UK Nonlinear News, August 2001

Stable and Random Motions in Dynamical Systems

By Jürgen Moser

Reviewed by Sebastian Reich

Princeton University Press
Hardcover - 210 pages 
ISBN: 0691089108

Jürgen Moser, who died in 1999, was one of the leading mathematicians of the twenties century, whose deep and important contributions range over many different fields such as dynamical systems and celestial mechanics, partial differential equations, nonlinear functional analysis, complex geometry, and the calculus of variations. He is perhaps best known through the'M' in the celebrated KAM - Kolmogorov/Arnold/Moser - theorem.

The book under review first appeared in Princeton Landmarks in Mathematics and Physics a few years after the publication of the English translation of the classic  Lectures on Celestial Mechanics by Siegel and Moser. It is the outgrowth of five lectures given by Moser at the Institute of Advanced Studies, Princeton, in 1972 and is witness to the enormous advance achieved in the qualitative theory of dynamical systems during the late 1950ies and 1960ies. It has now been reprinted for it reflects two persistent themes in dynamical systems and celestial mechanics, namely stable and random motion, and it represents an important step in the development of the qualitative theory of celestial mechanics. A foreword by Philip Holmes has been added to the new addition, which nicely summarizes the history and the background of Moser's contribution.

The stability problem in celestial mechanics can be traced back as far as Newton. Its modern geometric formulation is due to Poincar\'e. However, it was believed till the early 1950ies that invariant tori of integrable systems would generically be destroyed by most perturbations. This was stunningly disproved in a theorem announced by Kolmogorov in 1954, proved for analytic flows by Arnold, and for sufficiently smooth maps by Moser in the early 1960ies. The material on normal forms, a version of the KAM theorem on the persistence of invariant tori, and Moser's twist theorem for area-preserving maps form Chapter II of the book. The proofs and technical details are provided in Chapter IV. To keep the material in Chapter IV as accessible as possible, the KAM theorem from Chapter II is first reformulated in terms of reversible equations. The key idea of fast iteration methods is very carefully explained.

The second strand that makes up Moser's book is randomness. This type of behavior is typically assumed in molecular and statistical mechanics. It can be shown rigorously that ergodic behavior arise in the context of certain geodesic flows. However, as Moser points out in the Introduction, ergodic behavior on the complete phase space is a rather exceptional case and one has to expect a mixture of ``stable'' and ``random'' motion, in general. Chapter III of the book is devoted to this aspect of dynamical systems. In particular, Moser gave the first complete and accessible account of the Smale-Birkhoff homoclinic theorem and a fully-worked example provided by solutions to the restricted three-body problem. Again, technical details are given in an appendix. Till today the Smale-Birkhoff theorem coupled with Melnikov's method provides a most useful tool to prove and analyse the presence of chaotic motion.

Since the first publication of this 'Princeton Landmark in Mathematics and Physics' progress has been achieved in many different areas of classical mechanics. We mention the work of Nekhoroshev on exponentially slow Arnold diffusion for nearly integrable systems, recent contributions to the ergodic behavior of non-uniformly hyperbolic systems, Aubry's and Mather's work on closed invariant Cantor sets and the break-up of KAM tori, integrable PDEs, and, finally, the idea of transport in Hamiltonian systems. Most of these contributions either take stable or random motion as a starting point. Moser's book gives an elegant and very readable introduction to these two opposite, but ultimately linked, strands of dynamical systems and celestial mechanics. Its style and presentation make it ideally suited for a reading course. The book is a landmark indeed.

A listing of books reviewed in UK Nonlinear News is available.

UK Nonlinear News thanks Princeton University Press for providing a review copy of this book

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Last Updated: 6th August 2001.