On the superintegrability of a rational oscillator with nonlinear terms: Euclidean and non-Euclidean cases with nonlinear terms : Euclidean and non-Euclidean cases Manuel F. Ranada
(mfran@posta.unizar.es)
Abstract In the first part, the superintegrability of a Euclidean $n=2$ rational Harmonic Oscillator with nonlinear (centrifugal) terms is studied. It is proved that inversely quadratic nonlinearities modify the solutions but preserves superintegrability. The constants of motion of the nonlinear system are explicetely obtained. The second part is devoted to the study of the curvature dependent versions of these systems. We study, first the $n=2$ Harmonic Oscillator, and then, the Oscillator with nonlinear (centrifugal) terms, on the two-dimensional constant cusvature spaces (sphere $S^2$ and hyperbolic plane $H^2$). All the mathematical expressions are presented using the curvature ${\kappa}$ as a parameter, in such a way that particularizing for ${\kappa}>0$, ${\kappa}=0$, or ${\kappa}<0$, the corresponding properties are obtained for the system on the sphere $S^2$, the euclidean plane $E^2$, or the hyperbolic plane $H^2$, respectively. |