An equivariant Reeb–Beltrami correspondence and the Kepler–Euler flow
We prove that the correspondence between Reeb and Beltrami vector fields presented in Etnyre & Ghrist (Etnyre, Ghrist 2000 Nonlinearity 13, 441–458 (doi:10.1088/0951-7715/13/2/306)) can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2024 |
| País: | España |
| Recursos: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositório: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/405239 |
| Acesso em linha: | http://hdl.handle.net/10261/405239 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85184035966&doi=10.1098%2frspa.2023.0499&partnerID=40&md5=c6ef4b53add4d8eea16a2be8bd38f58b |
| Access Level: | Acceso aberto |
| Palavra-chave: | Beltrami vector field Contact geometry Euler equations Lifted metric Reeb vector field Astrophysics Flow of fluids Geometry Hamiltonians Beltrami Beltramus vector field Euler flows Stationary fluid flow Potential energy Vector fields |
| Resumo: | We prove that the correspondence between Reeb and Beltrami vector fields presented in Etnyre & Ghrist (Etnyre, Ghrist 2000 Nonlinearity 13, 441–458 (doi:10.1088/0951-7715/13/2/306)) 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. ©2024 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited. |
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