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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.22251 |
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| _version_ | 1866909712099835904 |
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| author | Rivin, Igor |
| author_facet | Rivin, Igor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22251 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computing Periodic Billiard Orbits in $L^p$ Balls via Newton's Method and Smale's $α$-Criterion Rivin, Igor Dynamical Systems 37D50, 37M21, 65P10, 70H12, 49M15 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. |
| title | Computing Periodic Billiard Orbits in $L^p$ Balls via Newton's Method and Smale's $α$-Criterion |
| topic | Dynamical Systems 37D50, 37M21, 65P10, 70H12, 49M15 |
| url | https://arxiv.org/abs/2507.22251 |