The anisotropic oscillator on the 2D sphere and the hyperbolic plane

An integrable generalization on the 2D sphere S2 and the hyperbolic plane H2 of the Euclidean anisotropic oscillator Hamiltonian with 'centrifugal' terms given by is presented. The resulting generalized Hamiltonian Hκ depends explicitly on the constant Gaussian curvature κ of the underlying space, in such a way that all the results here presented hold simultaneously for S2 (κ > 0), H2 (κ < 0) and E2 (κ = 0). Moreover, Hκ is explicitly shown to be integrable for any values of the parameters δ, Ω, λ1 and λ2. Therefore, Hκ can also be interpreted as an anisotropic generalization of the curved Higgs oscillator, that is recovered as the isotropic limit Ω → 0 of Hκ. Furthermore, numerical integration of some of the trajectories for Hκ are worked out and the dynamical features arising from the introduction of a curved background are highlighted. The superintegrability issue for Hκ is discussed by focusing on the value Ω = 3δ, 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 Ω = 3δ the curved Hamiltonian Hκ presents nonperiodic bounded trajectories, which seems to indicate that Hκ provides a non-superintegrable generalization of H even for values of Ω that lead to commensurate frequencies in the Euclidean case. We compare this result with a previously known superintegrable curved analogue Hκ' of the 1 : 2 Euclidean oscillator, which is described in detail, showing that the Ω = 3δ specialization of Hκ does not coincide with Hκ'. Hence we conjecture that Hκ would be an integrable (but not superintegrable) curved generalization of the anisotropic oscillator that exists for any value of Ω and has constants of the motion that are quadratic in the momenta. Thus each commensurate Euclidean oscillator could admit another specific superintegrable curved Hamiltonian which would be different from Hκ and endowed with higher order integrals. Finally, the geometrical interpretation of the curved 'centrifugal' terms appearing in Hκ is also discussed in detail. (paper)