Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Rolling of a Rigid Body Without Slipping and Spinning: Kinematics and Dynamics

Journal of Applied Nonlinear Dynamics 2(2) (2013) 161--173 | DOI:10.5890/JAND.2013.04.005

A.V. Borisov$^{1}$, I.S. Mamaev$^{2}$, D.V. Treschev$^{2}$,$^{3}$

$^{1}$ Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles,Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia

$^{2}$ Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia

$^{3}$ Steklov Mathematical Institute, Gubkina st. 8, Moscow, 119991, Russia

Download Full Text PDF



In this paper we investigate various kinematic properties of rolling of one rigid body on another both for the classical model of rolling without slipping (the velocities of bodies at the point of contact coincide) and for the model of rubber-rolling (with the additional condition that the spinning of the bodies relative to each other be excluded). Furthermore, in the case where both bodies are bounded by spherical surfaces and one of them is fixed, the equations of motion for a moving ball are represented in the form of the Chaplygin system. When the center of mass of the moving ball coincides with its geometric center, the equations of motion are represented in conformally Hamiltonian form, and in the case where the radii of the moving and fixed spheres coincides, they are written in Hamiltonian form.


This research was done at the Udmurt State University and was supported by the Grant Program of the Government of the Russian Federation for state support of scientific research conducted under the supervision of leading scientists at Russian institutions of higher professional education (Contract No11.G34.31.0039). The study was supported by the Ministry of education and science of Russian Federation, project 14.B37.21.1935.


  1. [1]  Borisov, A.V., Mamaev, I.S. (2008), Conservation Laws, Hierarchy of Dynamics and Explicit Integration of Nonholonomic Systems,Regular and Chaotic Dynamics, 13(5), 443-490.
  2. [2]  Borisov, A.V. and Mamaev, I.S. (2003), The Rolling of Rigid Body on a Plane and Sphere, Hierarchy of Dynamics, Regular and Chaotic Dynamics, 7(1), 177-200.
  3. [3]  Borisov, A.V., Kilin A.A., and Mamaev, I.S. (2003), Rolling of a Ball on a Surface, New Integrals and Hierarchy of Dynamics, Regular and Chaotic Dynamics, 7(1), 201-220.
  4. [4]  Hadamard, J. (1895), Sur les mouvements de roulement, Mémoires de la Société des sciences physiques et naturelles de Bordeaux, 4e serie, 397-417.
  5. [5]  Beghin, H. (1929), Sur les conditions d'application des équations de Lagrange à un système non holonome, Bulletin de la S.M.F., 57, 118-124.
  6. [6]  Li, Z. and Canny, J. (1990), Motion of two rigid bodies with rolling constraint, IEEE Transactions on Robotics and Automation, 6(1), 62-72.
  7. [7]  Marigo, A. and Bicchi, A. (1999), Rolling bodies with regular surface: the holonomic case. Differential geometry and control (Boulder, CO, 1997), 241-256, Proc. Sympos. Pure Math., 64, Amer. Math. Soc., Providence, RI.
  8. [8]  Marigo A. and Bicchi A. (2000), Rolling bodies with regular surface: controllability theory and applications, IEEE Tranasactions on Automatic Control, 45 (9), 1586-1599.
  9. [9]  Borisov, A.V., Kilin A.A., and Mamaev, I.S. (2012), How to Control Chaplygin Sphere Using Rotors, Regular and Chaotic Dynamics, 17(3-4), 258-272.
  10. [10]  Borisov, A.V. and Mamaev, I.S. (2007), Rolling of a non-homogeneous ball over a sphere without slipping and twisting, Regular and Chaotic Dynamics, 2007, 12(2), 153-159.
  11. [11]  Borisov, A.V., Mamaev, I.S. (2007) Isomorphism and Hamiltonian presentation of some nonholonomic systems, Russia Journal Siberian Mathamatical Journal, 48(1), 33-45.
  12. [12]  Ehlers, K. and Koiller, J. (2006), Rubber rolling: Geometry and dynamics of 2-3-5 distributions, Proceedings IUTAM symposium 2006 on Hamiltonian Dynamics, Vortex Structures, Turbulence Moscow, Russia, 25-30 August, 469-480.
  13. [13]  Agrachev, A.A. and Sachkov, Yu.L. (1999), An intrinsic approach to the control of rolling bodies, Proceedings of the 38-th IEEE Conference on Decision and Control, 1, Phoenix, Arizona, USA, December 7-10, 431-435.
  14. [14]  Levi, M. (1993), Geometric phases in the motion of rigid bodies. Archive for Rational Mechanics and Analysis, 122, 213-229.
  15. [15]  Koiler J. and Ehlers K.M. (2007), Rubber rolling over a sphere,Regular and Chaotic Dynamics , 12 (2), 127-152.
  16. [16]  Bolsinov A.V., Borisov A.V., and Mamaev I.S. (2011), Hamiltonization of nonholonomic systems in the neighborhood of invariant manifolds, Regular and Chaotic Dynamics, 16 (5), 443-464.
  17. [17]  Bolsinov, A.V., Borisov, A.V., and Mamaev, I.S. (2012), Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regular and Chaotic Dynamics, 17(6), 571-579.
  18. [18]  Chaplygin, S.A. (2008), On the theory of motion of nonholonomic systems. The reducing-multiplier theorem, Regular and Chaotic Dynamics, 13(4), 369-376.