An equivariant Reeb-Beltrami correspondence and the Kepler-Euler flow
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918132223836160 |
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| author | Fontana-McNally, Josep Miranda, Eva Peralta-Salas, Daniel |
| author_facet | Fontana-McNally, Josep Miranda, Eva Peralta-Salas, Daniel |
| contents | We prove that the correspondence between Reeb and Beltrami vector fields can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the $n$-body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler-Euler flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09898 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An equivariant Reeb-Beltrami correspondence and the Kepler-Euler flow Fontana-McNally, Josep Miranda, Eva Peralta-Salas, Daniel Symplectic Geometry Mathematical Physics Analysis of PDEs Dynamical Systems We prove that the correspondence between Reeb and Beltrami vector fields can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the $n$-body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler-Euler flow. |
| title | An equivariant Reeb-Beltrami correspondence and the Kepler-Euler flow |
| topic | Symplectic Geometry Mathematical Physics Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2306.09898 |