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Publisher:

Princeton University Press

Publication Date:

2001

Number of Pages:

200

Format:

Paperback

Series:

Princeton Landmarks in Mathematics and Physics

Price:

34.95

ISBN:

978-0691089102

Category:

General

[Reviewed by , on ]

Fernando Q. Gouvêa

05/30/2001

"Of the making of new books there is no end," said the Preacher. He (she?) might have added "and of the reprinting of old books there is not enough." All too often great old books become unavailable and students are denied the opportunity to learn from them. This is all the more serious when the books in question are written by the original creators of the topics they discuss.

Here's one. Jürgen Moser's *Stable and Random Motions in Dynamical Systems* is motivated by the stability problem in celestial mechanics: can we prove that the solar system is stable? In the first chapter, Moser explains the historical roots of the question, makes it precise, and sets up the mathematical questions that the rest of the book will address. The other chapters center on two big theorems: the Kolmogorov-Arnold-Moser (KAM) theorem, dealing with quasi-periodic motions, and the Smale-Birkhoff theorem, connecting dynamical systems with Bernoulli processes. These are, respectively, the "stable" and "random" aspects mentioned in the title.

The book is part of Princeton University Press's *Landmarks in Mathematics* series. The series consists of paperback reprints of old mathematical classics, ranging from Milnor's *Topology from the Differentiable Viewpoint* to Cartan and Eilenberg's *Homological Algebra*. All in all, a worthy endeavor.

Fernando Q. Gouvêa (fqgouvea@colby.edu) is the editor of **FOCUS** and **MAA Online**. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.

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

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