Closedness of orbits in a space with SU(2) Poisson structure

TL;DR

This paper investigates the conditions under which orbits are closed in a three-dimensional space with an SU(2) Poisson structure, showing that only the Kepler problem has all bounded orbits closed, similar to classical mechanics.

Contribution

It demonstrates that in SU(2) Poisson space, only the Kepler potential yields closed bounded orbits and constructs an analog of the Laplace-Runge-Lenz vector.

Findings

Only Kepler problem has all bounded orbits closed in SU(2) Poisson space

An explicit conserved vector analogous to Laplace-Runge-Lenz vector is constructed

The algebra of constants of motion is characterized

Abstract

The closedness of orbits of central forces is addressed in a three dimensional space in which the Poisson bracket among the coordinates is that of the SU(2) Lie algebra. In particular it is shown that among problems with spherically symmetric potential energies, it is only the Kepler problem for which all of the bounded orbits are closed. In analogy with the case of the ordinary space, a conserved vector (apart from the angular momentum) is explicitly constructed, which is responsible for the orbits being closed. This is the analog of the Laplace-Runge-Lenz vector. The algebra of the constants of the motion is also worked out.