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Nelin. Dinam., 2012, Volume 8, Number 3, Pages 605–616 (Mi nd346)  

This article is cited in 9 scientific papers (total in 9 papers)

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

Alexey V. Bolsinovab, Alexey V. Borisovbcd, Ivan S. Mamaevbcd

a School of Mathematics, Loughborough University United Kingdom, LE11 3TU, Loughborough, Leicestershire
b Laboratory of nonlinear analysis and the design of new types of vehicles, Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia
c A. A. Blagonravov Mechanical Engineering Institute of RAS, Bardina str. 4, Moscow, 117334, Russia
d Institute of Mathematics and Mechanics of the Ural Branch of RAS S. Kovalevskaja str. 16, Ekaterinburg, 620990, 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

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Document Type: Article
UDC: 517.925
MSC: 37J60, 37J35, 70H45
Received: 04.08.2012
Revised: 19.10.2012

Citation: Alexey V. Bolsinov, Alexey V. Borisov, Ivan S. Mamaev, “Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals”, Nelin. Dinam., 8:3 (2012), 605–616

Citation in format AMSBIB
\by Alexey~V.~Bolsinov, Alexey~V.~Borisov, Ivan~S.~Mamaev
\paper Rolling without spinning of a ball on a plane: absence of an invariant measure in a system with a complete set of first integrals
\jour Nelin. Dinam.
\yr 2012
\vol 8
\issue 3
\pages 605--616

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    This publication is cited in the following articles:
    1. Alexey V. Borisov, Ivan S. Mamaev, Dmitrii V. Treschev, “Rolling of a rigid body without slipping and spinning: kinematics and dynamics”, 2, no. 2, 2013, 161–173  mathnet  crossref
    2. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Ierarkhiya dinamiki pri kachenii tverdogo tela bez proskalzyvaniya i vercheniya po ploskosti i sfere”, Nelineinaya dinam., 9:2 (2013), 141–202  mathnet
    3. A. O. Kazakov, “Fenomeny khaoticheskoi dinamiki v zadache o kachenii rok-n-rollera bez vercheniya”, Nelineinaya dinam., 9:2 (2013), 309–325  mathnet
    4. A. A. Kilin, Yu. L. Karavaev, A. V. Klekovkin, “Kinematicheskaya model upravleniya vysokomanevrennym mobilnym sferorobotom s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 10:1 (2014), 113–126  mathnet
    5. A. V. Borisov, I. S. Mamaev, “Invariantnaya mera i gamiltonizatsiya negolonomnykh sistem”, Nelineinaya dinam., 10:3 (2014), 355–359  mathnet
    6. A. A. Kilin, Yu. L. Karavaev, “Kinematicheskaya model upravleniya sferorobotom s neuravnoveshennoi omnikolesnoi platformoi”, Nelineinaya dinam., 10:4 (2014), 497–511  mathnet
    7. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    8. Yury L. Karavaev, Alexander A. Kilin, “The Dynamics and Control of a Spherical Robot with an Internal Omniwheel Platform”, Regul. Chaotic Dyn., 20:2 (2015), 134–152  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    9. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Dynamics and Control of an Omniwheel Vehicle”, Regul. Chaotic Dyn., 20:2 (2015), 153–172  mathnet  crossref  mathscinet  zmath  adsnasa
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