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Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipse

Klaudiusz Czudek

TL;DR

This work analyzes the second-order Lazutkin-coordinate expansion for maximal symmetric $q$-periodic billiard orbits on ellipses, within the broader context of deformational spectral rigidity near the circle. It combines action-angle methods with elliptic parametrizations to compute the corrections $\alpha(t)$ and $\beta(t)$ that govern the Lazutkin coordinates and reflection angles of these orbits. The explicit expressions for $\alpha$ and $\beta$ rely on Jacobi elliptic functions and the elliptic invariants $e$ and $K(e)$, enabling a precise characterization of the orbit geometry in Lazutkin coordinates. These results supply a crucial step in extending rigidity arguments for $\mathbb{Z}_2$-symmetric domains close to the circle by incorporating a broader class of symmetric periodic orbits.

Abstract

In De Simoi J., Kaloshin V., Wei Q. "Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex domains close to a circle" (Appendix B coauthored with H. Hezari) Ann. of Math. 186.1 (2017), pp. 277-314 deformational spectral rigidity of $\mathbb{Z}_2$ symmetric domains close to the circle has been shown. One of the steps of the proof was to express the maximal symmetric periodic orbits in the Lazutkin parametrization. Here using the action-angle variables we find the second order approximation of Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipses.

Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipse

TL;DR

This work analyzes the second-order Lazutkin-coordinate expansion for maximal symmetric -periodic billiard orbits on ellipses, within the broader context of deformational spectral rigidity near the circle. It combines action-angle methods with elliptic parametrizations to compute the corrections and that govern the Lazutkin coordinates and reflection angles of these orbits. The explicit expressions for and rely on Jacobi elliptic functions and the elliptic invariants and , enabling a precise characterization of the orbit geometry in Lazutkin coordinates. These results supply a crucial step in extending rigidity arguments for -symmetric domains close to the circle by incorporating a broader class of symmetric periodic orbits.

Abstract

In De Simoi J., Kaloshin V., Wei Q. "Dynamical spectral rigidity among -symmetric strictly convex domains close to a circle" (Appendix B coauthored with H. Hezari) Ann. of Math. 186.1 (2017), pp. 277-314 deformational spectral rigidity of symmetric domains close to the circle has been shown. One of the steps of the proof was to express the maximal symmetric periodic orbits in the Lazutkin parametrization. Here using the action-angle variables we find the second order approximation of Lazutkin coordinates of the maximal symmetric periodic orbits on the ellipses.
Paper Structure (8 sections, 2 theorems, 59 equations)

This paper contains 8 sections, 2 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Notation and basic facts
  3. Calculation of $\alpha$
  4. The Taylor expansion of $\omega_\lambda$ and $\lambda_q$
  5. Elliptic coordinates of the maximal symmetric periodic orbit
  6. The Lazutkin coordinates of the maximal symmetric periodic orbit
  7. The calculation of $\beta$
  8. Acknowledgments

Key Result

Theorem 1

Let us consider the ellipse where $e=\sqrt{1-b^2}\in (0,1)$ represents the eccentricity. Then on $\Omega=\mathcal{E}_e$ the functions $\alpha$ and $\beta$ defined by E:def_alpha and E:def_beta take the forms and where Here $K$ is the complete elliptic integral of the first kind, $\textrm{dn}$ is the Jacobi delta function, $\textrm{am}$ is the Jacobi amplitudeThe definitions are given in Sectio

Theorems & Definitions (2)

  • Theorem 1
  • Proposition 2: Chang_Friedberg_88