Regular and Chaotic Dynamics

, Volume 19, Issue 6, pp 601–606 | Cite as

On rational integrals of geodesic flows

Article

Abstract

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.

Keywords

conformal coordinates rational integral irreducible integrals Cauchy-Kovalevskaya theorem 

MSC2010 numbers

34A34 58E10 

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References

  1. 1.
    Whittaker, E. T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., New York: Cambridge Univ. Press, 1959.Google Scholar
  2. 2.
    Birkhoff, G.D., Dynamical Systems, Providence, RI: AMS, 1966.MATHGoogle Scholar
  3. 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.Google Scholar
  4. 4.
    Kozlov, V.V., Symmetries, Topology and Resonances in Hamiltonian Mechanics, Ergeb. Math. Grenzgeb. (3), vol. 31, Berlin: Springer, 1996.CrossRefGoogle Scholar
  5. 5.
    Ten, V.V., Local Integrals of Geodesic Flows, Regul. Chaotic Dyn., 1997, vol. 2, no. 2, pp. 87–89 (Russian).MATHMathSciNetGoogle Scholar
  6. 6.
    Poincaré, H., Sur le méthode de Bruns, C. R. Acad. Sci. Paris, 1896, vol. 123, pp. 1224–1228.Google Scholar
  7. 7.
    Albouy, A., Projective Dynamics and First Integrals, arXiv:1401.1509 (2006), 28 pp.Google Scholar
  8. 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, 1986, vol. 18, no. 2, pp. 207–214.MATHMathSciNetGoogle Scholar
  9. 9.
    Collinson, C. D. and O’Donnell, P. J., A Class of Empty Spacetimes Admitting a Rational First Integral of the Geodesic Equation, Gen. Relativity Gravitation, 1992, vol. 24, no. 4, pp. 451–455.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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