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.
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
\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}
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Linking options:
https://www.mathnet.ru/eng/rcd358
https://www.mathnet.ru/eng/rcd/v23/i6/p665
This publication is cited in the following 7 articles:
Alexander A. Kilin, Elena N. Pivovarova, “Bifurcation analysis of the problem of a “rubber” ellipsoid of revolution rolling on a plane”, Nonlinear Dyn, 2024
Ivan Bizyaev, “Classification of the trajectories of uncharged particles in the Schwarzschild-Melvin metric”, Phys. Rev. D, 110:10 (2024)
Alexander A. Kilin, Elena N. Pivovarova, “Dynamics of an Unbalanced Disk
with a Single Nonholonomic Constraint”, Regul. Chaotic Dyn., 28:1 (2023), 78–106
William Clark, Anthony Bloch, “Existence of invariant volumes in nonholonomic systems subject to nonlinear constraints”, JGM, 15:1 (2023), 256
E. A. Mikishanina, “Dinamika kacheniya diska s naklonnoi skolzyaschei oporoi”, Izvestiya vysshikh uchebnykh zavedenii. Povolzhskii region. Fiziko-matematicheskie nauki, 2021, no. 3, 45–56
A. V. Borisov, A. V. Tsyganov, “Vliyanie effektov Barnetta-Londona i Einshteina-de Gaaza na dvizhenie negolonomnoi sfery Rausa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:4 (2019), 583–598
I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dynamics of a Chaplygin sleigh with an unbalanced rotor: regular and chaotic motions”, Nonlinear Dyn., 98:3 (2019), 2277–2291
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