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

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 $H=1/2(p_1^2+p_2^2)+ δq_1^2+(δ+ Ω)q_2^2 +\frac{λ_1}{q_1^2}+\frac{λ_2}{q_2^2}$ is presented. The resulting generalized Hamiltonian H_κ 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 S^2 (κ>0), H^2 (κ<0) and E^2 (κ=0). Moreover, H_κ is explicitly shown to be integrable for any values of the parameters δ, Ω, λ_1 and λ_2. Therefore, H_κ can also be interpreted as an anisotropic generalization of the curved Higgs oscillator, that is recovered as the isotropic limit Ω=0 of H_κ. Furthermore, numerical integration of some of the trajectories for H_κ are worked out and the dynamical features arising from the introduction of a curved background are highlighted. The superintegrability issue for H_κ 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 H_κ presents nonperiodic bounded trajectories, which seems to indicate that H_κ provides a non-superintegrable generalization of H. We compare this result with a previously known superintegrable curved analogue H'_κ of the 1:2 Euclidean oscillator showing that the Ω=3δ specialization of H_κ does not coincide with H'_κ. Finally, the geometrical interpretation of the curved "centrifugal" terms appearing in H_κ is also discussed in detail.