Dynamics of a massive particle in Bunimovich stadium billiards: an elastic model approach

This study explores the dynamics of a massive particle in Bunimovich stadium billiards (BSB), focusing on the velocity variations influenced by an innovative elastic collision model. Our approach is rooted in the experimental analysis of a robot navigating in a BSB setup, controlled by a geometric parameter over a 3.5 meters-long course. Contrary to expectations of simple specular reflections at the billiards’s edge, the robot exhibits complex behaviors due to inertia, manifesting in four distinct regimes: straight stationary translation, deceleration, rotation, and acceleration around collision points. We introduce an elastic force to comprehensively model these observations, offering a nuanced understanding of the particle’s interactions with the billiard walls. Our investigation not only delineates the significant role of wall stiffness in system dynamics but also uncovers a critical spring constant threshold demarcating chaotic from regular regimes. Through meticulous calculations of the largest Lyapunov exponent and the construction of a phase diagram, we elucidate the intricate interplay between system parameters, ultimately revealing that the high wall rigidity limit recovers the expected specular results.