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

Stable and Random Motions in Dynamical Systems

With Special Emphasis on Celestial Mechanics (AM-77)


Published: 6th May 2001
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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, Jurgen 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 entrees to the fascinating worlds of order and chaos in dynamics.

Forewordp. ix
Introductionp. 3
The stability problemp. 3
Historical commentsp. 8
Other problemsp. 10
Unstable and statistical behaviorp. 14
Planp. 18
Stability Problemp. 21
A model problem in the complexp. 21
Normal forms for Hamiltonian and reversible systemsp. 30
Invariant manifoldsp. 38
Twist theoremp. 50
Statistical Behaviorp. 61
Bernoulli shift. Examplesp. 61
Shift as a topological mappingp. 66
Shift as a subsystemp. 68
Alternate conditions for C[superscript 1]-mappingsp. 76
The restricted three-body problemp. 83
Homoclinic pointsp. 99
Final Remarksp. 109
Existence Proof in the Presence of Small Divisorsp. 113
Reformulation of Theorem 2.9p. 113
Construction of the root of a functionp. 120
Proof of Theorem 5.1p. 127
Generalitiesp. 138
Appendix to Chapter Vp. 149
Rate of convergence for scheme of [section]2b)p. 149
The improved scheme by Haldp. 151
Proofs and Details for Chapter IIIp. 153
Outlinep. 153
Behavior near infinityp. 154
Proof of Lemmas 1 and 2 of Chapter IIIp. 160
Proof of Lemma 3 of Chapter IIIp. 163
Proof of Lemma 4 of Chapter IIIp. 167
Proof of Lemma 5 of Chapter IIIp. 171
Proof of Theorem 3.7, concerning homoclinic pointsp. 181
Nonexistence of integralsp. 188
Books and Survey Articlesp. 191
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780691089102
ISBN-10: 0691089108
Series: Princeton Landmarks in Mathematics
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 216
Published: 6th May 2001
Country of Publication: US
Dimensions (cm): 22.86 x 15.24  x 1.27
Weight (kg): 0.33
Edition Type: Revised