Regular and Chaotic Dynamics

, Volume 19, Issue 2, pp 198–213 | Cite as

The dynamics of nonholonomic systems consisting of a spherical shell with a moving rigid body inside

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

Abstract

In this paper we investigate two systems consisting of a spherical shell rolling without slipping on a plane and a moving rigid body fixed inside the shell by means of two different mechanisms. In the former case the rigid body is attached to the center of the ball on a spherical hinge. We show an isomorphism between the equations of motion for the inner body with those for the ball moving on a smooth plane. In the latter case the rigid body is fixed by means of a nonholonomic hinge. Equations of motion for this system have been obtained and new integrable cases found. A special feature of the set of tensor invariants of this system is that it leads to the Euler — Jacobi — Lie theorem, which is a new integration mechanism in nonholonomic mechanics. We also consider the problem of free motion of a bundle of two bodies connected by means of a nonholonomic hinge. For this system, integrable cases and various tensor invariants are found.

Keywords

nonholonomic constraint tensor invariants isomorphism nonholonomic hinge 

MSC2010 numbers

70E18 37J60 37J35 

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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
  • Alexey V. Borisov
    • 2
    • 3
    • 4
  • Ivan S. Mamaev
    • 1
  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.A.A. Blagonravov Mechanical Engineering Research Institute of RASMoscowRussia
  3. 3.National Research Nuclear University “MEPhI”MoscowRussia
  4. 4.Moscow Institute of Physics and TechnologyDolgoprudny, Moscow RegionRussia

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