Billiard trajectories inside Cones

TL;DR

The paper investigates Birkhoff billiards inside cones and demonstrates that while $C^3$ convex cones force finite reflection counts, there exist $C^2$ convex cones with billiard trajectories that reflect infinitely many times in finite time. It provides a constructive method to realize such trajectories in $\mathbb{R}^3$ and extends the construction to higher dimensions, highlighting that no universal bound on reflections exists for smooth cones. In the elliptic cone case $K_e\subset\mathbb{R}^3$, the authors identify two first integrals $I_1$ and $I_2$ in involution, enabling a sharp bound $N_{c_1,c_2}$ on the number of reflections when $I_1=c_1$ and $I_2=c_2$ are fixed. The work thus blends explicit geometric constructions with integrable structure to delineate when and how billiard trajectories can have many reflections, including finite-time blow-up scenarios and finite reflection bounds. The results impact the understanding of billiard dynamics in smooth versus non-smooth cone geometries and provide tools for estimating reflection counts in structured three-dimensional cones.

Abstract

Recently it was proved that every billiard trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by a $C^3$ convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed $C^3$ submanifold of the hyperplane with nondegenerate second fundamental form. In this paper, we prove the existence of $C^2$ convex cones admitting billiard trajectories with infinitely many reflections in finite time. We also estimate the number of reflections of billiard trajectories in elliptic cones in $\mathbb{R}^3$ using two first integrals.

Figures (6)

• Figure 1: For an angle $\theta$ in $\mathbb{R}^2$, a trajectory has $n$ or $n-1$ reflections, where $\frac{\pi}{n} \leq \theta < \frac{\pi}{n-1}$.
• Figure 2: The angle $\beta_{k}$ between $l_{k-1}$ and $Op_{k}$. (Lines $l_k$, $l_{k-1}$ and $Op_{k}$ do not lie in the same plane.)
• Figure 3: The oriented angle $\sigma_k$ measured counterclockwise from $z_k$ to $w_k$.
• Figure 4: The arc $S_k$ near $q_k$.
• Figure 5: The circle $r= g(\xi, \sigma)$.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Lemma 1: Theorem 1, Lemma 2 in MY
• Lemma 2
• Lemma 3
• Remark 1
• Lemma 4
• Remark 2
• Lemma 5
• Lemma 6