Kustaanheimo–Stiefel Regularization and the Quadrupolar Conjugacy

In this article, we first present the Kustaanheimo–Stiefel regularization of the spatial Kepler problem in a symplectic and quaternionic approach. We then establish a set of action-angle coordinates, the so-called LCF coordinates, of the Kustaanheimo–Stiefel regularized Kepler problem, which is consequently used to obtain a conjugacy relation between the integrable approximating “quadrupolar” system of the lunar spatial three-body problem and its regularized counterpart. This result justifies the study of Lidov and Ziglin [14] of the quadrupolar dynamics of the lunar spatial three-body problem near degenerate inner ellipses.