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On rational integrals of geodesic flows. (English) Zbl 1343.37047
Summary: This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy-Kovalevskaya theorem.

##### MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37C10 Dynamics induced by flows and semiflows 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 35A10 Cauchy-Kovalevskaya theorems
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##### References:
 [1] Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., New York: Cambridge Univ. Press, 1959. [2] Birkhoff, G.D., Dynamical Systems, Providence, RI: AMS, 1966. · Zbl 0171.05402 [3] Kozlov, V.V., Integrable and Nonintegrable Hamiltonian Systems, Soviet Sci. Rev., Sect. C. Math. Phys. Rev., vol. 8, part 1, Chur: Harwood Acad. Publ., 1989. [4] Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996. [5] Ten, VV, Local integrals of geodesic flows, Regul. Chaotic Dyn., 2, 87-89, (1997) · Zbl 1083.37510 [6] Poincaré, H., Sur le méthode de Bruns, C. R. Acad. Sci. Paris, 1896, vol. 123, pp. 1224-1228. [7] Albouy, A., Projective Dynamics and First Integrals, arXiv:1401.1509 (2006), 28 pp. · Zbl 1378.70015 [8] Collinson, C D, A note on the integrability conditions for the existence of rational first integrals of the geodesic equations in a Riemannian space, Gen. Relativity Gravitation, 18, 207-214, (1986) · Zbl 0581.53010 [9] Collinson, C D; O’Donnell, P J, A class of empty spacetimes admitting a rational first integral of the geodesic equation, Gen. Relativity Gravitation, 24, 451-455, (1992) · Zbl 0756.53033
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