Regular and Chaotic Dynamics

, Volume 14, Issue 4–5, pp 571–579 | Cite as

Further classification of 2D integrable mechanical systems with quadratic invariants



Four new integrable classes of mechanical systems on Riemannian 2D manifolds admitting a complementary quadratic invariant are introduced. Those systems have quite rich structure. They involve 11–12 arbitrary parameters that determine the metric of the configuration space and forces with scalar and vector potentials. Interpretations of special versions of them are pointed out as problems of motions of rigid body in a liquid or under action of potential and gyroscopic forces and as motions of a particle on the plane, sphere, ellipsoid, pseudo-sphere and other surfaces.

Key words

integrable Lagrangian systems quadratic invariants time-irreversible systems 

MSC2000 numbers

70H06 70E40 


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  1. 1.
    Bertrand, J., Sur les intégrales commune à plusieurs problémes de mécanique, J. Math. Pures Appl., 1852, vol. 17, pp. 121–174.Google Scholar
  2. 2.
    Bertrand, J., Memoire sur quelque-unes des formes les plus simple que puissent présenter les intégrals des équations différentielles du movement d’un point matériel, J. Math. Pures Appl., Ser. II, 1857, vol. 2, pp. 113–140.Google Scholar
  3. 3.
    Darboux, G., Archives Néerlandaises, Vol. 6(ii), 1901, pp. 371–376.Google Scholar
  4. 4.
    Winternitz, P., Smorodinsky, J.A., Uhlir, M., and Fris, I., Symmetry Groups in Classical and Quantum Mechanics, Jadernaja Fiz., 1967, vol. 4, pp. 625–635 [Soviet J. Nuclear Phys., 1967, vol. 4, pp. 444–450].Google Scholar
  5. 5.
    Stäckel P.G., Ann. Mat. Pura, 26, 55–60 (1897).CrossRefGoogle Scholar
  6. 6.
    Eisenhart, L.P., Separable systems of Stackel, Ann. Math., 1934, vol. 35, no. 2, pp. 284–305.CrossRefMathSciNetGoogle Scholar
  7. 7.
    Landau, L.D. and Lifshitz, E.M., Mechanics, 1969, Pergamon, Oxford.Google Scholar
  8. 8.
    Yehia, H.M., Atlas of Two-dimensional Irreversible Conservative Lagrangian Mechanical Systems with a Second Quadratic Integral, J. Math. Phys., 2007, vol. 48, 082902, 32 pp.Google Scholar
  9. 9.
    Vandervoort, P.O., Isolating integrals of the motion for stellar orbits in a rotating galactic bar, Ap. J., 1979, vol. 232, pp. 91–105.CrossRefGoogle Scholar
  10. 10.
    Dorizzi, B., Grammaticos, B., Ramani, A., and Winternitz P., Integrable Hamiltonian Systems with Velocity-Dependent Potentials, J. Math. Phys., 1985, vol. 26, pp. 3070–3079.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Yehia, H.M., Generalized Natural Mechanical Systems of Two Degrees of Freedom with Quadratic Integrals, J. Phys A: Math. Gen., 1992, vol. 25, pp. 197–221.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Yehia, H.M., Motions of a Particle on a Sphere and Integrable Motions of a Rigid Body, J. Phys A: Math. Gen., 2000, vol. 33, pp. 5945–5949.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yehia, H.M., An Integrable Motion of a Particle on a Smooth Ellipsoid, Regul. Chaotic Dyn., 2003, vol. 8, pp. 463–468.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Contopoulos, G. and Vandervoort, P.O., A rotating Stáckel potential, Astron. J., 1992, vol. 389, pp. 118–128.Google Scholar
  15. 15.
    Stewart, P., Quadratic integrals in uniformly rotating Hamiltonian systems, Astron. Astrophys., 1994, vol. 287, pp. 757–760.Google Scholar
  16. 16.
    McSween, E. and Winternitz, P., Integrable and Superintegrable Hamiltonian Systems in Magnetic Fields, J. Math. Phys., 2000, vol. 41, pp. 2957–2967.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pucacco, G., On Integrable Hamiltonians with Velocity Dependent Potentials, Celestial Mech. Dynam. Astronom., 2004, vol. 90, pp. 111–125.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Pucacco, G. and Rosquist, K., Integrable Hamiltonian Systems with Vector Potentials, J. Math. Phys., 2005, vol. 46, 012701, 25 pp.Google Scholar
  19. 19.
    Benenti, S., Chanu, C., and Rastelli, G., Variable Separation for Natural Hamiltonians with Scalar and Vector Potentials on Riemannian Manifolds, J. Math. Phys., 2001, vol. 42, pp. 2065–2091.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Ferapontov, E.V. and Fordy, A.P., Commuting Quadratic Hamiltonians with Velocity Dependent Potentials, Rep. Math. Phys., 1999, vol. 44, pp. 71–80.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Marikhin, V.G. and Sokolov, V.V., On Quasi-Stäckel Hamiltonians, Uspekhi Mat. Nauk, 2005, vol. 60, no. 5(365), pp. 175–176 [Russian Math. Surveys, 2005, vol. 60, no. 5, pp. 981–983].MathSciNetGoogle Scholar
  22. 22.
    Ranada, M. and Santander, M., Superintegrable Systems on the Two-dimensional Sphere S 2 and the Hyperbolic Plane H 2, J. Math. Phys. 1999, vol. 40, pp. 5026–5057.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

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