Richard Moeckel
Publications:
Moeckel R.
Embedding the Kepler Problem as a Surface of Revolution
2018, vol. 23, no. 6, pp. 695703
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 threesphere or hyperbolic space, but succeed in constructing an embedding in fourdimensional Minkowski spacetime. Indeed, there are many such embeddings.

Moeckel R.
Symbolic dynamics in the planar threebody problem
2007, vol. 12, no. 5, pp. 449475
Abstract
A chaotic invariant set is constructed for the planar threebody problem. The orbits in the invariant set exhibit many close approaches to triple collision and also excursions near infinity. The existence proof is based on finding appropriate "windows" in the phase space which are stretched across one another by flowdefined Poincaré maps.
