Embedding the Kepler Problem as a Surface of Revolution

    2018, Volume 23, Number 6, pp.  695-703

    Author(s): Moeckel R.

    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
    Citation: Moeckel R., Embedding the Kepler Problem as a Surface of Revolution, Regular and Chaotic Dynamics, 2018, Volume 23, Number 6, pp. 695-703

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