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Discrete Lagrange-d’Alembert-Poincaré equations for Euler’s disk

  • Cédric M. Campos
  • Hernán Cendra
  • Viviana Alejandra DíazEmail author
  • David Martín de Diego
Original Paper

Abstract

Nonholonomic systems are described by the Lagrange-d’Alembert principle. The presence of symmetry leads to a reduced d’Alembert principle and to the Lagrange-d’Alembert-Poincaré equations. First, we briefly recall from previous works how to obtain these reduced equations for the case of a thick disk rolling on a rough surface using a three-dimensional abelian group of symmetries. The main results of the present paper are the calculation of the discrete Lagrange-d’Alembert-Poincaré equations for an Euler’s disk and the numerical simulation of a trajectory and its energy behavior.

Keywords

Euler’s disk Discrete Lagrange-d’Alembert-Poincaré equations 

Mathematics Subject Classification (2000)

53Z 70F25 70H 37J 

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References

  1. 1.
    Bildsten L.: Viscous dissipation for Euler disk. Phys. Rev. E. 66, 056309 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bloch, A.M.: Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics Series, vol. 24. Springer, New-York (2003)Google Scholar
  3. 3.
    Bloch A.M., Krishnaprasad P.S., Marsden J.E., Murray R.M: Nonholonomic mechanical systems with symmetry. Arch. Ration. Mech. Anal. 136, 21–99 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Borisov A.V., Mamaev I.S., Kilin A.A.: Dynamics of rolling disk. Regul. Chaotic Dyn. 8, 201–212 (2002)CrossRefGoogle Scholar
  5. 5.
    Caps H., Dorbolo S., Ponte S., Croisier H., Vandewalle N.: Rolling and slipping motion of Euler’s disk. Phys. Rev. E 69, 056610 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cendra H., Marsden J.E., Ratiu T.: Geometric Mechanics, Lagrangian reduction and Nonholonomic Systems Mathematics. Unlimited and Beyond. Springer, Berlin (2001)Google Scholar
  7. 7.
    Cendra H., Díaz V.A.: The Lagrange-d’Alembert-Poincaré equations and Integrability for the rolling disk. Regul. Chaotic Dyn. 11(1), 67–81 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cendra H., Díaz V.A: The Lagrange-d’Alembert-Poincaré equations and Integrability for the Euler’s Disk. Regul. Chaotic Dyn. 12(1), 56–67 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cortés J.: Geometric, control and numerical aspects of nonholonomic systems. Lecture Notes in Mathematics, vol. 1793. Springer, Berlin (2002)Google Scholar
  10. 10.
    Cortés J., Martí nez S.: Nonholonomic integrators. Nonlinearity 14(5), 1365–1392 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cushman R., Duistermaat H., Sniatycki J.: Geometry of nonholonomically constrained systems. Adv. Ser. Nonlinear Dyn. 26, (2010)Google Scholar
  12. 12.
    Cushman R., Hermans J., Kemppainen D.: The rolling disk. Nonlinear Differ. Equ. Appl. 19, (1996)Google Scholar
  13. 13.
    de León M., Martínde Diego D., Santamaría-Merino A.: Geometric integrators and nonholonomic mechanics. J. Math. Phys. 45(3), 1042–1064 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Easwar, K., Rouyer, F., Menon, N.: Speeding to a stop: the finite-time singularity of a spinning disk. Phys. Rev. E 66, 045102(R) (2002)Google Scholar
  15. 15.
    Fedorov Y.N., Zenkov D.V.: Discrete nonholonomic LL systems on Lie groups. Nonlinearity 18, 2211–2241 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Fedorov, Y.N. and Zenkov, D.V.: Dynamics of the discrete Chaplygin sleigh. Discret. Contin. Dyn. Syst., 258–267 (2005)Google Scholar
  17. 17.
    Hairer E., Lubich C., Wanner G.: Geometric Numerical Integration, Structure-Preserving Algorithms for Ordinary Differential Equations Springer Series in Computational Mathematics, vol 31. Springer, Berlin (2002)Google Scholar
  18. 18.
    Iglesias D., Marrero J.C., Martínde Diego D., Martínez E.: Discrete nonholonomic Lagrangian systems on Lie groupoids. J. Nonlinear Sci. 18, 221–276 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Kessler P., O’Reilly O.M.: The ringing of Euler’s disk. Regul. Chaotic Dyn. 7, 49–60 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Marsden J.E., West M.: Discrete Mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    McDonald, A.J., McDonald, K.T.: The rolling motion of a disk on a horizontal plane. e-print available at http://www.puhep1.princeton.edu/mcdonald/examples/rollingdisk.pdf and at http://www.lanl.gov/abs/physics/0008227 (2001)
  22. 22.
    McLachlan R., Perlmutter M.: Integrators for nonholonomic mechanical systems. J. Nonlinear Sci. 16, 283–328 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Moffatt H.K.: Euler’s disk and its finite-time singularity. Nature 404, 833–834 (2000)CrossRefGoogle Scholar
  24. 24.
    Moffat H.K.: Reply to G. van den Engh et al. Nature 408, 540 (2000)CrossRefGoogle Scholar
  25. 25.
    Petrie D., Hunt J.L., Gray C.G.: Does the Euler disk slip during its motion?. Am. J. Phys. 70, 1025 (2002)CrossRefGoogle Scholar
  26. 26.
    Stanislavsky A.A., Weron K.: Nonlinear oscillations in the rolling motion of Euler’s disk. Phys. D 156, 247–259 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    van den Engh G., Nelson P., Roach J.: Numismatic gyrations. Nature 408, 540 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Cédric M. Campos
    • 1
  • Hernán Cendra
    • 2
    • 3
  • Viviana Alejandra Díaz
    • 2
    Email author
  • David Martín de Diego
    • 1
  1. 1.Instituto de Ciencias Matemáticas, Campus de Cantoblanco, UAMMadridSpain
  2. 2.Departamento de MatemáticaUniversidad Nacional del SurBahía BlancaArgentina
  3. 3.CONICETBuenos AiresArgentina

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