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Mathematical aspects of classical and celestial mechanics. Transl. from the Russian by E. Khukhro. 3rd revised ed. (English) Zbl 1105.70002
Encyclopaedia of Mathematical Sciences 3. Dynamical Systems 3. Berlin: Springer (ISBN 3-540-28246-7/hbk). xiii, 518 p. (2006).
This beautiful book is the third edition of previous ones published by the same authors [see for the review Zbl 0623.00023, Zbl 0679.00016, Zbl 0785.00010, Zbl 0885.70001] and devoted to an overview of classical mechanics. It emphasizes some fundamental aspects of the mathematical formulation of systems with a finite number of degrees of freedom (however, some problems of fluid mechanics are considered, too). It gives a panorama of classical mathematical approaches and some open questions, more than enter in specific technicalities. For these interested readers can refer to the quoted literature in the rich bibliography (627 references)). The book splits in nine chapters. 1. Basic principles of classical mechanics. 2. The \(n\)-body problem. 3. Symmetry groups and order reduction. 4. Variational principles and methods. 5. Integrable systems and integration methods. 6. Perturbation theory for integrable systems. 7. Non-integrable systems. 8. Theory of small oscillations. 9. Tensor invariants of equations of dynamics.
Reviewer’s remark. The style is simple and clear, and does not require a particular mathematical background. So, the impression is to read some of one classical books written in the past by giants of the mathematics. In fact, reading this book I remembered another one where I had the same impression, i.e., the book by David Hilbert and Stefan Cohn-Vossen, entitled ”Anschauliche Geometrie”, with an appendix by Pavel Sergeevich Aleksandrov, entitled ”Einfachste Grundbegriffe der Topologie”, published by Julius Springer in Berlin in 1932 [Zbl 0005.11202; for the 2nd German ed. (1996) see Zbl 0847.01034]. Both books, even if devoted to different areas of mathematics, appear do mathematics without mathematics. Certainly this is art by geniuses.

70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70Hxx Hamiltonian and Lagrangian mechanics
70Gxx General models, approaches, and methods in mechanics of particles and systems
70Fxx Dynamics of a system of particles, including celestial mechanics
37N05 Dynamical systems in classical and celestial mechanics
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