The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

An integrable generalization on the two-dimensional sphere S^2 and the hyperbolic plane H^2 of the Euclidean anisotropic oscillator Hamiltonian with “centrifugal” terms given by $H=1/2(p_1^2+p_2^2)+ \delta q_1^2+(\delta + \Omega)q_2^2 +\frac{\lambda_1}{q_1^2}+\frac{\lambda_2}{q_2^2}$ is presented. The resulting generalized Hamiltonian H_\kappa\ depends explicitly on the constant Gaussian curvature \kappa\ of the underlying space, in such a way that all the results here presented hold simultaneously for S^2 (\kappa>0), H^2 (\kappa<0) and E^2 (\kappa=0). Moreover, H_\kappa\ is explicitly shown to be integrable for any values of the parameters \delta, \Omega, \lambda_1 and \lambda_2. Therefore, H_\kappa\ can also be interpreted as an anisotropic generalization of the curved Higgs oscillator, that is recovered as the isotropic limit \Omega=0 of H_\kappa. Furthermore, numerical integration of some of the trajectories for H_\kappa\ are worked out and the dynamical features arising from the introduction of a curved background are highlighted. The superintegrability issue for H_\kappa\ is discussed by focusing on the value \Omega=3\delta, which is one of the cases for which the Euclidean Hamiltonian H is known to be superintegrable (the 1:2 oscillator). We show numerically that for \Omega=3\delta\ the curved Hamiltonian H_\kappa\ presents nonperiodic bounded trajectories, which seems to indicate that H_\kappa
provides a non-superintegrable generalization of H. We compare this result with a previously known superintegrable curved analogue H’\kappa\ of the 1:2 Euclidean oscillator showing that the \Omega=3\delta\ specialization of H\kappa\ does not coincide with H’\kappa. Finally, the geometrical interpretation of the curved “centrifugal” terms appearing in H\kappa\ is also discussed in detail.