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Static and time-dependent perturbations of the classical elliptical billiard. (English) Zbl 1081.37530

Summary: The elliptical billiard problem defines a two-dimensional integrable discrete dynamical system. Integrability not being a robust property, we study some static and time-dependent perturbations of this problem. For the static case, we observe the transition from integrability to chaos, on some perturbations of the ellipse. Then we study time-dependent perturbations, supposing that the boundary deforms periodically with the time, remaining always an ellipse. We investigate numerically the now four-dimensional phase space, looking mainly at the question of whether or not the velocity of a given trajectory may increase indefinitely.

MSC:

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
70H99 Hamiltonian and Lagrangian mechanics
70K20 Stability for nonlinear problems in mechanics
82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
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