Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, Volume 27, Issue 4, Pages 576–582
Invariant measure in the problem of a disk rolling on a plane
I. A. Bizyaev
Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia
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.
nonholonomic mechanics, Schwarzschild–Littlewood theorem, manifold of falls, chaotic dynamics.
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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
\paper Invariant measure in the problem of a disk rolling on a plane
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
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