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Mathematical methods of classical mechanics. (Математические методы классической механики.) (Russian) Zbl 0647.70001
Moskva: Nauka. 431 p. R. 1.10 (1974).
Ten years ago, while visiting Moscow, I discovered this book by chance; it has become a classic in the meantime and is translated as far as I know, into English (Zbl 0386.70001), Polish, French and German (review below). At the time of its appearance, it was the first book, covering modern geometric methods in mechanics, which could serve as a textbook at the upper undergraduate - lower graduate level, as compared to the well known books of R. Abraham and J. E. Marsden [Foundations of mechanics (1967; Zbl 0158.42901)], J.-M. Souriau [Structure des systèmes dynamiques (1970; Zbl 0186.58001)], J. C. Godbillon [Géométrie différentielle et méchanique analytique (1969; Zbl 0174.24602)], all written for, and digestible by, professionals. In the meantime, this list can be completed, e.g., by W. Thirring’s Lehrbuch der mathematischen Physik, Band I (1977; Zbl 0364.70001).
Arnol’d’s book covers in three parts Newtonian mechanics, Lagrangian mechanics and Hamiltonian mechanics with emphasize on the latter (differential forms, symplectic manifolds, canonical formalism, introduction to perturbation theory are the chapter headings there). Moreover, it contains a lot of appendices being related more or less closely to the main body of the text, beginning with Riemannian curvature and ending with the Korteweg-de Vries-equation, to give an impression what is going on there.
All is written in the author’s well known and celebrated style, which makes the book a pleasure to read. I think, this book is a must in everybody’s library, who is interested in the relations between mathematics and physics.

MSC:
70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70Hxx Hamiltonian and Lagrangian mechanics
37N05 Dynamical systems in classical and celestial mechanics