### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 096, 23 pages      arXiv:1504.06228      https://doi.org/10.3842/SIGMA.2015.096
Contribution to the Special Issue on Analytical Mechanics and Differential Geometry in honour of Sergio Benenti

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

Davit R. Petrosyan a and George S. Pogosyan bc
a) Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow Region, 141980, Russia
c) International Center for Advanced Studies, Yerevan State University, A. Manoogian 1, Yerevan, 0025, Armenia

Received April 24, 2015, in final form November 20, 2015; Published online November 25, 2015

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.

Key words: superintegrable systems; harmonic oscillator; hyperbolic space; Hamilton-Jacobi equation.

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