Abstract
In this article the generalization of the motion of a particle in a central field to the case of a constant curvature space is investigated. We found out that orbits on a constant curvature surface are closed in two cases: when the potential satisfies Iaplace-Beltrami equation and can be regarded as an analogue of the potential of the gravitational interaction, and in the case when the potential is the generalization of the potential of an elastic spring. Also the full integrability of the generalized two-centre problem on a constant curvature surface is discovered and it is shown that integrability remains even if elastic “forces” are added.
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References
Abdrakhmanov, A.M.: 1990, ‘On Integrable Rebounding Reflection Systems’, Vestn. Mosk. un-ta, Ser. 1, 5, 85–88 (in Russian).
Birkhoff, G.D.: 1927, Dynamical Systems, Amer. Math. Soc. Colloq. Publ., Vol. 9.
Jacobi, C.G.J.: 1884, Vorlesungen über Dynamik, Berlin.
Moser, J.: 1978, ‘Various Aspects of Integrable Hamiltonian Systems’, Dynamical Systems, C.I.M.E. Lectures, Bressanone, Italy, Progress in Mathematics, Vol. 8, p. 273.
Wintner, A.: 1941, The Analytical Foundations of Celestial Mechanics, Princeton University Press.
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Kozlov, V.V., Harin, A.O. Kepler's problem in constant curvature spaces. Celestial Mech Dyn Astr 54, 393–399 (1992). https://doi.org/10.1007/BF00049149
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DOI: https://doi.org/10.1007/BF00049149