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Nelin. Dinam., 2013, Volume 9, Number 2, Pages 141–202 (Mi nd385)  

This article is cited in 19 scientific papers (total in 19 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. Mamaevabc, Ivan A. Bizyaeva

a Institute of Computer Science; Laboratory of nonlinear analysis and the design of new types of vehicles, Udmurt State University, Izhevsk, Russia
b A. A. Blagonravov Mechanical Engineering Research Institute of RAS, Moscow, Russia
c Institute of Mathematics and Mechanics of the Ural Branch of RAS, Ekaterinburg, 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.

Full text: PDF file (8099 kB)
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Document Type: Article
UDC: 531.38
MSC: 37J60, 37J35
Received: 12.03.2013
Revised: 08.05.2013

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”, Nelin. Dinam., 9:2 (2013), 141–202

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 Nelin. Dinam.
\yr 2013
\vol 9
\issue 2
\pages 141--202
\mathnet{http://mi.mathnet.ru/nd385}


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    This publication is cited in the following articles:
    1. I. A. Bizyaev, A. O. Kazakov, “Integriruemost i stokhastichnost nekotorykh zadach negolonomnoi mekhaniki”, Nelineinaya dinam., 9:2 (2013), 257–265  mathnet
    2. A. O. Kazakov, “Fenomeny khaoticheskoi dinamiki v zadache o kachenii rok-n-rollera bez vercheniya”, Nelineinaya dinam., 9:2 (2013), 309–325  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. 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
    7. A. V. Borisov, A. O. Kazakov, S. P. Kuznetsov, “Nonlinear dynamics of the rattleback: a nonholonomic model”, Phys. Usp., 57:5 (2014), 453–460  mathnet  crossref  crossref  adsnasa  isi  elib  elib
    8. E. V. Vetchanin, A. O. Kazakov, “Bifurkatsii i khaos v zadache o dvizhenii dvukh tochechnykh vikhrei v akusticheskoi volne”, Nelineinaya dinam., 10:3 (2014), 329–343  mathnet
    9. A. V. Borisov, I. S. Mamaev, “Invariantnaya mera i gamiltonizatsiya negolonomnykh sistem”, Nelineinaya dinam., 10:3 (2014), 355–359  mathnet
    10. A. V. Borisov, A. O. Kazakov, I. R. Sataev, “Regulyarnye i khaoticheskie attraktory v negolonomnoi modeli volchka Chaplygina”, Nelineinaya dinam., 10:3 (2014), 361–380  mathnet
    11. A. A. Kilin, Yu. L. Karavaev, “Kinematicheskaya model upravleniya sferorobotom s neuravnoveshennoi omnikolesnoi platformoi”, Nelineinaya dinam., 10:4 (2014), 497–511  mathnet
    12. 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
    13. A. V. Borisov, Yu. L. Karavaev, I. S. Mamaev, N. N. Erdakova, T. B. Ivanova, V. V. Tarasov, “Experimental Investigation of the Motion of a Body with an Axisymmetric Base Sliding on a Rough Plane”, Regul. Chaotic Dyn., 20:5 (2015), 518–541  mathnet  crossref
    14. A. A. Kilin, Yu. L. Karavaev, “Eksperimentalnye issledovaniya dinamiki sfericheskogo robota kombinirovannogo tipa”, Nelineinaya dinam., 11:4 (2015), 721–734  mathnet
    15. I. A. Bizyaev, A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and bifurcations in nonholonomic mechanics”, International Journal of Bifurcation and Chaos, 25:10 (2015), 15300–21  mathnet  crossref  isi
    16. A. V. Borisov, I. S. Mamaev, “Symmetries and reduction in nonholonomic mechanics”, Regul. Chaotic Dyn., 20:5 (2015), 553–604  mathnet  crossref
    17. 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
    18. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    19. I. R. Sataev, A. O. Kazakov, “Stsenarii perekhoda k khaosu v negolonomnoi modeli volchka Chaplygina”, Nelineinaya dinam., 12:2 (2016), 235–250  mathnet  elib
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