Regular and Chaotic Dynamics

, 14:615 | Cite as

Superintegrable system on a sphere with the integral of higher degree

Research Articles

Abstract

We consider the motion of a material point on the surface of a sphere in the field of 2n + 1 identical Hooke centers (singularities with elastic potential) lying on a great circle. Our main result is that this system is superintegrable. The property of superintegrability for this system has been conjectured by us in [1], where the structure of a superintegral of arbitrarily high odd degree in momemnta was outlined. We also indicate an isomorphism between this system and the one-dimensional N-particle system discussed in the recent paper [2] and show that for the latter system an analogous superintegral can be constructed.

Key words

superintegrable systems systems with a potential Hooke center 

MSC2000 numbers

70Hxx 70H06 70G65 37J35 70F10 

References

  1. 1.
    Borisov, A.V., Kilin, A.A., Mamaev, I.S., Multiparticle Systems. The Algebra of Integrals and Integrable Cases, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 18–41.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Chanu, C., Degiovanni, L., and Rastelli, G. Superintegrable Three-body Systems on the Line, Journal of Math. Phys., 2008, vol. 49, 112901, 10 pp.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Borisov A.V. and Mamaev I.S. (Eds.), Classical Dynamics in non-Eucledian Spaces, Moscow-Izhevsk: Inst. komp. issled., RCD, 2004 (in Russian).Google Scholar
  4. 4.
    Borisov, A.V. and Mamaev, I.S., Generalized Problem of Two and Four Newtonian Centers, Celestial Mech. Dynam. Astronom., 2005, vol. 92, no. 4, pp. 371–380.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Borisov, A.V. and Mamaev, I.S., The Restricted Two-Body Problem in Constant Curvature Spaces, Celestial Mech. Dynam. Astronom., 2006, vol. 96, no. 1, pp. 1–17.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Borisov, A.V., Mamaev, I.S., and Kilin, A.A., Two-Body Problem on a Sphere. Reduction, Stochasticity, Periodic Orbits, Regul. Chaotic Dyn., 2004, vol. 9, no. 3, pp. 265–280.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Borisov, A.V. and Mamaev, I.S., Superintegrable Systems on a Sphere, Regul. Chaotic Dyn., 2005, vol. 10, no. 3, pp. 257–266.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Borisov, A.V. and Mamaev, I.S., Non-linear Poisson Brackets and Isomorphisms in Dynamics, Regul. Chaotic Dyn., 1997, vol. 2, no. 3–4, pp. 72–89 (in Russian).MATHMathSciNetGoogle Scholar
  9. 9.
    Kozlov, V.V., Some Integrable Extensions of the Jacobi’s Problem of Geodesics on an Ellipsoid, Prikl. Mat. Mekh., 1995, vol. 59, no. 1, pp. 3–9 [J. Appl. Math. Mech., 1995, vol. 59, no. 1, pp. 1–7].MathSciNetGoogle Scholar
  10. 10.
    Kozlov, V.V. and Fedorov, Y.N., Integrable Systems on a Sphere with Potentials of Elastic Interaction., Mat. Zametki, 1994, vol. 56, no. 3, pp. 74–79 [Math. Notes 56 (1994), 1994, vol. 56, no. 3–4, pp. 927–930].MathSciNetGoogle Scholar
  11. 11.
    Tsiganov, A.V. and Grigoriev, Yu.A., On Abel’s Equations and Richelot’s Integrals, Rus. J. Nonlin. Dyn, 2010, vol.6 (in press).Google Scholar
  12. 12.
    Tsiganov, A.V., Leonard Euler: Addition Theorems and Superintegrable Systems, Regul. Chaotic Dyn., 2009, vol. 14, no. 3, pp. 389–406.MathSciNetGoogle Scholar
  13. 13.
    Borisov A.V. and Mamaev I.S., Rigid Body Dynamics. Hamiltonian Methods, Integrability, Chaos, Moscow-Izhevsk: Inst. komp. issled., RCD, 2005 (in Russian).Google Scholar
  14. 14.
    Darboux, G., Etude d’une question relative au mouvement d’un point sur une surface de revolution, Bull. S.M.F., 1877, T. 5, pp. 100–113.MathSciNetGoogle Scholar
  15. 15.
    Kolokol’tsov, V.N., Geodesic Flows on Two-dimensional Manifolds with an Additional First Integral Polynomial in Velocities, Izv. Akad. Nauk. SSSR, 1982, vol. 46, pp. 994–1010 (in Russian).MATHMathSciNetGoogle Scholar
  16. 16.
    Bolsinov, A.V. and Fomenko, A.T., Integrable Geodesic Flows on a Sphere Generated by Goryachev-Chaplygin and Kovalevskaya Systems in the Dynamics of a rigid body, Mat. Zametki, 1994, vol. 56, pp. 139–142 [Math. Notes, 1994, vol. 56, pp. 859–861].MathSciNetGoogle Scholar
  17. 17.
    Kiyohara, K., Two-dimensional Geodesic Flows Having First Integrals of Higher Degree, Math. Ann., 2001, vol. 320, pp. 487–505.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Institute of Computer ScienceUdmurt State UniversityIzhevskRussia

Personalised recommendations