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Regular and Chaotic Dynamics

, Volume 23, Issue 6, pp 665–684 | Cite as

An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane

  • Ivan A. BizyaevEmail author
  • Alexey V. Borisov
  • Ivan S. Mamaev
Article

Abstract

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.

Keywords

nonholonomic mechanics regularization blowing-up invariant measure ergodic theorems normal hyperbolic submanifold Poincaré map first integrals 

MSC2010 numbers

37J60 34A34 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Ivan A. Bizyaev
    • 1
    Email author
  • Alexey V. Borisov
    • 1
  • Ivan S. Mamaev
    • 1
  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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