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

TL;DR

This paper investigates quasi-bi-Hamiltonian structures of the 2D Kepler problem, linking superintegrability with complex Poisson structures, and introduces new structures using polar and parabolic coordinates.

Contribution

It demonstrates the existence of quasi-bi-Hamiltonian structures for the Kepler problem using complex functions and coordinate separability, advancing understanding of its integrability.

Findings

Established quasi-bi-Hamiltonian structure using polar coordinates.

Derived a new quasi-bi-Hamiltonian structure via parabolic coordinates.

Linked superintegrability with complex Poisson bracket properties.

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.