Motion On the Sphere: Integrability and Families of Orbits

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Abstract

We study the problem of the motion of a unit mass on the unit sphere \({\mathbb S}^2\) and examine the relation between integrability and certain monoparametric families of orbits. In particular we show that if the potential is compatible with a family of meridians, it is integrable with an integral linear in the velocities, while a family of parallels guarantees integrability with an integral quadratic in the velocities.

Keywords

motion on the sphere integrability inverse problem 
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References

  1. Borisov, A. V., Mamaev, I. S., Kilin, A. A. 2004‘Two-body problem on a sphere. Reduction, stochasticity, periodic orbits’Reg. Ch. Dyn.9265279MathSciNetMATHGoogle Scholar
  2. Bozis, G. 1984‘Szebehely’s inverse problem for finite symmetrical material concentration’Astron. Astrophys.134360364ADSMATHMathSciNetGoogle Scholar
  3. Bozis, G. 1995‘The inverse problem of dynamics: basic facts’Inverse Prob.11687708ADSMATHMathSciNetGoogle Scholar
  4. Bozis, G., Mertens, R. 1985‘On Szebehely’s inverse problem for a particle describing orbits on a given surface’Z. Angew. Math. Mech.65383384MathSciNetMATHGoogle Scholar
  5. Darboux, G. 1901‘Sur un problème de méchanique’Arch. Neerland.6371MATHGoogle Scholar
  6. Dorizzi, B., Grammaticos, B., Ramani, A. 1983‘A new class of integrable systems’J. Math. Phys.2422822288CrossRefADSMathSciNetGoogle Scholar
  7. Hietarinta, J. 1987‘Direct methods for the search of the second invariant’Phys. Rep.14787154CrossRefADSMathSciNetGoogle Scholar
  8. Higgs, P.W. 1979‘Dynamical symmetries in a spherical geometry I’J. Phys. Math. Gen.12309323ADSMATHMathSciNetGoogle Scholar
  9. Ichtiaroglou, S., Meletlidou, E. 1990‘On monoparametric families of orbits sufficient for integrability of planar potential with linear or quadratic invariants’J. Phys. A: Math. Gen.2336733679CrossRefADSMathSciNetMATHGoogle Scholar
  10. Kozlov, V.V., Harin, A.O. 1992‘Kepler’s problem in constant curvature spaces’Celest. Mech. Dyn. Astron.54393399CrossRefADSMathSciNetMATHGoogle Scholar
  11. Puel, F. 1995‘Separable and partially separable systems in the light of the inverse problem of dynamics’Celest. Mech. Dyn. Astron.634157ADSMATHMathSciNetGoogle Scholar
  12. Szebehely V.: 1974, ‘On the determination of the potential by satellite observations’. In Proverbio E. (ed.), Proceedings of the International Meeting ‘Earth’s Rotation by Satellite Observations’, Bologna, pp. 31–35.Google Scholar
  13. Voyatzi, D., Ichtiaroglou, S. 2003‘Two-parametric families of orbits and multiseparability of planar potentials’Celest. Mech. Dyn. Astr.86209221ADSMathSciNetMATHGoogle Scholar
  14. Vozmishcheva, T. G. 2003‘Some integrable problems in Celestial Mechanics in spaces of constant curvature’J. Math. Sci.11410251066CrossRefMATHMathSciNetGoogle Scholar
  15. Whittaker, E. T. 1944A Treatise on Analytical Dynamics of Particles and Rigid BodiesCambridge University PressCambridgeMATHGoogle Scholar
  16. Ziglin, S. L. 2001‘Nonintegrability of a bounded two-body problem’Doklady Phys.46570571ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Physics DepartmentUniversity of ThessalonikiThessalonikiGreece

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