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This article is cited in 37 scientific papers (total in 37 papers)
Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals
Alexey V. Bolsinovab, Alexey V. Borisovb, Ivan S. Mamaevb a School of Mathematics, Loughborough University, Loughborough, Leicestershire, UK
b Institute of Computer Science, Udmurt State University, Izhevsk, Russia
Abstract:
In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.
Keywords:
non-holonomic constraint, Liouville foliation, invariant torus, invariant measure,
integrability.
Received: 04.08.2012 Accepted: 19.10.2012
Citation:
Alexey V. Bolsinov, Alexey V. Borisov, Ivan S. Mamaev, “Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals”, Regul. Chaotic Dyn., 17:6 (2012), 571–579
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https://www.mathnet.ru/eng/rcd350 https://www.mathnet.ru/eng/rcd/v17/i6/p571
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