On the regularization of the Kepler problem. (English) Zbl 1266.37022

Heckman, Gert; de Laat, Tim

J. Symplectic Geom. 10, No. 3, 463-473 (2012).

The Kepler problem is the study of the motion of two masses subject to an attractive inverse-square force. A key element in modern treatments of the Kepler problem in \(\mathbb{R}^n\) is the regularization of collision orbits. Moser’s fundamental work here related the flow for a fixed negative energy level to the geodesic flow on the sphere \(S^n\) by stereographic projection on the punctured cotangent bundle of the sphere.
An alternative approach to regularization – with the advantage of canonically transforming the whole negative energy part of the \(\mathbb{R}^{2n}\) phase space for the Kepler Hamiltonian into the punctured cotangent bundle of \(S^n\) – was developed by T. Ligon and M. Schaaf [Rep. Math. Phys. 9, 281–300 (1976; Zbl 0347.58005)]. Unfortunately, the Ligon-Schaaf approach requires a significant computational effort.
The authors of this paper show that the Ligon-Schaaf regularization is closely related to the Moser regularization map. It turn out that the Ligon-Schaaf map is the natural adaptation of the Moser map intertwining the Kepler flow on the negative energy part of the phase space \(\mathbb{R}^{2n}\) and the geodesic Delaunay flow on the punctured cotangent bundle of \(S^n\) in a canonical way. The authors’ approach makes the role of the hidden Kepler symmetry quite apparent.

Reviewer: William J. Satzer jun. (St. Paul)

MSC:

37J15	Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70F05	Two-body problems

Keywords:

Kepler problem; regularization; hidden symmetry

Citations:

Zbl 0347.58005