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Modern celestial mechanics. Aspects of solar system dynamics. (English) Zbl 1411.70019
Advances in Astronomy and Astrophysics 5. London: Taylor & Francis (ISBN 978-0-415-27938-3). xi, 356 p. (2002).
From the preface: In the last 20 years, Celestial Mechanics has achieved spectacular results on the structure and evolution of our Solar System. The discovery of the chaotic dynamics of the planets, the identification of the main transport routes by which asteroids escape from the main belt and reach Earth-threatening orbits, and the understanding of the dynamical mechanisms at the origin of the internal heating of the Galilean satellites, are just a few examples of the results that have made Celestial Mechanics a respectable branch of Solar System science. As a consequence, worldwide congresses on planetology, such as the annual meetings of the Division for Planetary Sciences of the American Astronomical Society or those of the Asteroids Comets & Meteoroids series, always schedule long and well-attended sessions on dynamics.
This is what I call Modern Celestial Mechanics, although one should not forget that other facets of Celestial Mechanics, concentrated more on the rigorous mathematical study of toy models rather than on the realistic description of Solar System evolution, have also recently achieved innovative results.
Several books have been published on Celestial Mechanics, but none of them describe the recent results on Solar System dynamics, with a solid theoretical basis. They are either focused on the general theory of dynamical systems or limited to the fundamentals of Celestial Mechanics. The present book should fill this gap, in the hope of helping advanced students and young researchers, who are nowadays compelled to find their way through a vast literature of scientific papers without the aid of a guide book presenting the state of the art in a unified form. The goal is to take the reader to a point where he can start developing his own original contribution.
Modern Celestial Mechanics is intimately related to the theory of Hamiltonian systems. Most analytic studies make an essential use of Hamiltonian perturbation techniques; but also the correct interpretation of the results of numerical simulations often requires a good theoretical knowledge of Hamiltonian dynamics. Because Hamiltonian theory does not enter in the usual cultural baggage of people approaching Celestial Mechanics from the astronomical side, the first part of this book revisits its fundamental concepts. Without the pretence of being exhaustive, the first six chapters present what one should know of Hamiltonian theory to work at ease in Celestial Mechanics.
Details and technical mathematical proofs are skipped, while only the guiding ideas are given, with specific references for those who would like to enter deeper into the subject. After the first chapter on the basic concepts of Celestial and Hamiltonian Mechanics, Chapter 2 explains Hamiltonian perturbation theory based on Lie series. Chapters 3 and 4 illustrate the properties of invariant tori and resonances, respectively. Particular care is paid in Chapter 5 to discussing the numerical tools that are useful for the detection of chaos. Chapter 6 discusses the possible dynamical structures of Hamiltonian systems that result from the interaction of its resonances, and details how these structures can be identified with numerical explorations.
The second part of the book is devoted to the subject of Celestial Mechanics itself. This part is more technical, although care is taken to concentrate on the procedures and on the ideas, and not on the most technical details. Thus, the book should also be pleasant for reading experts in Hamiltonian theory, who are curious to know what is done in Modern Celestial Mechanics. In particular, this book does not go into detail on the techniques for practical computations (series expansion or close evaluation of the perturbation functions, numerical calculation of the action integrals, etc.). In fact, practical techniques evolve very fast, in parallel with the evolution of computer power, while the ideas and the conceptual approaches stay valid for a much longer time. Chapters 7 and 8 are respectively devoted to the secular motion of planets and of small bodies. The chaotic dynamics of the terrestrial planets, the theories for the computation of an asteroid’s proper elements and the dynamics of secular resonances are among the issues treated in these chapters.
Chapters 9-12, conversely, are devoted to the difficult subject of mean motion resonances. In particular, Chapter 9 outlines the structure of mean motion resonances, first in the framework of the simple planar restricted three-body problem, and then in more realistic models. Chapter 10 is on three-body resonances, whose importance has been recently understood both for small body dynamics and for planetary dynamics. Chapter 11 discusses the secular dynamics inside mean motion resonances, which in my opinion is one of the most complicated topics of Celestial Mechanics. Finally Chapter 12 investigates the global dynamical structure of the regions of the Solar System that are densely inhabited by small bodies, and discusses the fashionable subject of slow chaotic diffusion.
As one can see, the book is strongly focused on the dynamics of planets and of small bodies. This is mainly due to the limitation of my knowledge. Important problems related to the dynamics of natural satellites and of planetary rings (see the Introduction for an overview) are therefore not discussed. However, most of the concepts developed here apply also to satellite and ring dynamics, so that I hope that this book can serve as an introduction also for people interested in these fields. Moreover, I have decided to exclude from this book the dynamics of bodies that have close encounters with the planets, because it does not fall into the category of quasi-integrable dynamics, and its relationship with the rest of the book would be very weak. In fact, the study of planet crossing dynamics is essentially numeric, the possibilities of analytic and semi-analytic theories being very limited.

MSC:
70F15 Celestial mechanics
70-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mechanics of particles and systems
37N05 Dynamical systems in classical and celestial mechanics
65P20 Numerical chaos
70Hxx Hamiltonian and Lagrangian mechanics
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