Orlando Ragnisco

In this paper we tersely recall the main algebraic and geometric properties of the maximally superintegrable system known as "Perlick System Tipe I", considering all possible values of the relevant parameters. We will follow a classical variant of the so called factorization method, emphasizing the role played the Poisson Algebra of the constants of motion in sheding light on the geometric features of the trajectories.

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.

In a recent paper the so-called Spectrum Generating Algebra (SGA) technique has been applied to the N-dimensional Taub-NUT system, a maximally superintegrable Hamiltonian system which can be interpreted as a one-parameter deformation of the Kepler-Coulomb system. Such a Hamiltonian is associated to a specific Bertrand space of non-constant curvature. The SGA procedure unveils the symmetry algebra underlying the Hamiltonian system and, moreover, enables one to solve the equations of motion. Here we will follow the same path to tackle the Darboux III system, another maximally superintegrable system, which can indeed be viewed as a natural deformation of the isotropic harmonic oscillator where the flat Euclidean space is again replaced by another space of non-constant curvature.

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: 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.