Regular and Chaotic Dynamics

, Volume 18, Issue 3, pp 277–328 | Cite as

The hierarchy of dynamics of a rigid body rolling without slipping and spinning on a plane and a sphere

  • Alexey V. Borisov
  • Ivan S. Mamaev
  • Ivan A. Bizyaev
Article

Abstract

In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.

Keywords

nonholonomic constraint tensor invariant first integral invariant measure integrability conformally Hamiltonian system rubber rolling reversible involution 

MSC2010 numbers

37J60 37J35 

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Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • Alexey V. Borisov
    • 1
    • 2
    • 3
  • Ivan S. Mamaev
    • 1
    • 2
    • 3
  • Ivan A. Bizyaev
    • 1
  1. 1.Institute of Computer ScienceLaboratory of Nonlinear Analysis and the Design of New Types of Vehicles Udmurt 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 RASEkaterinburgRussia

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