Abstract
We consider a dynamical system that describes a convex planer body with a smooth boundary, which falls under the influence of gravity and is elastically reflected upon hitting a wall. Under suitable restriction on the system’s parameters, the system is equivalent to a billiard problem within a cylinder. We prove that the dynamics of the system is intimately linked to a variational problem, corresponding to a twist map. Moreover, we utilise the established variational method to investigate the stability of trivial periodic orbits and near-integrable dynamics when this planar body closely approximates a disc with its centre of mass at the geometric centre. Numerical results illustrates our mathematical findings and highlight the system’s intricate dynamics.
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