Computing Periodic Billiard Orbits in $L^p$ Balls via Newton's Method and Smale's $\alpha$-Criterion

TL;DR

This paper introduces a computational approach using Newton's method and Smale's alpha-criterion to find and verify numerous periodic billiard orbits in $L^p$ balls, revealing complex patterns and large orbit counts.

Contribution

It develops an efficient computational framework for discovering and certifying periodic billiard orbits in $L^p$ spaces, surpassing traditional guarantees and uncovering intricate orbit structures.

Findings

Discovered thousands of certified periodic orbits in $L^3$ balls.

Revealed power-law growth and clustering patterns in orbit distributions.

Identified dependence of orbit structures on parity and primality of bounce counts.

Abstract

We present a computational method for finding and verifying periodic billiard orbits in $L^{p}$ balls ($p>2$) using Newton's method applied to a variational formulation. The orbits are verified with Smale's alpha-criterion, which provides a rigorous certificate of existence. We implement efficient batched computations in JAX and present systematic results for various $p$ and bounce counts $N$. Our experiments reveal striking patterns in the critical-point structure, including a predominance of specific Morse signatures and rotation numbers that depend on the parity and primality of $N$. Notably, our method routinely finds many more than the two periodic orbits per rotation number guaranteed by Birkhoff's theorem -- a large-scale run with five bounces in the $L^{3}$ ball produced 8,927 distinct certified orbits from 30,000 random seeds, uncovering power-law growth and intricate clustering visualised with UMAP.