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

Abstract

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: $${\mathcal{H}}_{\eta }(\mathbf{q},\mathbf{p})={\mathcal{T}}_{\eta }(\mathbf{q},\mathbf{p})+{\mathcal{U}}_{\eta }(\mathbf{q})=\frac{|\mathbf{q}|{\mathbf{p}}^2}{2m(\eta +|\mathbf{q}|)}-\frac{k}{\eta +|\mathbf{q}|}(k>0,\eta >0).$$ 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 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.

11 pages, 6 figures. Version essentially extended: three new sections added, some notations changed, typos corrected and four new figures included

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Cited by in corpus (1)

• The Perlick system type I: from the algebra of symmetries to the geometry of the trajectories