Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics (AM-77)

by Jurgen Moser

Overview

For centuries, astronomers have been interested in the motions of the planets and in methods to calculate their orbits. Since Newton, mathematicians have been fascinated by the related N-body problem. They seek to find solutions to the equations of motion for N masspoints interacting with an inverse-square-law force and to determine whether there are quasi-periodic orbits or not. Attempts to answer such questions have led to the techniques of nonlinear dynamics and chaos theory. In this book, a classic work of modern applied mathematics, Jürgen Moser presents a succinct account of two pillars of the theory: stable and chaotic behavior. He discusses cases in which N-body motions are stable, covering topics such as Hamiltonian systems, the (Moser) twist theorem, and aspects of Kolmogorov-Arnold-Moser theory. He then explores chaotic orbits, exemplified in a restricted three-body problem, and describes the existence and importance of homoclinic points. This book is indispensable for mathematicians, physicists, and astronomers interested in the dynamics of few- and many-body systems and in fundamental ideas and methods for their analysis. After thirty years, Moser's lectures are still one of the best entrées to the fascinating worlds of order and chaos in dynamics.

Product Details

ISBN-13:	9780691089102
Publisher:	Princeton University Press
Publication date:	05/06/2001
Series:	Princeton Landmarks in Mathematics and Physics , #31
Edition description:	With a New foreword by Philip J. Holmes
Pages:	216
Product dimensions:	6.00(w) x 9.25(h) x (d)

About the Author

Jürgen Moser, who died in 1999, was one of the most influential mathematicians of his generation, making important contributions in dynamical systems and nonlinear analysis. Among other posts, he was variously Director of the New York University Courant Institute, Director of the Research Institute for Mathematics at Switzerland's
Federal Institute of Technology, and President of the International Mathematical Union. He was awarded the 1994/95 Wolf Prize.

Table of Contents

Foreward ix

I. INTRODUCTION 3

1. The stability problem 3

2. Historical comments 3

3. Other problems 8

4. Unstable and statistical behavior 14

5. Plan 18

II. STABILITY PROBLEM 21

1. A model problem in the complex 21

2. Normal forms for Hamiltonian and reversible systems 30

3. Invariant manifolds 38

4. Twist theorem 50

III. STATISTICAL BEHAVIOR 61

1. Bernoulli shift. Example 61

2. Shift as a topological mapping 66

3. Shift as a subsystem 68

4. Alternate conditions for C'-mappings 76

5. The restricted three-body problem 83

6. Homoclinic points 99

IV. FINAL REMARKS 113

V. EXISTENCE PROOF IN THE PRESENCE OF SMALL DIVISORS 113

1. Reformulation of Theorem 2.9 113

2. Construction of the root of a function 120

3. Proof of Theorem 5.1 127

4. Generalities 138

A. Appendix to Chapter V 149

a. Rate of convergence for scheme of s.2b) 149

b. The improved scheme by Hald 151

VI. PROOFS AND DETAILS FOR CHAPTER III 153

1. Outline 153

2. Behavior near infinity 154

3. Proof of Lemmas 1 and 2 of Chapter III 160

4. Proof of Lemma 3 of Chapter III 163

5. Proof of Lemma 4 of Chapter III 167

6. Proof of Lemma 5 of Chapter III 171

7. Proof of Theorem 3.7, concerning homoclinic points 181

8. Nonexistence of intergals 188

BOOKS AND SURVEY ARTICLES 191