Notes on Dynamical System s

Courant Lecture Notes in Mathematics

Executive Editor Jalal Shatah

Managing Editor Paul D. Monsour

Assistant Editor Reeva Goldsmith Suzan Toma

Copy Editor Josh Singer

Jtirgen Moser

Eduard J. Zehnder ETH Zurich

12 Note s on Dynamical Systems

Courant Institute of Mathematical Science s New York University New York, New York

American Mathematical Societ y Providence, Rhode Island

http://dx.doi.org/10.1090/cln/012

2000 Mathematics Subject Classification. P r imar y 37-01 , 37Kxx , 53Dxx , 58Exx , 70Fxx , 70H05.

For addi t iona l informatio n an d upda t e s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / c l n - 1 2

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Moser, Jiirgen , 1928 -Notes o n dynamica l system s / Jiirge n Moser , Eduar d J . Zehnder .

p. cm . — (Couran t lectur e note s i n mathematic s ; 12 ) Includes bibliographica l references . ISBN 0-8218-3577- 7 (alk . paper ) 1. Differentiabl e dynamica l systems . 2 . Hamiltonia n systems . 3 . Transformation s (Mathe -

matics) 4 . Combinatoria l dynamics . I . Zehnder , Eduard , 1940 - II . Title . III . Series .

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Contents

Preface vi i

Chapter 1 . Transformatio n Theor y 1 1.1. Differentia l Equation s and Vector Fields 1 1.2. Variationa l Principles, Hamiltonian Systems 1 1 1.3. Canonica l Transformations 1 9 1.4. Hamilton-Jacob i Equations 2 9 1.5. Integral s and Group Actions 4 1 1.6. Th e SO(4) Symmetry of the Kepler Problem 5 2 1.7. Symplecti c Manifolds 6 2 1.8. Hamiltonia n Vector Fields on Symplectic Manifolds 7 3

Chapter 2. Periodi c Orbits 8 5 2.1. Poincare' s Perturbation Theory of Periodic Orbits 8 5 2.2. A Theorem by Lyapunov 9 4 2.3. A Theorem by E. Hopf 9 8 2.4. Th e Restricted 3-Body Problem 10 2 2.5. Reversibl e Systems 10 8 2.6. Th e Plane 3- and 4-Body Problems 11 6 2.7. Poincare-Birkhof f Fixe d Point Theorem 12 0 2.8. Variation s on the Fixed Point Theorems 13 3 2.9. Th e Billiard Ball Problem 14 0 2.10. A Theorem by Jacobowitz and Hartman 14 9 2.11. Close d Geodesies on a Riemannian Manifold 16 0 2.12. Periodi c Orbits on a Convex Energy Surface 17 2 2.13. Periodi c Orbits Having Prescribed Periods 18 1

Chapter 3. Integrabl e Hamiltonian Systems 18 7 3.1. A Theorem of Arnold and Jost 18 7 3.2. Delauna y Variables 19 9 3.3. Integral s via Asymptotics; the Stormer Problem 20 7 3.4. Th e Toda Lattice 21 3 3.5. Separatio n of Variables 22 8 3.6. Constraine d Vector Fields 23 4 3.7. Isospectra l Deformations 24 2

Bibliography 25 5

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Preface

Written i n 1979-80 , thes e note s constitut e th e first three chapter s o f a book that was never finished. It was planned as an introduction to the field of dynamical systems, i n particular , o f the specia l clas s o f Hamiltonia n systems . W e aimed a t keeping the requirements o f mathematical techniques minima l but giving detaile d proofs an d many examples and illustrations from physic s and celestial mechanics . After all , the celestial iV-body problem is the origin of dynamical systems and gave rise in the past to many mathematical developments.

The first chapter is about the transformation theory of systems and also contains the so-called Hamiltonian formalism. The second chapter is devoted to periodic phe-nomena and starts with perturbation methods going back to H. Poincare and local existence results due to Lyapunov and E. Hopf. Classica l periodic solutions are es-tablished in the restricted 3-body problem and the celestial 3- and 4-body problems. Variational technique s the n allo w searchin g fo r globa l periodi c orbit s lik e close d geodesies o n Riemannian manifold s an d closed orbit s o n convex energ y surface s of general Hamiltonian systems . Th e Poincare-Birkhoff fixed point theorem of an area-preserving annulus map in the plane is also proven in the second chapter. Thi s theorem led to the V. Arnold conjectures abou t forced oscillations of time-periodic Hamiltonian systems on symplectic manifolds. Incidentally , afte r these notes were written, th e Arnold conjecture s triggere d ne w development s i n symplecti c geom -etry an d Hamiltonia n systems . Also , i t turne d ou t tha t th e periodi c phenomen a of Hamiltonian systems are intimately related to symplectic invariants and surpris-ing symplecti c rigidit y phenomen a discovere d b y Y . Eliashberg an d M. Gromov . These more recent development s ar e presented i n the book Symplectic Invariants and Hamiltonian Dynamics by H. Hofer and E. Zehnder.

The third chapte r i s devoted to a special an d interesting clas s o f Hamiltonia n systems possessing many integrals. Following the construction of the so-called ac-tion and angle variables, illustrated by the Delaunay variables, several examples of integrable systems are described in detail. Chapter s 4 and 5 should have dealt with the analytically subtle stability problems in Hamiltonian systems close to integrable systems known as KAM theory, and with unstable hyperbolic solutions , which, in general, d o coexist wit h the stabl e solutions . Unfortunately , thes e chapter s wer e never completed.

These notes owe much to Jiirgen Moser's deep insight into dynamical system s and his broad view of mathematics. They also reflect his specific approach to math-ematics by singling out inspiring typical phenomena rather than designing abstrac t theories.

Finally, I would like to thank Paul Wright for carefully checkin g these notes.

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Bibliography

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