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

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.

### MSC:

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

### Keywords:

Kepler problem; regularization; hidden symmetry

Zbl 0347.58005
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