Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 010, 16 pages      arXiv:1509.07493      https://doi.org/10.3842/SIGMA.2016.010
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

Quasi-Bi-Hamiltonian Structures of the 2-Dimensional Kepler Problem

Jose F. Cariñena and Manuel F. Rañada
Departamento de Física Teórica and IUMA, Universidad de Zaragoza, 50009 Zaragoza, Spain

Received September 29, 2015, in final form January 25, 2016; Published online January 27, 2016

Abstract
The existence of quasi-bi-Hamiltonian structures for the Kepler problem is studied. We first relate the superintegrability of the system with the existence of two complex functions endowed with very interesting Poisson bracket properties and then we prove the existence of a quasi-bi-Hamiltonian structure by making use of these two functions. The paper can be considered as divided in two parts. In the first part a quasi-bi-Hamiltonian structure is obtained by making use of polar coordinates and in the second part a new quasi-bi-Hamiltonian structure is obtained by making use of the separability of the system in parabolic coordinates.

Key words: Kepler problem; superintegrability; complex structures; bi-Hamiltonian structures; quasi-bi-Hamiltonian structures.

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