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This article is cited in 2 scientific papers (total in 2 papers)
Embedding the Kepler Problem as a Surface of Revolution
Richard Moeckel School of Mathematics, University of Minnesota, Minneapolis, MN 55455
Abstract:
Solutions of the planar Kepler problem with fixed energy $h$ determine geodesics of the corresponding
Jacobi–Maupertuis metric. This is a Riemannian metric on $\mathbb{R}^2$ if $h\geqslant 0$ or on a disk $\mathcal{D}\subset \mathbb{R}^2$ if $h<0$. The metric is singular at the origin (the collision singularity) and also on the boundary of the disk when $h<0$. The Kepler problem and the corresponding metric are invariant under rotations of the plane and it is natural to wonder whether the metric can be realized as a surface of revolution in $\mathbb{R}^3$ or some other simple space. In this note, we use elementary methods to study the geometry of the Kepler metric and the embedding problem. Embeddings of the metrics with $h\geqslant0$ as surfaces of revolution in $\mathbb{R}^3$ are constructed explicitly but no such embedding exists for $h<0$ due to a problem near the boundary of the disk. We prove a theorem showing that the same problem occurs for every analytic central force potential. Returning to the Kepler metric, we rule out embeddings in the three-sphere or hyperbolic space, but succeed in constructing an embedding in four-dimensional Minkowski spacetime. Indeed, there are many such embeddings.
Keywords:
celestial mechanics, Jacobi–Maupertuis metric, surfaces of revolution.
Received: 09.08.2018 Accepted: 21.09.2018
Citation:
Richard Moeckel, “Embedding the Kepler Problem as a Surface of Revolution”, Regul. Chaotic Dyn., 23:6 (2018), 695–703
Linking options:
https://www.mathnet.ru/eng/rcd360 https://www.mathnet.ru/eng/rcd/v23/i6/p695
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Abstract page: | 145 | References: | 26 |
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