ON THE STABILITY OF 2-PERIODIC ORBITS IN MINKOWSKI BILLIARD SYSTEMS

Abstract

We investigate the fundamental properties of Minkowski billiards and introduce a new coordinate system $(s,u)$ on the phase space $\mathcal{M}$, under which the Minkowski billiard map $\mathcal{T}$ preserves the standard area form $\omega = ds \wedge du$. After, we derive an expression for the tangent map $D\mathcal{T}$ thus enabling the classification of periodic orbits as elliptic, parabolic, or hyperbolic. Then for a specific family of Minkowski billiards, we derive formulas for the first twist coefficient $\tau_1$ for elliptic 2-periodic orbits in terms of the Minkowski norm and the geometric quantities of the billiard table. Additionally, we analyze the stability properties of these elliptic periodic orbits and provide some numerical simulations.