they represent, in particular, obstructions to symplectic embeddings. The

Gromov non-squeezing phenomenon is an immediate consequence. A symplectic

capacity of dynamical nature (Hofer–Zehnder capacity) measures the minimal oscil-

lations of Hamiltonian functions needed to conclude the existence of a fast periodic

solution of the associated Hamiltonian vector ﬁeld. Its construction is based on

a variational principle for the action functional of classical mechanics. The tools

from the calculus of variations are developed from scratch.

ChapterVIII deals with applications of dynamical symplectic capacity to Hamil-

tonian systems. It turns out that a compact and regular energy surface gives rise to

an abundance of periodic orbits nearby, if a neighborhood of the surface possesses

a ﬁnite Hofer–Zehnder capacity. The existence of a periodic solution on the given

energy surface (and not only nearby) necessarily requires additional properties of

the surface. The contact type property, for example, immediately leads to the solu-

tion of the Weinstein conjecture due to C. Viterbo which generalizes the pioneering

results of P. Rabinowitz and A. Weinstein. Finally, a classical result of H. Poincaré

Preface vii

shows that a periodic solution of a Hamiltonian system is, in general, not isolated

but belongs to a smooth family of periodic solutions parametrized by the energy

and having similar periods.

The chapters devoted to Hamiltonian systems and their global periodic orbits

related to the symplectic capacities rely heavily on the book Symplectic Invariants

and Hamiltonian Dynamics [52] by H. Hofer and E. Zehnder and on the Notes in

Dynamical Systems [74] by J. Moser and E. Zehnder.

Each chapter begins with a short survey of its contents and ends with a brief

selection of references to literature giving an alternative view on the subjects or

describing related and more advanced topics.

I would like to thank Marcel Nutz for setting the German text into L

X, for

many improvements and for producing the beautiful ﬁgures. I also would like

to thank Manfred Karbe and Irene Zimmermann for their encouragement, their

patience, their careful editing and for many helpful suggestions. For his generous

help in preparing and improving the text of these lectures, I am very grateful to

Felix Schlenk. I thank Beata and Kris Wysocki for their patient assistance with

technical problems. For her translation of the German text into English, I am very

grateful to Jeannette Zehnder.

Zürich, March, 2010

Contents

Preface v

I Introduction 1

I.1 N -body problem of celestial mechanics ............ 1

I.2 Mappings as dynamical systems ................ 3

I.3 Transitive dynamical systems ................. 14

I.4 Structural stability ....................... 18

I.5 Measure preserving maps and the ergodic theorem ...... 26

II Invariant manifolds of hyperbolic ﬁxed points 46

II.1 Hyperbolic ﬁxed points .................... 47

II.2 Local invariant manifolds ................... 58

II.3 Stable and unstable invariant manifolds ............ 69

III Hyperbolic sets 81

III.1 Deﬁnition of a hyperbolic set ................. 81

III.2 The shadowing lemma ..................... 89

III.3 Orbit structure near a homoclinic orbit, chaos ......... 95

III.4 Existence of transversal homoclinic points .......... 107

III.5 Torus automorphisms ..................... 116

III.6 Invariant manifolds of ƒ .................... 121

III.7 Structural stability on hyperbolic sets ............. 124

IV Gradientlike ﬂows 127

IV.1 Flow of a vector ﬁeld, recollections from ODE ........ 127

IV.2 Limit sets, attractors and Lyapunov functions ......... 136

IV.3 Gradient systems ........................ 149

IV.4 Gradient systems on manifolds and Morse theory ....... 161

V Hamiltonian vector ﬁelds and symplectic diffeomorphisms 182

V.1 Symplectic vector spaces .................... 182

V.2 The exterior derivative d .................... 194

V.3 The Lie derivative L

of forms ................ 198

V.4 The Lie derivative L

of vector ﬁelds ............. 201

V.5 Commuting vector ﬁelds .................... 203

V.6 The exterior derivative d on manifolds ............ 203

V.7 Symplectic manifolds ..................... 206

V.8 Symplectic maps ........................ 209

x Contents

V.9 Generating functions of symplectic maps in R

....... 218

V.10 Integrable systems, action–angle variables .......... 225

VI Questions, phenomena, results 239

VI.1 Geometric questions ...................... 239

VI.2 Approximation of measure preserving diffeomorphisms . . . 243

VI.3 A dynamical question ..................... 244

VI.4 A connection between geometry and Hamiltonian dynamics . 247

VII Symplectic invariants 252

VII.1 Symplectic capacities and ﬁrst applications .......... 252

VII.2 The Hofer–Zehnder capacity c

................ 260

VII.3 Minimax principles ....................... 270

VII.4 The functional analysis of the action functional ........ 276

VII.5 Existence of a critical point of ˆ ................ 292

VIII Applications of the capacity c

in Hamiltonian systems 307

VIII.1 Global periodic solutions on prescribed energy surfaces . . . 307

VIII.2 Hypersurfaces of contact type ................. 317

VIII.3 Examples from classical mechanics .............. 322

VIII.4 Poincaré’s continuation method ................ 327

VIII.5 Transversal sections on energy surfaces ............ 333

Bibliography 339

List of Symbols 347

Index 349