Birkhoff Normal Form and Twist Coefficients of Periodic Orbits of Billiards

TL;DR

This paper explicitly constructs the Birkhoff Normal Form for elliptic periodic points in billiards, providing formulas for twist coefficients and characterizing stability and integrability based on geometric parameters.

Contribution

It offers an explicit construction of the Birkhoff transformation and formulas for twist coefficients in billiards, advancing understanding of local dynamics.

Findings

Explicit formulas for twist coefficients in billiards.

Characterization of nonlinear stability around elliptic points.

Conditions for local integrability based on geometry.

Abstract

In this paper we study the Birkhoff Normal Form around elliptic periodic points for a variety of dynamical billiards. We give an explicit construction of the Birkhoff transformation and obtain explicit formulas for the first two twist coefficients in terms of the geometric parameters of the billiard table. As an application, we obtain characterizations of the nonlinear stability and local analytic integrability of the billiards around the elliptic periodic points.