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Regular and Chaotic Dynamics, 2021, Volume 26, Issue 6, Pages 775–786
DOI: https://doi.org/10.1134/S1560354721060149
(Mi rcd1146)
 

This article is cited in 4 scientific papers (total in 4 papers)

Regular Papers

A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane

Alexander A. Kilin, Elena N. Pivovarova

Ural Mathematical Center, Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia
Citations (4)
References:
Abstract: In this paper we investigate the motion of a Chaplygin sphere rolling without slipping on a plane performing horizontal periodic oscillations. We show that in the system under consideration the projections of the angular momentum onto the axes of the fixed coordinate system remain unchanged. The investigation of the reduced system on a fixed level set of first integrals reduces to analyzing a three-dimensional period advance map on $SO(3)$. The analysis of this map suggests that in the general case the problem considered is nonintegrable. We find partial solutions to the system which are a generalization of permanent rotations and correspond to nonuniform rotations about a body- and space-fixed axis. We also find a particular integrable case which, after time is rescaled, reduces to the classical Chaplygin sphere rolling problem on the zero level set of the area integral.
Keywords: Chaplygin sphere, rolling motion, nonholonomic constraint, nonautonomous dynamical system, periodic oscillations, permanent rotations, integrable case, period advance map.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-02-2021-1383
Russian Science Foundation 20-71-00053
The work of A.A.Kilin (Sections 1, 2, 4, 7 and 8) was performed at the Ural Mathematical Center (agreement no. 075-02-2021-1383). The work of E.N. Pivovarova (Sections 3, 5 and 6) is supported by the Russian Science Foundation under grant no. 20-71-00053.
Received: 15.03.2021
Accepted: 11.11.2021
Bibliographic databases:
Document Type: Article
Language: English
Citation: Alexander A. Kilin, Elena N. Pivovarova, “A Particular Integrable Case in the Nonautonomous Problem of a Chaplygin Sphere Rolling on a Vibrating Plane”, Regul. Chaotic Dyn., 26:6 (2021), 775–786
Citation in format AMSBIB
\Bibitem{KilPiv21}
\by Alexander~A. Kilin, Elena N. Pivovarova
\paper A Particular Integrable Case in the Nonautonomous Problem
of a Chaplygin Sphere Rolling on a Vibrating Plane
\jour Regul. Chaotic Dyn.
\yr 2021
\vol 26
\issue 6
\pages 775--786
\mathnet{http://mi.mathnet.ru/rcd1146}
\crossref{https://doi.org/10.1134/S1560354721060149}
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\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85120883007}
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  • https://www.mathnet.ru/eng/rcd1146
  • https://www.mathnet.ru/eng/rcd/v26/i6/p775
  • This publication is cited in the following 4 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    Abstract page:107
    References:26
     
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