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On Moser’s regularization of the Kepler system: Positive and negative energies

Part of: Hamiltonian and Lagrangian mechanics

Published online by Cambridge University Press:  18 December 2020

Sebastián Ferrer  and
Francisco Crespo
Sebastián Ferrer
Affiliation:
Grupo de Dinámica Espacial, Departamento de Ingeniería y Tecnología de Computadores, Facultad de Informática, Universidad Murcia, Murcia30100, Spaine-mail:sferrer@um.es
Francisco Crespo*
Affiliation:
Grupo de Investigación en Sistemas Dinámicos y Aplicaciones, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Collao1202, Concepción, Casilla 5-C, Chile
*
e-mail:fcrespo@ubiobio.cl
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Abstract

We present a generalization of Moser’s theorem on the regularization of Keplerian systems that include positive and negative energies. Our approach does not consider the geodesics of the hyperboloid embedded in a Lorentz space for the unbounded orbits, as it is previously done in the literature. Instead, we connect the Keplerian positive and negative energy orbits with the harmonic oscillator with negative and positive frequencies. The connection is established through the canonical extension of the stereographic projection, as it is done in Moser’s original paper. How we base our study reveals that Kustaanheimo–Stiefel map KS and Moser regularizations are alternative ways of showing the spatial Kepler system as a subdynamics of the 4D harmonic oscillator.

Keywords

Stereographic-type transformationKepler systemharmonic oscillator

MSC classification

Primary: 70F15: Celestial mechanics
Secondary: 70F16: Collisions in celestial mechanics, regularization 70H05: Hamilton's equations
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 4 , December 2021 , pp. 970 - 983
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Copyright
© Canadian Mathematical Bulletin 2020

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