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Regul. Chaotic Dyn., 2018, Volume 23, Issue 6, Pages 665–684 (Mi rcd358)  

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

Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev

Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

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

Funding Agency Grant Number
Russian Science Foundation 14-50-00005
This work was carried out at the Steklov Mathematical Institute and was supported by RSF under grant No. 14-50-00005.


DOI: https://doi.org/10.1134/S1560354718060035

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MSC: 37J60, 34A34
Received: 03.10.2018
Accepted:05.11.2018
Language:

Citation: Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane”, Regul. Chaotic Dyn., 23:6 (2018), 665–684

Citation in format AMSBIB
\Bibitem{BizBorMam18}
\by Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev
\paper An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 6
\pages 665--684
\mathnet{http://mi.mathnet.ru/rcd358}
\crossref{https://doi.org/10.1134/S1560354718060035}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3890806}
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