Exploring the Fascinating World of Bouncing Balls: Adventures in Two Time-Dependent Billiards

Abstract: This paper explores two instances where dissipation plays a crucial role in curbing the unbounded energy growth of particles in time-dependent billiards. The first example involves an elliptical-like billiard with inelastic collisions between the particle and the boundary. This scenario introduces a fractional loss of energy upon collision, resulting in the suppression of unbounded energy growth. The second example examines an oval-like billiard, where a particle undergoes continuous energy reduction due to a viscous drag force. Both findings emerged in 2009 from collaborative research with Lyonia Bunimovich during a visit to the Georgia Institute of Technology.