Twist Coefficients of Periodic Orbits of Minkowski Billiards

TL;DR

This paper introduces a new coordinate system for Minkowski billiards, classifies period-2 orbits, derives formulas for twist coefficients, and analyzes the stability of elliptic periodic orbits.

Contribution

It provides a novel coordinate framework and explicit formulas for twist coefficients in Minkowski billiards, enhancing understanding of orbit stability.

Findings

Preserves area form in the new coordinate system

Classifies period-2 orbits in Minkowski billiards

Derives explicit formulas for twist coefficients

Abstract

We investigate the fundamental properties of Minkowski billiards and introduce a new coordinate system $(s,u)$ on the phase space $\mathcal{M}$. In this coordinate system, the Minkowski billiard map $\mathcal{T}$ preserves the standard area form $\omega = ds \wedge du$. We then classify the periodic orbits of Minkowski billiards with period $2$ and derive formulas for the twist coefficient $\tau_1$ for elliptic periodic orbits, expressed in terms of the geometric characteristics of the billiard table. Additionally, we analyze the stability properties of these elliptic periodic orbits.