## 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 ℝ^{2} if h ⩾ 0 or on a disk D ⊂ 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 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 ⩾ 0 as surfaces of revolution in 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.

Original language | English (US) |
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Pages (from-to) | 695-703 |

Number of pages | 9 |

Journal | Regular and Chaotic Dynamics |

Volume | 23 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2018 |

### Bibliographical note

Funding Information:This paper was motivated by discussions with Richard Montgomery and Doug Arnold. The author was supported by NSF grant DMS-1712656.

Publisher Copyright:

© 2018, Pleiades Publishing, Ltd.

## Keywords

- 53A05
- 53C42
- 53C80
- 70F05
- 70F15
- 70G45
- Jacobi–Maupertuis metric
- celestial mechanics
- surfaces of revolution