Harmonic Oscillator on the $\mathrm{SO}(2,2)$ Hyperboloid

Abstract

In the present work the classical problem of harmonic oscillator in the hyperbolic space $H_2^2$: $z_0^2+z_1^2-z_2^2-z_3^2=R^2$ has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on $H_2^2$, as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the bounded classical trajectories are closed and periodic. The orbits of motion are ellipses or circles for bounded motion and ultraellipses or equidistant curve for infinite ones.

References in corpus (4)

• Superintegrability on Three-Dimensional Riemannian and Relativistic Spaces of Constant Curvature
• Maximally superintegrable Smorodinsky-Winternitz systems on the N-dimensional sphere and hyperbolic spaces
• The harmonic oscillator on Riemannian and Lorentzian configuration spaces of constant curvature
• Hamiltonians for the Quantum Hall Effect on Spaces with Non-Constant Metrics

Cited by in corpus (2)

• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• Curvature as an integrable deformation