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Birkhoff Normal Form and Twist Coefficients of Periodic Orbits of Billiards

Xin Jin, Pengfei Zhang

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.

Birkhoff Normal Form and Twist Coefficients of Periodic Orbits of Billiards

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.

Paper Structure

This paper contains 18 sections, 5 theorems, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Symmetric billiards
  3. Applications
  4. Asymmetric billiards
  5. Ellipse-hyperbola lens billiards
  6. Preliminaries
  7. Taylor polynomial of the function at the point $P$
  8. Birkhoff Normal Form with First Twist Coefficient
  9. Introduce complex coordinates
  10. An intermediate transformation
  11. A symplectic coordinate transformation
  12. Birkhoff Normal Form with First Two Twist Coefficients
  13. Twist coefficients of the periodic orbits along the minor axis of an ellipse
  14. Twist Coefficients for Periodic Orbits of Asymmetric Billiards
  15. The polynomial $N$
  16. ...and 3 more sections

Key Result

Theorem 1.3

Assuming (A1), the first twist coefficient $\tau_1(F,P)$ of the one-step billiard map $F$ at $P$ is given by Assuming (A1) and (A2), the second twist coefficient $\tau_2(F,P)$ of the one-step billiard map $F$ at $P$ is given by

Figures (4)

  • Figure 1: $\gamma_0(t)= (t^2 +0.5 t^4, t)$ and $\gamma_1(t)=(4 - t^2 +0.3 t^4, -t)$, $t\in(-1, 1)$.
  • Figure 2: The lemon table $Q(1)$ with two families of periodic points of period 4. Left: periodic points on the table; right: periodic points on the phase space.
  • Figure 3: The asymmetric lemon billiards $Q(r,B_{+}(r,R),R)$ when $R=2r$. Left: the table; Right: the phase portrait.
  • Figure 4: Some ellipse-hyperbola lenses with fixed ellipse. (1) the confocal hyperbola with $p=p_c$ (black) and two shifts (gray); (2) a confocal hyperbola with $p>p_c$ (red) and two deformations $q=q_2$ (green) and $q>q_2$ (blue).

Theorems & Definitions (27)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 1.5
  • Remark 1.6
  • Example 1.7
  • Remark 1.8
  • Proposition 1.9
  • Example 1.10
  • ...and 17 more