Ligon–Schaaf regularization map revisited

Abstract

The phase space of the classical Kepler problem is incomplete, as certain Kepler motions do not continue indefinitely in time. To address this issue, a method called “regularization” was introduced in the 1960s and 1970s. While there are several versions of this scheme, the one developed by Ligon and Schaaf [Rep. Math. Phys. 9, 281–300 (1976)] is often preferred due to its global nature and lack of time reparameterization requirement. In this article, we present a new regularization method that is both global and requires no time reparameterization, while also exhibiting a higher degree of symmetry. Our method emphasizes symmetries over spaces and exhibits a duality similar to the compact-noncompact duality observed in symmetric spaces.