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

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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: Hη(q, p) = Tη(q, p) + Uη(q) = |q| p²2m(η+ |q|) - kη+ |q| (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→ 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→ 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 R³ 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.