Embedding the Kepler Problem as a Surface of Revolution

• Published in: Regular & chaotic dynamics 2018-11, Vol.23 (6), p.695-703
• Main Author: Moeckel, Richard
• Format: Article
• Language: English
• Subjects: Dynamical Systems and Ergodic Theory; Embedding; Geodesy; Mathematics; Mathematics and Statistics
• ISSN: 1560-3547; 1560-3547; 1468-4845
• DOI: 10.1134/S1560354718060059

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.