Stability and Chaos in Celestial Mechanics

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Springer Science & Business Media, Mar 10, 2010 - Science - 264 pages
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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.

 

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To the memory of Vladimir Igorevich Arnol'd
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
 

Contents

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About the author (2010)

ALESSANDRA CELLETTI was born in 1962 (Rome, Italy). She obtained a master degree in Mathematics at the University of Rome "La Sapienza" in 1984 and a Ph.D. at the Polytechnical Institute of Technology (ETH) in ZA1/4rich (Switzerland). Since 1999 she is Associate Professor at Dept. of Mathematics of the University of Rome "Tor Vergata." Her primary area of research concerns Mathematical Physics and Celestial Mechanics. She is the author of scientific papers published in high-level international journals and she has been involved in several international meetings and schools. She is president of the Italian Society of Celestial Mechanics and Astrodynamics.
SYLVIO FERRAZ-MELLO was born in 1936 (Sao Paulo, Brazil). He obtained a BSc in Physics in 1959 and a Dr. dAEtat in Paris in 1967. Then he became Professor at the University of Sao Paulo. He was former president of the Commission 7 (IAU).
His working subjects refer to the dynamics of natural and artificial satellites, variable stars, asteroid dynamics, extra-solar planetary systems and dynamical systems. He is the author of scientific papers published in high-level international journals and he has been involved in several international meetings and schools.

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