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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bildsten L.: Viscous dissipation for Euler disk. Phys. Rev. E. 66, 056309 (2002)
Bloch, A.M.: Nonholonomic Mechanics and Control. Interdisciplinary Applied Mathematics Series, vol. 24. Springer, New-York (2003)
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)
Borisov A.V., Mamaev I.S., Kilin A.A.: Dynamics of rolling disk. Regul. Chaotic Dyn. 8, 201–212 (2002)
Caps H., Dorbolo S., Ponte S., Croisier H., Vandewalle N.: Rolling and slipping motion of Euler’s disk. Phys. Rev. E 69, 056610 (2004)
Cendra H., Marsden J.E., Ratiu T.: Geometric Mechanics, Lagrangian reduction and Nonholonomic Systems Mathematics. Unlimited and Beyond. Springer, Berlin (2001)
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)
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)
Cortés J.: Geometric, control and numerical aspects of nonholonomic systems. Lecture Notes in Mathematics, vol. 1793. Springer, Berlin (2002)
Cortés J., Martí nez S.: Nonholonomic integrators. Nonlinearity 14(5), 1365–1392 (2001)
Cushman R., Duistermaat H., Sniatycki J.: Geometry of nonholonomically constrained systems. Adv. Ser. Nonlinear Dyn. 26, (2010)
Cushman R., Hermans J., Kemppainen D.: The rolling disk. Nonlinear Differ. Equ. Appl. 19, (1996)
de León M., Martínde Diego D., Santamaría-Merino A.: Geometric integrators and nonholonomic mechanics. J. Math. Phys. 45(3), 1042–1064 (2004)
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)
Fedorov Y.N., Zenkov D.V.: Discrete nonholonomic LL systems on Lie groups. Nonlinearity 18, 2211–2241 (2005)
Fedorov, Y.N. and Zenkov, D.V.: Dynamics of the discrete Chaplygin sleigh. Discret. Contin. Dyn. Syst., 258–267 (2005)
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)
Iglesias D., Marrero J.C., Martínde Diego D., Martínez E.: Discrete nonholonomic Lagrangian systems on Lie groupoids. J. Nonlinear Sci. 18, 221–276 (2008)
Kessler P., O’Reilly O.M.: The ringing of Euler’s disk. Regul. Chaotic Dyn. 7, 49–60 (2002)
Marsden J.E., West M.: Discrete Mechanics and variational integrators. Acta Numer. 10, 357–514 (2001)
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)
McLachlan R., Perlmutter M.: Integrators for nonholonomic mechanical systems. J. Nonlinear Sci. 16, 283–328 (2006)
Moffatt H.K.: Euler’s disk and its finite-time singularity. Nature 404, 833–834 (2000)
Moffat H.K.: Reply to G. van den Engh et al. Nature 408, 540 (2000)
Petrie D., Hunt J.L., Gray C.G.: Does the Euler disk slip during its motion?. Am. J. Phys. 70, 1025 (2002)
Stanislavsky A.A., Weron K.: Nonlinear oscillations in the rolling motion of Euler’s disk. Phys. D 156, 247–259 (2001)
van den Engh G., Nelson P., Roach J.: Numismatic gyrations. Nature 408, 540 (2000)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Campos, C.M., Cendra, H., Díaz, V.A. et al. Discrete Lagrange-d’Alembert-Poincaré equations for Euler’s disk. RACSAM 106, 225–234 (2012). https://doi.org/10.1007/s13398-011-0053-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13398-011-0053-3