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Integrable systems of classical mechanics and Lie algebras. (Интегрируемые системы классической механики и алгебры Ли.) (Russian. English summary) Zbl 0717.70003
Moskva: Nauka. 240 p. R. 3.50 (1990).
Summary: The book is devoted to completely integrable systems in classical mechanics. The results and methods of the last century are demonstrated in detail, as well as data obtained during the last decade with the aid of the universe dispersion technique. Multiparticle systems are addressed, along with systems of the Todd chain type, geodesic systems on an ellipsoid. The monograph includes an advanced treatment of solid body movement around a stationary point and also solid body movement in an ideal liquid. Periodic problems are given throughout.
Contents: Chapter 1 contains the necessary background material and outlines the isospectral deformation method in a Lie-algebraic form. Chapter 2 gives an account of numerous previously known integrable systems. Chapter 3 deals with many-body systems of generalized Calogero-Moser type, related to root systems of simple Lie algebras. Chapter 4 is devoted to the Toda lattice and its various modifications seen from the group-theoretical point of view. Chapter 5 investigates some additional topics related to many-body systems.

70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
17B81 Applications of Lie (super)algebras to physics, etc.
70H05 Hamilton’s equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics