On the Regularization of the Kepler Problem

Gert Heckman; Tim de Laat

On the Regularization of the Kepler Problem

Gert Heckman, Tim de Laat

Institut for Matematiske Fag

8 Citationer (Scopus)

Abstract

In 1970, Moser showed that the Hamiltonian flow of the Kepler problem in ℝⁿ for a fixed negative energy level is regularized via stereographic projection to the geodesic flow on the punctured cotangent bundle of the unit sphere in ℝⁿ⁺¹, in such a way that the time parameter in the Kepler problem and the arc length for the geodesic flow are related by the Kepler equation. Ligon and Schaaf gave an alternative regularization of the Kepler problem, treating the whole negative energy part of the phase space at once, such that the Kepler flow and the Delaunay flow on the punctured cotangent bundle of the sphere become related by a canonical transformation. The rather elaborate calculations of Ligon and Schaaf were simplified by Cushman and Duistermaat. In this paper, we derive the Ligon-Schaaf regularization as an almost trivial adaptation of the Moser regularization. As a consequence, the hidden symmetry of the Kepler problem becomes naturally visible.

Originalsprog	Engelsk
Tidsskrift	Journal of Symplectic Geometry
Vol/bind	10
Udgave nummer	3
Sider (fra-til)	463-474
ISSN	1527-5256
Status	Udgivet - mar. 2012

Citationsformater

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abstract = "In 1970, Moser showed that the Hamiltonian flow of the Kepler problem in ℝn for a fixed negative energy level is regularized via stereographic projection to the geodesic flow on the punctured cotangent bundle of the unit sphere in ℝn+1, in such a way that the time parameter in the Kepler problem and the arc length for the geodesic flow are related by the Kepler equation. Ligon and Schaaf gave an alternative regularization of the Kepler problem, treating the whole negative energy part of the phase space at once, such that the Kepler flow and the Delaunay flow on the punctured cotangent bundle of the sphere become related by a canonical transformation. The rather elaborate calculations of Ligon and Schaaf were simplified by Cushman and Duistermaat. In this paper, we derive the Ligon-Schaaf regularization as an almost trivial adaptation of the Moser regularization. As a consequence, the hidden symmetry of the Kepler problem becomes naturally visible.",

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T1 - On the Regularization of the Kepler Problem

AU - Heckman, Gert

AU - de Laat, Tim

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N2 - In 1970, Moser showed that the Hamiltonian flow of the Kepler problem in ℝn for a fixed negative energy level is regularized via stereographic projection to the geodesic flow on the punctured cotangent bundle of the unit sphere in ℝn+1, in such a way that the time parameter in the Kepler problem and the arc length for the geodesic flow are related by the Kepler equation. Ligon and Schaaf gave an alternative regularization of the Kepler problem, treating the whole negative energy part of the phase space at once, such that the Kepler flow and the Delaunay flow on the punctured cotangent bundle of the sphere become related by a canonical transformation. The rather elaborate calculations of Ligon and Schaaf were simplified by Cushman and Duistermaat. In this paper, we derive the Ligon-Schaaf regularization as an almost trivial adaptation of the Moser regularization. As a consequence, the hidden symmetry of the Kepler problem becomes naturally visible.

AB - In 1970, Moser showed that the Hamiltonian flow of the Kepler problem in ℝn for a fixed negative energy level is regularized via stereographic projection to the geodesic flow on the punctured cotangent bundle of the unit sphere in ℝn+1, in such a way that the time parameter in the Kepler problem and the arc length for the geodesic flow are related by the Kepler equation. Ligon and Schaaf gave an alternative regularization of the Kepler problem, treating the whole negative energy part of the phase space at once, such that the Kepler flow and the Delaunay flow on the punctured cotangent bundle of the sphere become related by a canonical transformation. The rather elaborate calculations of Ligon and Schaaf were simplified by Cushman and Duistermaat. In this paper, we derive the Ligon-Schaaf regularization as an almost trivial adaptation of the Moser regularization. As a consequence, the hidden symmetry of the Kepler problem becomes naturally visible.

M3 - Journal article

SN - 1527-5256

VL - 10

SP - 463

EP - 474

JO - Journal of Symplectic Geometry

JF - Journal of Symplectic Geometry

IS - 3

ER -

On the Regularization of the Kepler Problem

Abstract

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