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: $$ mathcal{H}_eta ({mathbf{q}}, {mathbf{p}}) = mathcal{T}_eta ({mathbf{q}}, {mathbf{p}}) + mathcal{U}_eta({mathbf{q}}) = frac{|{mathbf{q}}| {mathbf{p}}^2}{2m(eta + |{mathbf{q}}|)} - frac{k}{eta + |{mathbf{q}}|} quad (k>0, eta>0) , .$$ 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 Keplers law is also recovered. The closing section is devoted to a discussion of the $eta<0$ case, where new and partly unexpected features arise.