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

Abstract

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:  &eegr; ( q , p ) =  &eegr; ( q , p ) +  &eegr; ( q ) = ∣ q ∣ p 2 2 m ( &eegr; + ∣ q ∣ ) − k &eegr; + ∣ q ∣ ( k > 0 , &eegr; > 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 , and for which a ‘SUSYQM’ approach has been recently introduced by Kuru and 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 . 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 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 case, where new and partly unexpected features arise.