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Regular and Chaotic Dynamics

, Volume 18, Issue 1–2, pp 166–183 | Cite as

Quaternion solution for the rock’n’roller: Box orbits, loop orbits and recession

  • Peter LynchEmail author
  • Miguel D. Bustamante
Article

Abstract

We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed in quaternionic form and a complete analytical solution is obtained. Both types of orbit, boxes and loops, are found, the particular form depending on the initial conditions. We interpret the motion in terms of epi-elliptic orbits. The phenomenon of recession, or reversal of precession, is associated with box orbits. The small-amplitude solutions for the symmetric case, or Routh sphere, are expressed explicitly in terms of epicycles; there is no recession in this case.

Keywords

rolling body dynamics nonholonomic constraints Hamiltonian dynamics 

MSC2010 numbers

70E18 70E20 70H07 

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References

  1. 1.
    Altmann, S. L., Rotations, Quaternions and Double Groups, New York: Dover, 1986.zbMATHGoogle Scholar
  2. 2.
    Arnold, V. I., Mathematical Methods of Classical Mechanics, Berlin: Springer, 1978.zbMATHGoogle Scholar
  3. 3.
    Binney, J. and Scott, T., Galactic Dynamics, Princeton: Princeton Univ. Press, 2008.zbMATHGoogle Scholar
  4. 4.
    Bobylev, D.K., On a Ball with a Gyroscope inside Rolling without Sliding on the Horizontal Plane, Mat. Sb., 1892, vol. 16, no. 3, pp. 544–581 (Russian).Google Scholar
  5. 5.
    Borisov, A. V., Kilin, A.A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, pp. 272–275.zbMATHCrossRefGoogle Scholar
  6. 6.
    Borisov, A. V. and Mamaev, I. S., Rolling of a Rigid Body on a Plane and a Sphere: Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 177–200.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chaplygin, S.A., On a Motion of a Heavy Body of Revolution on a Horizontal Plane, in Collected Works: Vol. 1, Moscow: Gostekhizdat, 1948, pp. 51–57 [Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 119–130].Google Scholar
  8. 8.
    Chaplygin, S. A., On a Ball’s Rolling on a Horizontal Plane, Math. Sb., 1903, vol. 24, no. 1, pp. 139–168 [Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131–148].Google Scholar
  9. 9.
    Cushman, R., Routh’s Sphere, Rep. Math. Phys., 1998, vol. 42, nos. 1–2, pp. 47–70.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Duistermaat, J. J., Chaplygin’s Sphere, arXiv:math/0409019v1 [math.DS] 1 Sep 2004.Google Scholar
  11. 11.
    Goldstein, H., Poole, C., and Safko, J., Classical Mechanics, 3rd ed., San Fransisco, CA: Addison-Wesley, 2002.Google Scholar
  12. 12.
    Gray, C.G. and Nickel, B.G., Constants of the Motion for Nonslipping Tippe Tops and Other Tops with Round Pegs, Amer. J. Phys., 2000, vol. 68, no. 9, pp. 821–828.CrossRefGoogle Scholar
  13. 13.
    Holm, D.D., Geometric Mechanics: P. 2. Rotating, Translating and Rolling, 2nd ed., London: Imperial Coll. Press, 2011.CrossRefGoogle Scholar
  14. 14.
    Kim, B., Routh Symmetry in the Chaplygin’s Rolling Ball, Regul. Chaotic Dyn., 2011, vol. 16, no. 6, pp. 663–670.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kilin, A.A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291–306.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Koslov, V.V., On the Integration Theory of Equations of Nonholonomic Mechanics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 161–176.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kuipers, J. B., Quaternions and Rotation Sequences, Princeton: Princeton Univ. Press, 1999.zbMATHGoogle Scholar
  18. 18.
    Landau, L. D. and Lifshitz, E. M., Course of Theoretical Physics: Vol. 1. Mechanics, 3rd ed., Oxford: Pergamon, 1976.Google Scholar
  19. 19.
    Lynch, P. and Bustamante, M.D., Precession and Recession of the Rock’n’roller, J. Phys. A, 2009, vol. 42, no. 42, 425203, 25 pp.MathSciNetCrossRefGoogle Scholar
  20. 20.
    Marsden, J. E. and Ratiu, T. S., Introduction to Mechanics and Symmetry, 2nd ed., Texts Appl. Math., vol. 17, New York: Springer, 1999.zbMATHCrossRefGoogle Scholar
  21. 21.
    Routh, E. J., A Treatise on the Dynamics of a System of Rigid Bodies: P. 2. The Advanced Part, 6th ed., New York: Macmillan, 1905. See also: New York: Dover, 1955 (reprint).Google Scholar
  22. 22.
    Synge, J. L. and Griffith, B. A., Principles of Mechanics, 3rd ed., New York: McGraw-Hill, 1959.Google Scholar
  23. 23.
    Whitham, G. B., Linear and Nonlinear Waves, New York: Wiley-Interscience, 1974.zbMATHGoogle Scholar
  24. 24.
    Whittaker, E.T., A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., Cambridge: Cambridge Univ. Press, 1937.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesUCDBelfield, Dublin 4Ireland

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