Stability and Chaos in Celestial Mechanics

Alessandra Celletti

Mar 2010 · Springer Science & Business Media

About this eBook

This overview of classical celestial mechanics focuses the interplay with dynamical systems. Paradigmatic models introduce key concepts – order, chaos, invariant curves and cantori – followed by the investigation of dynamical systems with numerical methods.

Ratings and reviews

1937-2010 Stability and Chaos in Celestial Mechanics, Springer, Jointly published with Praxis Publishing, UK 2010, 264 p., 2010 Professor Alessandra Celletti Soa June 2010 Review report The aim of this book is to demonstrate a modern aspects of celestial mechanics. After giving to the reader the pedagogical introduction, needed for a basic understanding of the underlying physical phenomena, based on the interplay of classical celestial mechanics with dynamical systems, she uses paradigmatic models, such as the logistic map or the standard map, to introduce the reader to the concepts of order, chaos, invariant curves, cantori, etc. The second chapter presents numerical methods to investigate a dynamical systems: Poincare’ mapping, Lyapunov exponents, frequency analysis and fast Lyapunov indicators for distinguishing between regular and chaotic motions, etc. Then she reviews the classical two-body problem and proceeds to explore the three-body model in order to investigate orbital resonances and Lagrange solutions. A perturbative approach to find periodic orbits is presented together with an application to the computation of the libration in longitude of the Moon. The study of collisions in the solar system is approached through regularization theory. The main ideas of Kolmogorov-Arnold-Moser (KAM) [1, 2, 3] theorem is discussed, and also a dissipative version of the theorem is provided. The KAM theorem is proved in details for the specific case of spin orbit model. The author provide also a brief introduction to a dissipative KAM theorem and to non-existence criteria of invariant tori. The eight chapter of the book is devoted to the proof of Nekhoroshev’s theorem for the long-term stability of the near-integrable Hamiltonian systems. The book contain a set of Appendices (A–G) good for quick reference to the literature, see also [3]. The present excellent book is devoted to advance level undergraduate students as well as postgraduate students and researchers. The author provided comprehensive literature, 175 items. References [1] Vladimir Igorevich Arnol’d, Small denominators and problems of stability of motion in classical and celestial mechanics, U.S. Dept. of Commerce, Office of Technical Services, 1964. [2] Alessandra Celletti, Luigi Chierchia, KAM stability and celestial mechanics, AMS Bookstore, 2007. [3] Alessandra Celletti, Ettore Perozzi (Editors) , Celestial mechanics: the waltz of the planets, Springer Praxis Books Subseries: Popular Astronomy, Jointly published with Praxis Publishing, UK, 245p. 2007. Reviewer: Nikolay Asenov Kostov