Billiards. A genetic introduction to the dynamics of systems with impacts
Translations of Mathematical Monographs, 89. Providence, RI: American Mathematical Society (AMS), 1991, viii+171 pp.
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Billiard systems arise in many problems of mechanics and physics, some of them being described in the introduction. The authors have chosen a genetic approach, which consists in replacing a single-sided constraint imposed on the system by a field of elastic and dissipative forces. When the elastic and dissipative coefficients tend to infinity, the motion with fixed initial values tends on finite time intervals to a motion with impacts. The celebrated theorem of Birkhoff on the estimating of the number of distinct periodic trajectories of a convex elastic billiard is proved in a variational way, and some conditions for stability of the periodic trajectories are given. A chapter is dedicated to the Hill equation, obtained in the problem of the motion of the perigee of the moon and encountered also in problems of the stability of periodic motions with impacts. An important problem is the integrability of systems of billiard type. Old and new integrable problems, as the elliptic billiard, the billiard in affine Weyl cells, a harmonic oscillator inside an ellipse, billiards on surfaces of constant curvatures are described. The book is written in a clear and careful style and completely describes the problems from the physical phenomenon to the mathematical solution.
Kozlov V. V., Treschev D. V., Billiards. A genetic introduction to the dynamics of systems with impacts, Translations of Mathematical Monographs, 89. Providence, RI: American Mathematical Society (AMS), 1991, viii+171 pp.