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

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.

Noncommutative Classical Dynamics on Velocity Phase Space and Souriau Formalism

“…This would lead to unveil the close relation existing between the non-commutative geometry and the geometric phases. Quantization of these models could give rise to new physics at some very high energy scale [80,81,82,83,84,85,86,87,88,89,90,91].…”