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 Regul. Chaotic Dyn., 2012, Volume 17, Issue 6, Pages 571–579 (Mi rcd350)

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.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation No11.G34.31.0039 This research was done at the Udmurt State University and was supported by the Grant Program of the Government of the Russian Federation for state support of scientific research conducted under the supervision of leading scientists at Russian institutions of higher professional education (Contract No11.G34.31.0039).

DOI: https://doi.org/10.1134/S1560354712060081

Bibliographic databases:

Document Type: Article
MSC: 37J60, 37J35, 70H45
Accepted:19.10.2012
Language: English

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

Citation in format AMSBIB
\Bibitem{BolBorMam12} \by Alexey V.~Bolsinov, Alexey V.~Borisov, Ivan S.~Mamaev \paper Rolling of a Ball without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals \jour Regul. Chaotic Dyn. \yr 2012 \vol 17 \issue 6 \pages 571--579 \mathnet{http://mi.mathnet.ru/rcd350} \crossref{https://doi.org/10.1134/S1560354712060081} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3001102} \zmath{https://zbmath.org/?q=an:1267.37066} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?2012RCD....17..571B} 

• http://mi.mathnet.ru/eng/rcd350
• http://mi.mathnet.ru/eng/rcd/v17/i6/p571

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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2. A. O. Kazakov, “Fenomeny khaoticheskoi dinamiki v zadache o kachenii rok-n-rollera bez vercheniya”, Nelineinaya dinam., 9:2 (2013), 309–325
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