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Regul. Chaotic Dyn., 2013, Volume 18, Issue 3, Pages 277–328 (Mi rcd114)  

This article is cited in 47 scientific papers (total in 49 papers)

The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere

Alexey V. Borisovabc, Ivan S. Mamaevcba, Ivan A. Bizyaeva

a Institute of Computer Science; Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia
b A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Bardina str. 4, Moscow, 117334, Russia
c Institute of Mathematics and Mechanics of the Ural Branch of RAS, S. Kovalevskaja str. 16, Ekaterinburg, 620990, Russia

Abstract: In this paper, we investigate the dynamics of systems describing the rolling without slipping and spinning (rubber rolling) of various rigid bodies on a plane and a sphere. It is shown that a hierarchy of possible types of dynamical behavior arises depending on the body’s surface geometry and mass distribution. New integrable cases and cases of existence of an invariant measure are found. In addition, these systems are used to illustrate that the existence of several nontrivial involutions in reversible dissipative systems leads to quasi-Hamiltonian behavior.

Keywords: nonholonomic constraint, tensor invariant, first integral, invariant measure, integrability, conformally Hamiltonian system, rubber rolling, reversible, involution

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation NSh-2519.2012.1
1.1248.2011
1.7734.2013
14.A37.21.1935
This work was carried out at the Udmurt State University and was supported by Grant of the President of the Russian Federation for Support of Leading Scientific Schools NSh-2519.2012.1 “Dynamical Systems of Classical Mechanics and Control Problems”, Analytic Departmental Target Program “Development of Scientific Potential of Higher Schools” (1.1248.2011), Analytic Depart-mental Target Program ”Development of Scientific Potential of Higher Schools” (1.7734.2013), Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (Agreement ¹14.A37.21.1935).


DOI: https://doi.org/10.1134/S1560354713030064

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Document Type: Article
MSC: 37J60, 37J35
Received: 12.03.2013
Language: English

Citation: Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere”, Regul. Chaotic Dyn., 18:3 (2013), 277–328

Citation in format AMSBIB
\Bibitem{BorMamBiz13}
\by Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev
\paper The Hierarchy of Dynamics of a Rigid Body Rolling without Slipping and Spinning on a Plane and a Sphere
\jour Regul. Chaotic Dyn.
\yr 2013
\vol 18
\issue 3
\pages 277--328
\mathnet{http://mi.mathnet.ru/rcd114}
\crossref{https://doi.org/10.1134/S1560354713030064}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3061810}
\zmath{https://zbmath.org/?q=an:06197382}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000319763900006}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. O. Kazakov, “Fenomeny khaoticheskoi dinamiki v zadache o kachenii rok-n-rollera bez vercheniya”, Nelineinaya dinam., 9:2 (2013), 309–325  mathnet
    2. N. N. Erdakova, I. S. Mamaev, “Dinamika tela s osesimmetrichnym osnovaniem, skolzyaschego po sherokhovatoi ploskosti”, Nelineinaya dinam., 9:3 (2013), 521–545  mathnet
    3. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dinamika negolonomnykh sistem, sostoyaschikh iz sfericheskoi obolochki s podvizhnym tverdym telom vnutri”, Nelineinaya dinam., 9:3 (2013), 547–566  mathnet
    4. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Geometrizatsiya teoremy Chaplygina o privodyaschem mnozhitele”, Nelineinaya dinam., 9:4 (2013), 627–640  mathnet
    5. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Problema dreifa i vozvraschaemosti pri kachenii shara Chaplygina”, Nelineinaya dinam., 9:4 (2013), 721–754  mathnet
    6. Alexey O. Kazakov, “Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane”, Regul. Chaotic Dyn., 18:5 (2013), 508–520  mathnet  crossref  mathscinet  zmath
    7. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Problem of Drift and Recurrence for the Rolling Chaplygin Ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859  mathnet  crossref  mathscinet  zmath
    8. Luis C. García-Naranjo, Juan C. Marrero, “Non-Existence of an Invariant Measure for a Homogeneous Ellipsoid Rolling on the Plane”, Regul. Chaotic Dyn., 18:4 (2013), 372–379  mathnet  crossref  mathscinet  zmath
    9. 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
    10. A. V. Borisov, I. S. Mamaev, “Invariantnaya mera i gamiltonizatsiya negolonomnykh sistem”, Nelineinaya dinam., 10:3 (2014), 355–359  mathnet
    11. A. A. Kilin, Yu. L. Karavaev, “Kinematicheskaya model upravleniya sferorobotom s neuravnoveshennoi omnikolesnoi platformoi”, Nelineinaya dinam., 10:4 (2014), 497–511  mathnet
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    17. S. P. Kuznetsov, “On the validity of the nonholonomic model of the rattleback”, Phys. Usp., 58:12 (2015), 1223–1224  mathnet  crossref  crossref  adsnasa  isi  elib
    18. E. N. Pivovarova, A. V. Klekovkin, “Vliyanie treniya kacheniya na upravlyaemoe dvizhenie robota-kolesa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:4 (2015), 583–592  mathnet  elib
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    21. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
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    45. Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367  mathnet  crossref
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