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

TL;DR

This paper solves the classical harmonic oscillator problem on the hyperbolic space ${ m SO}(2,2)$ hyperboloid using Hamilton-Jacobi theory, demonstrating its superintegrability and characterizing all bounded and unbounded trajectories.

Contribution

It provides a complete solution to the harmonic oscillator on the ${ m SO}(2,2)$ hyperboloid, establishing its superintegrability and detailed trajectory classifications.

Findings

All bounded trajectories are closed and periodic.

Bounded orbits are ellipses or circles.

Unbounded orbits are ultraellipses or equidistant curves.

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.