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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2502.01997 |
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| _version_ | 1866929697178255360 |
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| author | Mironov, Andrey E. Yin, Siyao |
| author_facet | Mironov, Andrey E. Yin, Siyao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_01997 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Billiard trajectories inside Cones Mironov, Andrey E. Yin, Siyao Dynamical Systems Differential Geometry 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. |
| title | Billiard trajectories inside Cones |
| topic | Dynamical Systems Differential Geometry |
| url | https://arxiv.org/abs/2502.01997 |