The classical Taub-Nut System: factorization, spectrum generating algebra and solution to the equations of motion

TL;DR

This paper extends SUSYQM to analyze the classical Taub-Nut system, revealing its superintegrability, spectrum algebra, and solutions, and explores how it relates to the Kepler system and its deformations.

Contribution

It introduces a SUSYQM-based formalism for the classical Taub-Nut system, demonstrating its superintegrability and spectrum algebra, and analyzes the effects of the deformation parameter ta on dynamics.

Findings

The Taub-Nut system shares features with the Kepler system in the Euclidean limit.

Superintegrability is preserved under ta-deformation due to a deformed Runge-Lenz vector.

Periodic motion occurs for positive ta and negative energy, with a ta-dependent period.

Abstract

The formalism of SUSYQM (SUperSYmmetric Quantum Mechanics) is properly modified in such a way to be suitable for the description and the solution of a classical maximally superintegrable Hamiltonian System, the so-called Taub-Nut system, associated with the Hamiltonian: $$ \mathcal{H}_\eta ({\mathbf{q}}, {\mathbf{p}}) = \mathcal{T}_\eta ({\mathbf{q}}, {\mathbf{p}}) + \mathcal{U}_\eta({\mathbf{q}}) = \frac{|{\mathbf{q}}| {\mathbf{p}}^2}{2m(\eta + |{\mathbf{q}}|)} - \frac{k}{\eta + |{\mathbf{q}}|} \quad (k>0, \eta>0) \, .$$ In full agreement with the results recently derived by A. Ballesteros et al. for the quantum case, we show that the classical Taub-Nut system shares a number of essential features with the Kepler system, that is just its Euclidean version arising in the limit $\eta \to 0$, and for which a SUSYQM approach has been recently introduced by S. Kuru and J. Negro. In particular, for positive $\eta$ and negative energy the motion is always periodic; it turns out that the period depends upon $ \eta$ and goes to the Euclidean value as $\eta \to 0$. Moreover, the maximal superintegrability is preserved by the $\eta$-deformation, due to the existence of a larger symmetry group related to an $\eta$-deformed Runge-Lenz vector, which ensures that in $\mathbb{R}^3$ closed orbits are again ellipses. In this context, a deformed version of the third Kepler's law is also recovered. The closing section is devoted to a discussion of the $\eta<0$ case, where new and partly unexpected features arise.