On the regularization of the Kepler problem

Abstract

In 1970, Moser showed that the Hamiltonian flow of the Kepler problem in $\mathbb{R}^n$ for a fixed negative energy level is regularized via stereographic projection to the geodesic flow on the punctured cotangent bundle of the unit sphere in $\mathbb{R}^{n+1}$, in such a way that the time parameter in the Kepler problem and the arc length for the geodesic flow are related by the Kepler equation. Ligon and Schaaf gave an alternative regularization of the Kepler problem, treating the whole negative energy part of the phase space at once, such that the Kepler flow and the Delaunay flow on the punctured cotangent bundle of the sphere become related by a canonical transformation. The rather elaborate calculations of Ligon and Schaaf were simplified by Cushman and Duistermaat.

In this paper, we derive the Ligon–Schaaf regularization as an almost trivial adaptation of the Moser regularization. As a consequence, the hidden symmetry of the Kepler problem becomes naturally visible.