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Billiards. A genetic introduction to the dynamics of systems with impacts. (Биллиарды. Генетическое введение в динамику систем с ударами.) (Russian) Zbl 0751.70009

Moskva: Izdatel’stvo Moskovskogo Universiteta. 168 p. (1991).
The book is devoted to the mathematical foundations of dynamical systems of the billiard type. The historical development of the main ideas of the impact theory is presented. The treatment is based on the genetic approach to the foundations of constrained system dynamics, worked out by F. Klein and L. Prandtl. The basic idea of this approach is to replace a unilateral constraint imposed on the system by a field of elastic dissipative forces and to obtain the system with impacts as a result of the corresponding passage to the limit. The existence and stability of periodic trajectories in the systems with impacts are then investigated. The main attention is focused on the integrable billiards (including many-dimensional ones). Some results concerning the non-integrability of typical billiards are also given.
The book is written clearly and can be useful for specialists, who are dealing with analytical mechanics and dynamical systems.

MSC:

37N05 Dynamical systems in classical and celestial mechanics
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
70F35 Collision of rigid or pseudo-rigid bodies
74M20 Impact in solid mechanics
70H05 Hamilton’s equations
70H45 Constrained dynamics, Dirac’s theory of constraints