Regular and Chaotic Dynamics

, Volume 14, Issue 4–5, pp 495–505 | Cite as

Hamiltonization of the generalized Veselova LR system

  • Yu. N. FedorovEmail author
  • B. Jovanović


We revise the solution to the problem of Hamiltonization of the n-dimensional Veselova nonholonomic system studied previously in [1]. Namely, we give a short and direct proof of the hamiltonization theorem and also show the trajectorial equivalence of the problem with the geodesic flow on the ellipsoid.

Key words

nonholonomic systems integrability geodesic flows 

MSC2000 numbers

37J60 37J35 70H45 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Fedorov, Yu.N. and Jovanović, B., Nonholonomic LR Systems as Generalized Chaplygin Systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces, J. Nonlinear Sci., 2004, vol. 14, pp. 341–381; arXiv: math-ph/0307016.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Veselov, A. P. and Veselova, L.E., Integrable Nonholonomic Systems on Lie Groups, Mat. zametki, 1988, vol. 44, no. 5, pp. 604–619 [Mat. Notes, 1988, vol. 44, no. 5, pp. 810–819].zbMATHMathSciNetGoogle Scholar
  3. 3.
    Chaplygin, S. A., On the Theory of the Motion of Nonholonomic Systems: Theorem on the Reducing Multiplier, Mat. Sb., 1911, vol. 28, no. 2, pp. 303–314 [Regul. Chaotic Dyn., 2008, vol. 13, no. 4, pp. 369–376].Google Scholar
  4. 4.
    Stanchenko, S., Nonholonomic Chaplygin Systems, Prikl. Mat. Mekh., 1989, vol. 53, no. 1, pp. 16–23 [J. Appl. Math. Mech., 1989, vol. 53, no. 1, pp. 11–17].MathSciNetGoogle Scholar
  5. 5.
    Koiller, J., Reduction of Some Classical Non-Holonomic Systems with Symmetry, Arch. Ration. Mech. Anal., 1992, vol. 118, pp. 113–148.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bloch, A. M., Krishnaprasad, P. S., Marsden, J.E., and Murray, R. M., Nonholonomical Mechanical Systems with Symmetry, Arch. Ration. Mech. Anal., 1996, vol. 136, pp. 21–99.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fedorov, Yu.N. and Kozlov, V.V., Various Aspects of n-Dimensional Rigid Body Dynamics, Amer. Math. Soc. Transl. Ser. 2, vol. 168, 1995, pp. 141–171.MathSciNetGoogle Scholar
  8. 8.
    Ehlers, K. and Koiller, J., Rubber Rolling Over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, pp. 127–152; arXiv: math.SG/0612036.CrossRefMathSciNetGoogle Scholar
  9. 9.
    Jovanović, B., LR and L + R Systems, J. Phys. A: Math. Theor., 2009, vol. 42, 225202 (18 pp.); arXiv: 0902.1656 [math-ph].preprint, 2009.Google Scholar
  10. 10.
    Fedorov, Yu.N. and Jovanović, B., Quasi-Chaplygin Systems and Nonholonimic Rigid Body Dynamics, Lett. Math. Phys., 2006, vol. 76, pp. 215–230; arXiv: math-ph/0510088.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Borisov, A. V. and Mamaev, I. S., Chaplygin’s Ball Rolling Problem Is Hamiltonian, Mat. Zametki, 2001, vol. 70, no. 5, pp. 793–795 [Math. Notes, 2001, vol. 70, nos. 5–6, pp. 720–723].MathSciNetGoogle Scholar
  12. 12.
    Borisov, A.V. and Mamaev, I. S., Isomprphism and Hamiltonizations of Some Nonholonomic Systems, Sib. Mat. Zh., 2007, vol. 46, no. 1, pp. 33–45 [Siberian Math. J., 2007, vol. 48, no. 1, pp. 26–36]; extended version: arXiv:nlin/0509036.MathSciNetGoogle Scholar
  13. 13.
    Chaplygin, S. A., On a Rolling Sphere on a Horizontal Plane, Mat. Sb., 1903, vol. 24, pp. 139–168 [Regul. Chaotic Dyn., 2002, vol. 7, pp. 131–148].Google Scholar
  14. 14.
    Fedorov, Yu.N., A Complete Complex Solution of the Nonholonomic Chaplygin Sphere Problem, preprint, 2009.Google Scholar
  15. 15.
    Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557–571.CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hochgerner, S. and García-Naranjo, L., G-Chaplygin Systems with Internal Symmetries, Truncation, and an (Almost) Symplectic View of Chaplygin’s Ball, J. Geom. Mech., 2009, vol. 1, No. 1, pp. 35–53; arXiv:08105454 [math-ph].CrossRefGoogle Scholar
  17. 17.
    Jovanović, B., Hamiltonization and Integrability of the Chaplygin Sphere inn, 2009, arXiv: 0902.4397v1 [math-ph].Google Scholar
  18. 18.
    Topalov, P., Integrability Criterion of Geodesical Equavalence: Hierarchies, Acta Appl. Math., 1999, vol. 59, pp. 271–298.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Tabachnikov, S., Projectively Equivalent Metrics, Exact Transverse Line Field and Geodesic Flow on the Ellipsoid, Comment. Math. Helv., 1999, vol. 74, pp. 306–321.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fedorov, Yu.N. and Jovanović, B., Integrable Nonholonomic Geodesic Flows on Compact Lie Groups, in Topological Methods in the Theory of Integrable Systems, A. V. Bolsinov, A. T. Fomenko, A. A. Oshemkov (Eds.), Cambrige: Cambrige Intern. Sci., 2006, pp. 115–152; arXiv: math-ph/0408037.Google Scholar
  21. 21.
    Tatarinov, Ya.V., Nonholonomic Systems in Comarison with Hamiltonian Ones, Doctoral disseration, Moscow State Univ., Moscow, 1990.Google Scholar
  22. 22.
    Cantrijn, F., Cortes, J., de Leon, M., and Martin de Diego, D., On the Geometry of Generalized Chaplygin Systems, Math. Proc. Cambridge Philos. Soc., 2002, vol. 132, no. 2, pp. 323–351; arXiv: math.DS/0008141.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Dirac, P. A., On Generalized Hamiltonian Dynamics, Canad. J. Math., 1950, vol. 2, no. 2, pp. 129–148.zbMATHMathSciNetGoogle Scholar
  24. 24.
    Arnol’d, V. I., Kozlov, V.V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III, Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 1993.Google Scholar
  25. 25.
    Davison, C.M., Geodesic Flow on the Ellipsoid with Equal Semi-Axes, PhD. Thesis, Loughborough University, 2006.Google Scholar
  26. 26.
    Wojciechowski, S., Integrable One-Partical Potentials Related to the Neumann System and the Jacobi Problem of Geodesic Motion on an Ellipsoid, Phys. Lett. A, vol. 107, pp. 107–111.Google Scholar
  27. 27.
    Braden, H.W., A Completely Integrable Mechanical System, Lett. Math. Phys., 1982, vol. 6, no. 6, pp. 449–452.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Bogoyavlensky, O. I., Integrable Cases of Rigid Body Dynamics and Integrable Systems on Spheres S n, Izv. Akad. Nauk. SSSR, Ser. Mat., 1985, vol. 49, no. 5, pp. 899–915 [Math. USSR-Izv., 1986, vol. 27, no. 2, pp. 203–218].MathSciNetGoogle Scholar
  29. 29.
    Kalnins, E. G, Benenti, S., and Miller, W., Integrability, Stäckel Spaces, and Rational Potentials, J. Math. Phys., 1997, vol. 38, pp. 2345–2365.zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Jovanović, B., Integrable Perturbations of Billiards on Constant Curvature Surfaces, Phys. Lett. A, 1997, vol. 231, pp. 353–358.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Dragović, V., The Appel Hypergeometric Functions and Classical Separable Mechanical Systems, J. Phys. A: Math. Gen., 2002, vol. 35, pp. 2213–2221.zbMATHCrossRefGoogle Scholar
  32. 32.
    Neumann, C., De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatum, J. Reine Angew. Math., 1859, vol. 56, pp. 46–63.zbMATHGoogle Scholar
  33. 33.
    Knörrer, H., Geodesics on Quadrics and a Mechanical Problem of C. Neumann, J. Reine Angew. Math., 1982, vol. 334, pp. 69–78.zbMATHMathSciNetGoogle Scholar
  34. 34.
    Manakov, S. V., Note on the Integrability of the Euler Equations of n-Dimensional Rigid Body Dynamics, Funkts. Anal. Prilozh., 1976, vol. 10, no. 4, pp. 93–94.zbMATHMathSciNetGoogle Scholar
  35. 35.
    Brailov, A.V., Construction of Complete Integrable Geodesic Flows on Compact Symmetric Spaces, Izv. Acad. Nauk SSSR, Ser. Matem., 1986, vol. 50, no. 2, pp. 661–674 [Math. USSR-Izv., 1986, vol. 50, no. 4, pp. 19–31].MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Departament de Matemàtica IUniversitat Politecnica de CatalunyaBarcelonaSpain
  2. 2.Mathematical Institute SANUBelgradeSerbia

Personalised recommendations