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

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: $\mathcal{H}_η({\mathbf{q}}, {\mathbf{p}}) = \mathcal{T}_η({\mathbf{q}}, {\mathbf{p}}) + \mathcal{U}_η({\mathbf{q}}) = \frac{|{\mathbf{q}}| {\mathbf{p}}^2}{2m(η+ |{\mathbf{q}}|)} - \frac{k}{η+ |{\mathbf{q}}|} \quad (k>0, η>0) \, .$ 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 $η\to 0$, and for which a SUSYQM approach has been recently introduced by S. Kuru and J. Negro. In particular, for positive $η$ and negative energy the motion is always periodic; it turns out that the period depends upon $η$ and goes to the Euclidean value as $η\to 0$. Moreover, the maximal superintegrability is preserved by the $η$-deformation, due to the existence of a larger symmetry group related to an $η$-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 $η<0$ case, where new and partly unexpected features arise.

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