## Stability and Chaos in Celestial MechanicsThis 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

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

### Contents

LXX | 124 |

LXXI | 126 |

LXXIII | 131 |

LXXV | 140 |

LXXVI | 143 |

LXXVII | 145 |

LXXVIII | 148 |

LXXIX | 149 |

XII | 20 |

XIV | 23 |

XV | 25 |

XVI | 26 |

XVII | 31 |

XVIII | 32 |

XIX | 33 |

XX | 35 |

XXI | 39 |

XXII | 40 |

XXIII | 41 |

XXIV | 44 |

XXVI | 46 |

XXVII | 48 |

XXVIII | 49 |

XXIX | 50 |

XXX | 51 |

XXXI | 53 |

XXXIII | 57 |

XXXIV | 58 |

XXXV | 62 |

XXXVIII | 65 |

XXXIX | 67 |

XLI | 68 |

XLII | 73 |

XLIII | 76 |

XLIV | 83 |

XLVI | 85 |

XLVII | 87 |

XLVIII | 89 |

XLIX | 91 |

LI | 94 |

LII | 95 |

LIII | 96 |

LIV | 97 |

LV | 98 |

LVI | 99 |

LVII | 103 |

LVIII | 107 |

LX | 108 |

LXI | 110 |

LXII | 112 |

LXIV | 114 |

LXV | 115 |

LXVI | 116 |

LXVII | 118 |

LXVIII | 121 |

LXXX | 150 |

LXXXI | 152 |

LXXXII | 153 |

LXXXIII | 156 |

LXXXIV | 160 |

LXXXV | 162 |

LXXXVI | 165 |

LXXXVII | 168 |

LXXXVIII | 169 |

LXXXIX | 170 |

XC | 173 |

XCI | 177 |

XCIII | 178 |

XCIV | 182 |

XCV | 183 |

XCVII | 187 |

XCVIII | 191 |

CI | 193 |

CII | 194 |

CIII | 196 |

CIV | 198 |

CV | 199 |

CVI | 200 |

CVIII | 202 |

CIX | 203 |

CX | 204 |

CXI | 207 |

CXIII | 211 |

CXIV | 214 |

CXV | 215 |

CXVI | 218 |

CXVII | 222 |

CXVIII | 225 |

CXIX | 227 |

CXX | 229 |

CXXI | 232 |

CXXII | 233 |

CXXIII | 236 |

CXXIV | 239 |

CXXV | 240 |

CXXVI | 241 |

CXXVII | 245 |

CXXVIII | 247 |

250 | |

257 | |