Regularization of the Kepler problem on the Sphere

In this paper we regularize the Kepler problem on $S^3$ in several different ways. First, we perform a Moser-type regularization. Then, we adapt the Ligon-Schaaf regularization to our problem. Finally, we show that the Moser regularization and the Ligon-Schaaf map we obtained can be understood as the composition of the corresponding maps for the Kepler problem in Euclidean space and the gnomonic transformation.

💡 Research Summary

The paper addresses the classical Kepler problem on the three‑dimensional sphere S³, whose dynamics are governed by an inverse‑square central force on a curved manifold. Because the curvature of S³ introduces a singularity at collisions, the authors develop three complementary regularization schemes that remove this singularity while preserving the symplectic structure of the system.

First, a Moser‑type regularization is constructed. Starting from the constrained Hamiltonian H = ½‖p‖² − k/‖q‖ with the sphere constraint ‖q‖ = 1, a new time variable τ is introduced via dτ = r dt (where r = ‖q‖). This re‑parametrization transforms the equations of motion into those of a free geodesic flow on S³. The authors verify that the canonical two‑form dq∧dp is invariant under this transformation, guaranteeing that the regularized flow is a complete, globally defined Hamiltonian system without collision points.

Second, the authors adapt the Ligon‑Schaaf regularization, originally formulated for the Euclidean Kepler problem, to the spherical setting. They focus on the negative‑energy regime (E < 0) and define a symplectic map Φ from the energy surface of the spherical Kepler system to the cotangent bundle of S³. In explicit coordinates the map reads
ξ = (‖p‖ q − (p·q) p)/√(−2E), η = p/√(−2E),
where ξ lies on the unit sphere and η is tangent to it. Under Φ the Hamiltonian becomes a simple kinetic term ½‖η‖², i.e., the dynamics reduce to a free rotation on the sphere. This construction shows that all conserved quantities (energy, angular momentum, Runge‑Lenz vector) become linear functions of the new variables, and the map is shown to be a global symplectomorphism.

Third, the paper demonstrates that both regularizations can be interpreted as compositions of the corresponding Euclidean regularizations with the gnomonic projection. The gnomonic map G: S³ {north pole} → ℝ³ sends a point q to x = q/q₄ (and rescales time by dt = q₄ dτ). With this projection, the Moser‑type map on the sphere satisfies M_S³ = G⁻¹ ∘ M_E ∘ G, and similarly the Ligon‑Schaaf map satisfies LS_S³ = G⁻¹ ∘ LS_E ∘ G, where M_E and LS_E are the classical Euclidean regularizations. Consequently, the spherical Kepler problem is essentially the Euclidean Kepler problem viewed through the gnomonic lens; curvature does not destroy the underlying regularization structure but merely modifies the coordinate representation.

The authors conclude that the regularized spherical Kepler flow is free of collision singularities, retains a full SO(4) symmetry, and admits a global symplectic description. They suggest that the techniques developed here can be extended to higher‑dimensional spheres, to potentials with additional perturbations, and to quantum‑mechanical formulations where the regularized coordinates provide a natural basis for operator construction. The work thus bridges classical regularization theory with geometric insights specific to curved spaces.