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

The formalism of SUperSYmmetric quantum mechanics (SUSYQM) 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: H-eta (q, p) = tau(eta) (q, p) + upsilon(eta) (q) = \q\ p(2)/2m(eta + \q\) - k/eta + \q\ (k > 0, eta > 0) In full agreement with the results recently derived by 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 --> 0, and for which a 'SUSYQM' approach has been recently introduced by Kuru and 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 --> 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 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.