Kozlov, V. V.; Treshchev, D. V. 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. Reviewer: A. B. Borisov (Sverdlovsk) Cited in 1 ReviewCited in 162 Documents 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 Keywords:dynamical systems; constrained system dynamics; field of elastic dissipative forces; existence; stability; periodic trajectories; integrable billiards; non-integrability × Cite Format Result Cite Review PDF