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

References in corpus (12)

• Superintegrability on N-dimensional curved spaces: Central potentials, centrifugal terms and monopoles
• Factorizations of one dimensional classical systems
• A maximally superintegrable system on an n-dimensional space of nonconstant curvature
• Gravitomagnetism in the Kerr-Newman-Taub-NUT spacetime
• Hidden symmetry of hyperbolic monopole motion
• Hamiltonian systems admitting a Runge-Lenz vector and an optimal extension of Bertrand's theorem to curved manifolds
• Conformal Yano-Killing tensors for the Taub-NUT metric
• Circular holonomy in the Taub-NUT spacetime
• An exactly solvable deformation of the Coulomb problem associated with the Taub-NUT metric
• Quantum Integrals from Coalgebra Structure
• On two superintegrable nonlinear oscillators in N dimensions
• A maximally superintegrable deformation of the N-dimensional quantum Kepler-Coulomb system

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• The classical Darboux III oscillator: factorization, Spectrum Generating Algebra and solution to the equations of motion
• Superintegrable systems, polynomial algebra structures and exact derivations of spectra
• Superintegrable systems in non-Euclidean plane: hidden symmetries leading to linearity
• Examples of Complete Solvability of 2D Classical Superintegrable Systems
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Nonlinear Lie-Hamilton systems: $t$-Dependent curved oscillators and Kepler-Coulomb Hamiltonians