Anomaly Reparametrization of the Ligon--Schaaf Regularization in the Kepler problem
Li-Chun Hsu
TL;DR
This work uncovers the geometric origin of the Ligon–Schaaf rotation in the Kepler problem by showing the LS angle equals the difference between eccentric and mean anomalies, i.e., $\varepsilon-M$, and extends this anomaly-based reparametrization to all energy regimes. By combining Moser regularization with anomaly reparametrizations, it provides a unified framework that converts Kepler flow into uniform geodesic flow on $T^*S^3$ for negative energies, on $T^*H^3$ for positive energies, and along Euclidean geodesics at zero energy, with parallel momentum-map structures and integrals of motion across regimes. The paper also recasts these regularizations in quaternionic and group-action language, derives explicit symplectic maps, and clarifies the role of hyperbolic and parabolic anomalies in the energy-uniform descriptions. The resulting anomaly-based regularization unifies classical celestial-mechanics constructions (Moser, Belbruno, and LS) within a single geometric picture, advancing the understanding of Kepler flow across all energy levels and highlighting practical links to the Kepler equation and LRL symmetry. This provides a robust, symmetry-aware toolkit for analyzing regularizations in integrable Hamiltonian systems and for potential extensions to related central-force problems.
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.