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Regular and Chaotic Dynamics

, Volume 23, Issue 6, pp 695–703 | Cite as

Embedding the Kepler Problem as a Surface of Revolution

  • Richard MoeckelEmail author
Article
  • 21 Downloads

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 ℝ2 if h ⩾ 0 or on a disk D ⊂ ℝ2 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 ℝ3 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 ⩾ 0 as surfaces of revolution in ℝ3 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 

MSC2010 numbers

70F05 70F15 70G45 53A05 53C42 53C80 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisItaly

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