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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, Volume 27, Issue 4, Pages 576–582 (Mi vuu609)  

MECHANICS

Invariant measure in the problem of a disk rolling on a plane

I. A. Bizyaev

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: This paper addresses the dynamics of a disk rolling on an absolutely rough plane. It is proved that the equations of motion have an invariant measure with continuous density only in two cases: a dynamically symmetric disk and a disk with a special mass distribution. In the former case, the equations of motion possess two additional integrals and are integrable by quadratures by the Euler–Jacobi theorem. In the latter case, the absence of additional integrals is shown using a Poincaré map. In both cases, the volume of any domain in phase space (calculated with the help of the density) is preserved by the phase flow. Nonholonomic mechanics is populated with systems both with and without an invariant measure.

Keywords: nonholonomic mechanics, Schwarzschild–Littlewood theorem, manifold of falls, chaotic dynamics.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 1.2404.2017/4.6
Dynasty Foundation


DOI: https://doi.org/10.20537/vm170407

Full text: PDF file (312 kB)
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Document Type: Article
UDC: 517.925
MSC: 37J60
Received: 22.11.2017

Citation: I. A. Bizyaev, “Invariant measure in the problem of a disk rolling on a plane”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:4 (2017), 576–582

Citation in format AMSBIB
\Bibitem{Biz17}
\by I.~A.~Bizyaev
\paper Invariant measure in the problem of a disk rolling on a plane
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2017
\vol 27
\issue 4
\pages 576--582
\mathnet{http://mi.mathnet.ru/vuu609}
\crossref{https://doi.org/10.20537/vm170407}
\elib{http://elibrary.ru/item.asp?id=32248458}


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  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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