Harmonic Oscillator on the SO(2,2) Hyperboloid

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In the present work the classical problem of harmonic oscillator in the hyperbolic space H₂²: z₀²+z₁²-z₂²-z₃²=R² has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on H₂², 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.