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

TL;DR

This paper introduces an integrable anisotropic oscillator model on curved surfaces like the sphere and hyperbolic plane, extending Euclidean results and exploring its dynamical and superintegrability properties.

Contribution

It generalizes the anisotropic oscillator to curved spaces, explicitly shows integrability for all parameters, and analyzes its dynamical features and superintegrability aspects.

Findings

The Hamiltonian is integrable for all parameter values.

Numerical trajectories show effects of curvature on dynamics.

The model is a non-superintegrable generalization of the Euclidean oscillator.

Abstract

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.