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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.19281 |
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| _version_ | 1866911983498952704 |
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| author | Bolle, Philippe Mazzucchelli, Marco Venturelli, Andrea |
| author_facet | Bolle, Philippe Mazzucchelli, Marco Venturelli, Andrea |
| contents | A level orbit of a mechanical Hamiltonian system is a solution of Newton equation that is contained in a level set of the potential energy. In 2003, Mark Levi asked for a characterization of the smooth potential energy functions on the plane with the property that any point on the plane lies on a level orbit; we call such functions Levi potentials. The basic examples are the radial monotone increasing smooth functions. In this paper we show that any Levi potential that is analytic or has totally path-disconnected critical set must be radial. Nevertheless, we show that every compact convex subset of the plane is the critical set of a Levi potential. A crucial observation for these theorems is that, outside the critical set, the family of level sets of a Levi potential forms a solution of the inverse curvature flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19281 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On potentials whose level sets are orbits Bolle, Philippe Mazzucchelli, Marco Venturelli, Andrea Differential Geometry Analysis of PDEs Dynamical Systems 53E10, 37J06 A level orbit of a mechanical Hamiltonian system is a solution of Newton equation that is contained in a level set of the potential energy. In 2003, Mark Levi asked for a characterization of the smooth potential energy functions on the plane with the property that any point on the plane lies on a level orbit; we call such functions Levi potentials. The basic examples are the radial monotone increasing smooth functions. In this paper we show that any Levi potential that is analytic or has totally path-disconnected critical set must be radial. Nevertheless, we show that every compact convex subset of the plane is the critical set of a Levi potential. A crucial observation for these theorems is that, outside the critical set, the family of level sets of a Levi potential forms a solution of the inverse curvature flow. |
| title | On potentials whose level sets are orbits |
| topic | Differential Geometry Analysis of PDEs Dynamical Systems 53E10, 37J06 |
| url | https://arxiv.org/abs/2403.19281 |