Regular and Chaotic Dynamics

, Volume 18, Issue 5, pp 490–496 | Cite as

The dynamics of the chaplygin ball with a fluid-filled cavity

Article

Abstract

We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.

Keywords

vortex motion nonholonomic constraint Chaplygin ball invariant measure integrability rigid body ideal fluid 

MSC2010 numbers

70E18 76B47 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bolsinov, A.V., Borisov, A. V., and Mamaev, I. S., Hamiltonisation of Non-Holonomic Systems in the Neighborhood of Invariant Manifolds, Nelin. Dinam., 2010, vol. 6, no. 4, pp. 829–854 (Russian).Google Scholar
  2. 2.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Nelin. Dinam., 2010, vol. 6, no. 4, pp. 869–889 (Russian).MathSciNetGoogle Scholar
  3. 3.
    Borisov, A. V., Gazizullina, L. A., and Mamaev, I. S., On V. A. Steklov’s Legacy in Classical Mechanics, Nelin. Dinam., 2011, vol. 7, no. 2, pp. 389–403 (Russian).Google Scholar
  4. 4.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., Generalized Chaplygins Transformation and Explicit Integration of a System with a Spherical Support, Nelin. Dinam., 2011, vol. 7, no. 2, pp. 313–338 (Russian).MathSciNetGoogle Scholar
  5. 5.
    Borisov, A.V. and Mamaev, I. S., Rigid Body Dynamics, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2001 (Russian).Google Scholar
  6. 6.
    Zhukovskii, N.E., Motion of a Rigid Body Containing a Cavity Filled with a Homogeneous Continuous Liquid, in Collected Works: Vol. 2, Moscow: Gostekhteorizdat, 1949, pp. 31–152 (Russian).Google Scholar
  7. 7.
    Karapetyan, A.V. and Prokomina, O. V., The Stability of Permanent Rotations of a Top with a Cavity Filled with Liquid on a Plane with Friction, Prikl. Mat. Mekh., 2000, vol. 64, no. 1, pp. 85–91 [J. Appl. Math. Mech., 2000, vol. 64, no. 1, pp. 81–86].MathSciNetMATHGoogle Scholar
  8. 8.
    Markeev, A. P., The Stability of the Rotation of a Top with a Cavity Filled with Liquid, Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, 1985, vol. 20, no. 3, pp. 19–26 (Russian).Google Scholar
  9. 9.
    Moiseev, N.N. and Rumiantsev, V.V., Dynamic Stability of Bodies Containing Fluid, New York: Springer, 1968.CrossRefGoogle Scholar
  10. 10.
    Rudenko, T. V., The Stability of the Steady Motion of a Gyrostat with a Liquid in a Cavity, Prikl. Mat. Mekh., 2002, vol. 66, no. 2, pp. 183–191 [J. Appl. Math. Mech., 2002, vol. 66, no. 2, pp. 171–178].MathSciNetMATHGoogle Scholar
  11. 11.
    Steklov, V. A., Works on Mechanics 1902–1909: Translations from French, Moscow-Izhevsk: R&C Dynamics, Institute of Computer Science, 2011 (Russian).Google Scholar
  12. 12.
    Liu, Y. Z., The Stability of a Fluid-Filled Top Rotating on a Horizontal Plane, Arch. Appl. Mech., 1992, vol. 62, pp. 487–494.CrossRefMATHGoogle Scholar
  13. 13.
    Perry, J., Spinning Top and Gyroscopic Motions, New York: Dover, 1957.Google Scholar
  14. 14.
    Poincaré, H., Sur la précession des corps déformables, Bull. Astron., 1910, vol. 27, pp. 321–356.Google Scholar
  15. 15.
    Poincaré, H., Sur le forme nouvelle des equations de la mecanique, C. R. Acad. Sci. Paris, 1901, vol. 132, pp. 369–371.MATHGoogle Scholar
  16. 16.
    Rambaux, N., Van Hoolst, T., Dehant, V., and Bois, E., Inertial Core-Mantle Coupling and Libration of Mercury, Astron. Astrophys., 2007, vol. 468, no. 2, pp. 711–719.CrossRefGoogle Scholar
  17. 17.
    Stekloff, V. A., Sur la théorie des tourbillons, Ann. Fac. Sci. Toulouse Math. (2), 1908, vol. 10, pp. 271–334.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Stekloff, V. A., Sur le movement d’un corps solide ayant une cavité de forme ellipsoidale remplie par un liquide incompressible et sur les variations des latitudes, Ann. Fac. Sci. Toulouse Math. (3), 1909, vol. 1, pp. 145–256.MathSciNetCrossRefGoogle Scholar
  19. 19.
    Thompson, S.Ph., The Life of William Thomson, Baron Kelvin of Largs: Vol. 2, London: McMillan, 1910, pp. 736–752.Google Scholar
  20. 20.
    Thomson, W., On the Precessional Motion of a Liquid, Nature, 1877, vol. 15, pp. 297–298.CrossRefGoogle Scholar
  21. 21.
    Volterra, V., Sur la théorie des variations des latitedes, Acta. Math., 1899, vol. 22, pp. 201–358.MathSciNetCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Institute of Computer Science; Laboratory of Nonlinear Analysis and the Design of New Types of VehiclesUdmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.Institute of Mathematics and Mechanics of the Ural Branch of RASYekaterinburgRussia

Personalised recommendations