The Kepler system as a reduced 4D harmonic oscillator

In this paper we review the connection between the Kepler problem and the harmonic oscillator. More specifically we consider how the Kepler system can be obtained through geometric reduction of the harmonic oscillator. We use the method of constructive geometric reduction and explicitly construct the reduction map in terms of invariants. The Kepler system is obtained in a particular chart on the reduced phase space. This reduction is the reverse of the well known KS regularization. Furthermore the reduced phase space connects to Moser's regularization. The integrals for the Kepler system given by the momentum and Laplace vectors, as well as the Delaunay elements, can now be easily related to symmetries of the harmonic oscillator.