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1937-2010
Stability and Chaos in Celestial Mechanics,
Springer, Jointly published with Praxis Publishing,
UK 2010, 264 p., 2010
Professor Alessandra Celletti
Soa 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