等变量里布-贝尔特拉米对应关系和开普勒-欧勒流

IF 2.9 3区 综合性期刊 Q1 MULTIDISCIPLINARY SCIENCES Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Pub Date : 2024-01-31 DOI:10.1098/rspa.2023.0499
Josep Fontana-McNally, Eva Miranda, Daniel Peralta-Salas
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引用次数: 0

摘要

我们证明,只要考虑底层几何结构的附加对称性,Etnyre & Ghrist (Etnyre, Ghrist 2000 Nonlinearity 13 , 441-458 ( doi:10.1088/0951-7715/13/2/306 ) 中提出的 Reeb 和 Beltrami 向量场之间的对应关系就可以等变。作为这一对应关系的推论,我们证明了机械哈密顿系统势能最大值以上的能级可视为静止流体流,尽管其度量没有规定。我们特别展示了 n 体问题的典型例子,并重点讨论了开普勒问题。我们明确构造了一个兼容的黎曼度量,使天体力学的开普勒问题成为一个合适流形上的静止流体流(贝尔特拉米类型),即开普勒-欧勒流。
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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 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 .
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来源期刊
CiteScore
6.40
自引率
5.70%
发文量
227
审稿时长
3.0 months
期刊介绍: Proceedings A has an illustrious history of publishing pioneering and influential research articles across the entire range of the physical and mathematical sciences. These have included Maxwell"s electromagnetic theory, the Braggs" first account of X-ray crystallography, Dirac"s relativistic theory of the electron, and Watson and Crick"s detailed description of the structure of DNA.
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