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Harmonic Oscillator on the $\mathrm{SO}(2,2)$ Hyperboloid
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: superintegrable systems; harmonic oscillator; hyperbolic space; Hamilton–Jacobi equation. 
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     title = {Harmonic {Oscillator} on the $\mathrm{SO}(2,2)$ {Hyperboloid}},
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}
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Davit R. Petrosyan; George S. Pogosyan. Harmonic Oscillator on the $\mathrm{SO}(2,2)$ Hyperboloid. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a95/

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