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Integrability of Billiards Inside Cones as a Discrete-Time Hamiltonian System

Andrey E. Mironov, Siyao Yin

Abstract

In this paper, we continue to study billiards inside cones $K\subset \mathbb{R}^n$ over strictly convex closed $C^3$ manifolds with non-degenerate second fundamental form. Recently we proved that the billiard is superintegrable, i.e., the billiard admits first integrals whose values uniquely determine all billiard trajectories. In this paper we prove that this billiard system admits $n-1$ independent first integrals in involution. Consequently, the system is completely integrable as a discrete-time Hamiltonian system. This provides an example of an integrable billiard where the billiard table is neither a quadric nor consists of pieces of quadrics.

Integrability of Billiards Inside Cones as a Discrete-Time Hamiltonian System

Abstract

In this paper, we continue to study billiards inside cones over strictly convex closed manifolds with non-degenerate second fundamental form. Recently we proved that the billiard is superintegrable, i.e., the billiard admits first integrals whose values uniquely determine all billiard trajectories. In this paper we prove that this billiard system admits independent first integrals in involution. Consequently, the system is completely integrable as a discrete-time Hamiltonian system. This provides an example of an integrable billiard where the billiard table is neither a quadric nor consists of pieces of quadrics.

Paper Structure

This paper contains 2 sections, 9 theorems, 47 equations, 2 figures.

Table of Contents

  1. Introduction and Main Results
  2. Proof of Theorem 1

Key Result

Theorem 1

The Birkhoff billiard inside $K \subset \mathbb{R}^n$ is completely integrable as a discrete-time Hamiltonian system. Specifically, the $n-1$ first integrals $\{\tilde{v}^i\}_{i=1}^{n-1}$ obtained via the lifting procedure eq:lift-def-intro of $\{{v}^i\}_{i=1}^{n-1}$ are in involution with respect t

Figures (2)

  • Figure 1: The sphere as a caustic of the billiard inside a cone.
  • Figure 2: Oriented lines in $\psi_-$, $\psi$, and $\psi_+$.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • proof
  • Corollary 1
  • Corollary 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 5 more