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Integrable Birkhoff Billiards inside Cones

Andrey E. Mironov, Siyao Yin

TL;DR

The paper proves that the Birkhoff billiard inside a convex $C^3$ cone in $\mathbb{R}^n$ is integrable by constructing a degree-two first integral in the velocity components, showing that the distance of billiard lines from the origin is preserved under reflection, and identifying spheres as caustics. It then demonstrates that every trajectory has a finite number of reflections, leveraging $C^3$ smoothness and convexity. The core integrability result is established through a phase-space analysis of the extended billiard map $\mu$ on $TS^{n-1}$ and the explicit construction of $2n-1$ continuous first integrals that—together with a boundary-vanishing family—completely determine trajectories in a dense subset of the phase space. This work provides the first example of an integrable billiard where the table is not a quadric, expanding the known landscape of integrable billiard systems.

Abstract

One of the most interesting problems in the theory of Birkhoff billiards is the problem of integrability. In all known examples of integrable billiards, the billiard tables are either conics, quadrics (closed ellipsoids as well as unclosed quadrics like paraboloids or cones), or specific configurations of conics or quadrics. This leads to the natural question: are there other integrable billiards? The Birkhoff conjecture states that if the billiard inside a convex, smooth, closed curve is integrable, then the curve is an ellipse or a circle. In this paper we study the Birkhoff billiard inside a cone in $\mathbb{R}^n$. We prove that the billiard always admits a first integral of degree two in the components of the velocity vector. Using this fact, we prove that every trajectory inside a $C^3$ convex cone has a finite number of reflections. Here, by $C^3$ 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. The main result of this paper is the following. We prove that the Birkhoff billiard inside a convex $C^3$ cone is integrable. This is the first example of an integrable billiard where the billiard table is neither a quadric nor composed of pieces of quadrics.

Integrable Birkhoff Billiards inside Cones

TL;DR

The paper proves that the Birkhoff billiard inside a convex cone in is integrable by constructing a degree-two first integral in the velocity components, showing that the distance of billiard lines from the origin is preserved under reflection, and identifying spheres as caustics. It then demonstrates that every trajectory has a finite number of reflections, leveraging smoothness and convexity. The core integrability result is established through a phase-space analysis of the extended billiard map on and the explicit construction of continuous first integrals that—together with a boundary-vanishing family—completely determine trajectories in a dense subset of the phase space. This work provides the first example of an integrable billiard where the table is not a quadric, expanding the known landscape of integrable billiard systems.

Abstract

One of the most interesting problems in the theory of Birkhoff billiards is the problem of integrability. In all known examples of integrable billiards, the billiard tables are either conics, quadrics (closed ellipsoids as well as unclosed quadrics like paraboloids or cones), or specific configurations of conics or quadrics. This leads to the natural question: are there other integrable billiards? The Birkhoff conjecture states that if the billiard inside a convex, smooth, closed curve is integrable, then the curve is an ellipse or a circle. In this paper we study the Birkhoff billiard inside a cone in . We prove that the billiard always admits a first integral of degree two in the components of the velocity vector. Using this fact, we prove that every trajectory inside a convex cone has a finite number of reflections. Here, by convex cone, we mean a cone whose section with some hyperplane is a strictly convex closed submanifold of the hyperplane with nondegenerate second fundamental form. The main result of this paper is the following. We prove that the Birkhoff billiard inside a convex cone is integrable. This is the first example of an integrable billiard where the billiard table is neither a quadric nor composed of pieces of quadrics.
Paper Structure (9 sections, 19 theorems, 213 equations, 18 figures)

This paper contains 9 sections, 19 theorems, 213 equations, 18 figures.

Table of Contents

  1. Introduction and Main Results
  2. Finite Number of Reflections inside $C^3$ Convex Cones
  3. Caustics and first integrals of degree two
  4. Geometry of billiard trajectories inside cones
  5. Trajectories inside $C^3$ cones
  6. Integrability
  7. The phase space
  8. The billiard map
  9. Construction of the first integrals

Key Result

Theorem 1

1. The Birkhoff billiard inside $K \subset \mathbb{R}^n$ admits the first integral where $m_{i,j} := x^i v^j - x^j v^i$ for $i < j$, $i, j = 1, \ldots, n$, and $v = (v^1, \ldots, v^n)$ is the velocity vector. 2. The spheres centered at the vertex $O \in \mathbb{R}^n$ of $K$ are caustics of the billiard inside $K$ (see Fig. fig:caustic).

Figures (18)

  • Figure 1: The sphere as a caustic of the billiard inside a cone.
  • Figure 2: 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 3: Oriented lines in $\psi_-$, $\psi$, and $\psi_+$, respectively.
  • Figure 4: The angles $\alpha_k, \theta_k$.
  • Figure 5: The angle $\beta_{k+1}$ between $l_{k}$ and $Op_{k+1}$. (Lines $l_k$, $l_{k+1}$ and $Op_{k+1}$ do not lie in the same plane.)
  • ...and 13 more figures

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Remark 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 19 more