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    Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals

    2012, Vol. 8, No. 3, pp.  605-616

    Author(s): Bolsinov A. V., Borisov A. V., Mamaev I. S.

    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
    Citation: Bolsinov A. V., Borisov A. V., Mamaev I. S., Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals, Rus. J. Nonlin. Dyn., 2012, Vol. 8, No. 3, pp.  605-616
    DOI:10.20537/nd1203013


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