Kustaanheimo-Stiefel Regularization and the Quadrupolar Conjugacy

TL;DR

This paper presents a symplectic quaternionic formulation of Kustaanheimo-Stiefel regularization, establishes its relation to Levi-Civita regularization, introduces generalized Darboux coordinates, and demonstrates a conjugacy between regularized and approximating dynamics in the spatial three-body problem.

Contribution

It introduces a symplectic quaternionic approach to Kustaanheimo-Stiefel regularization, links it with Levi-Civita regularization, and extends conjugacy results to the spatial three-body problem.

Findings

Symplectic and quaternionic formulation of Kustaanheimo-Stiefel regularization.

Relation established between Kustaanheimo-Stiefel and Levi-Civita regularizations.

Conjugacy between regularized and approximating dynamics in the spatial three-body problem.

Abstract

In this note, we present the Kustaanheimo-Stiefel regularization in a symplectic and quaternionic fashion. The bilinear relation is associated with the moment map of the $S^{1}$- action of the Kustaanheimo-Stiefel transformation, which yields a concise proof of the symplecticity of the Kustaanheimo-Stiefel transformation symplectically reduced by this circle action. The relation between the Kustaanheimo-Stiefel regularization and the Levi-Civita regularization is established via the investigation of the Levi-Civita planes. A set of Darboux coordinates (which we call Chenciner-F\'ejoz coordinates) is generalized from the planar case to the spatial case. Finally, we obtain a conjugacy relation between the integrable approximating dynamics of the lunar spatial three-body problem and its regularized counterpart, similar to the conjugacy relation between the extended averaged system and the averaged regularized system in the planar case.