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

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.

💡 Research Summary

The paper revisits the classic Ligon–Schaaf regularization of the Kepler problem and uncovers the geometric origin of the extra rotation that Ligon and Schaaf introduced. By exploiting the scaling symmetry of the Kepler Hamiltonian H(q,p)=½|p|²−μ/|q|, the authors first normalize the dynamics using auxiliary Hamiltonians K₊, K₋ and K₀, each corresponding respectively to negative, positive and zero energy regimes. For negative energies they review Moser’s construction, which identifies the energy surface H⁻¹(−½) with the unit cotangent bundle T*₁S³ via a stereographic projection combined with a symplectic exchange map. In quaternionic language this map is shown to be equivariant under a G₊=ℍ²_{unit} action, whose momentum map reproduces the angular momentum vector L and the Laplace–Runge–Lenz vector A.

The key insight is that the Ligon–Schaaf rotation angle is exactly the difference between the eccentric anomaly ε and the mean anomaly M, i.e. ε−M. Using the classical Kepler equation ε−e sin ε=M, the authors demonstrate that this angle naturally re‑parametrizes the non‑uniform Kepler flow into a uniform geodesic flow on S³. Consequently, the Ligon–Schaaf map is not a mere technical tweak but a geometrically natural “anomaly‑based re‑parametrization”.

Building on this, the paper extends the construction to positive energies by revisiting Belbruno’s hyperbolic regularization. The hyperbolic anomaly ψ_h satisfies the hyperbolic Kepler equation M_h=e sinh ψ_h−ψ_h, and the difference ψ_h−M_h again provides the required rotation. The resulting map sends each positive‑energy level H⁻¹(c>0) to an open subset of the cotangent bundle of hyperbolic 3‑space, preserving the G₊ symmetry and yielding a uniform geodesic flow.

For the zero‑energy (parabolic) case, a Euclidean regularization is introduced. The parabolic anomaly ψ_p obeys M_p=ψ_p+⅓ψ_p³, and the same ψ_p−M_p rotation produces a uniform flow on the appropriate limit manifold.

In all three regimes the authors formulate a unified principle: “Moser regularization + anomaly re‑parametrization = uniformization of the Kepler flow.” This principle clarifies the physical meaning of the auxiliary angle, ties the classical orbital anomalies directly to symplectic geometry, and provides a single, coherent geometric framework that works uniformly across elliptic, hyperbolic and parabolic motions. The paper also supplies detailed calculations of the G₊ action, momentum maps, Poisson brackets, and time‑reparametrization factors, offering a comprehensive and rigorous treatment of the global symplectic structure of the Kepler problem.