Anomaly Reparametrization of the Ligon--Schaaf Regularization in the Kepler problem

TL;DR

This paper provides a geometric reinterpretation of the Ligon--Schaaf regularization for the Kepler problem, revealing the role of eccentric, hyperbolic, and parabolic anomalies in unifying the flow across energy levels.

Contribution

It introduces a new geometric perspective on the regularization, linking the rotation to anomalies and extending the approach to all energy regimes.

Findings

Identifies the geometric origin of the rotation in Ligon--Schaaf regularization.

Provides a unified description of Kepler flow for all energies.

Extends the regularization to hyperbolic and parabolic anomalies.

Abstract

We revisit the Ligon--Schaaf regularization of the Kepler problem and identify the geometric origin of the rotation appearing in their transformation. We show that this rotation is determined by the eccentric anomaly of the Kepler motion, providing a transparent dynamical interpretation of the angle that renders the Kepler flow uniform on $T^{*}S^{3}$. Building on this insight, we extend the construction to positive and zero energies via the corresponding hyperbolic and parabolic anomalies, obtaining a unified geometric description of the Kepler flow across all energy levels.