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Nelin. Dinam., 2011, Volume 7, Number 2, Pages 313–338 (Mi nd261)  

This article is cited in 14 scientific papers (total in 14 papers)

Generalized Chaplygins transformation and explicit integration of a system with a spherical support

A. V. Borisov, A. A. Kilin, I. S. Mamaev

Institute of Computer Science, Izhevsk

Abstract: We consider the problem of explicit integration and bifurcation analysis for two systems of nonholonomic mechanics. The first one is the Chaplygin's problem on no-slip rolling of a balanced dynamically non-symmetrical ball on a horizontal plane. The second problem is on the motion of rigid body in a spherical support. We explicitly integrate this problem by generalizing the transformation which Chaplygin applied to the integration of the problem of the rolling ball at a non-zero constant of areas. We consider the geometric interpretation of this transformation from the viewpoint of a trajectory isomorphism between two systems at different levels of the energy integral. Generalization of this transformation for the case of dynamics in a spherical support allows us to integrate the equations of motion explicitly in quadratures and, in addition, to indicate periodic solutions and analyze their stability. We also show that adding a gyrostat does not lead to the loss of integrability.

Keywords: nonholonomic mechanics, spherical support, Chaplygin ball, explicit integration, isomorphism, bifurcation analysis.

Full text: PDF file (768 kB)
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Document Type: Article
UDC: 532.5
MSC: 37J60, 37J35, 70E18, 70F25, 70H45
Received: 22.04.2011
Revised: 23.06.2011

Citation: A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Generalized Chaplygins transformation and explicit integration of a system with a spherical support”, Nelin. Dinam., 7:2 (2011), 313–338

Citation in format AMSBIB
\Bibitem{BorKilMam11}
\by A.~V.~Borisov, A.~A.~Kilin, I.~S.~Mamaev
\paper Generalized Chaplygins transformation and explicit integration of a system with a spherical support
\jour Nelin. Dinam.
\yr 2011
\vol 7
\issue 2
\pages 313--338
\mathnet{http://mi.mathnet.ru/nd261}
\elib{http://elibrary.ru/item.asp?id=16515652}


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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. V. Borisov, I. S. Mamaev, “Dve negolonomnye integriruemye svyazki tverdykh tel”, Nelineinaya dinam., 7:3 (2011), 559–568  mathnet
    2. A. V. Tsyganov, “Deformatsii kanonicheskoi skobki Puassona dlya negolonomnykh sistem Chaplygina i Borisova–Mamaeva–Fëdorova pri nulevoi konstante ploschadei. I”, Nelineinaya dinam., 7:3 (2011), 577–599  mathnet
    3. A. V. Tsyganov, “O bigamiltonovoi strukture sistem Chaplygina i Borisova–Mamaeva–Fedorova pri nulevoi konstante ploschadei. II”, Nelineinaya dinam., 8:1 (2012), 43–55  mathnet
    4. A. V. Borisov, I. S. Mamaev, “Dinamika shara Chaplygina s polostyu, zapolnennoi zhidkostyu”, Nelineinaya dinam., 8:1 (2012), 103–111  mathnet
    5. A. V. Tsyganov, “O puassonovykh strukturakh, voznikayuschikh pri rassmotrenii shara Chaplygina i ego obobschenii”, Nelineinaya dinam., 8:2 (2012), 345–353  mathnet
    6. A. V. Tsiganov, “One family of conformally Hamiltonian systems”, Theoret. and Math. Phys., 173:2 (2012), 1481–1497  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    7. A. V. Tsyganov, “O negolonomnykh sistemakh Veselovoi i Chaplygina”, Nelineinaya dinam., 8:3 (2012), 541–547  mathnet
    8. E. N. Pivovarova, T. B. Ivanova, “Issledovanie ustoichivosti periodicheskikh reshenii v zadache o kachenii shara s mayatnikom”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2012, no. 4, 146–155  mathnet
    9. S. V. Bolotin, “Zadacha optimalnogo upravleniya kacheniem shara s rotorami”, Nelineinaya dinam., 8:4 (2012), 837–852  mathnet
    10. A. V. Borisov, I. S. Mamaev, “Topologicheskii analiz odnoi integriruemoi sistemy, svyazannoi s kacheniem shara po sfere”, Nelineinaya dinam., 8:5 (2012), 957–975  mathnet
    11. S. V. Bolotin, T. V. Popova, “Ob uravneniyakh dvizheniya sistemy vnutri katyaschegosya shara”, Nelineinaya dinam., 9:1 (2013), 51–58  mathnet
    12. 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
    13. Alexey V. Borisov, Ivan S. Mamaev, “The Dynamics of the Chaplygin Ball with a Fluid-filled Cavity”, Regul. Chaotic Dyn., 18:5 (2013), 490–496  mathnet  crossref  mathscinet  zmath
    14. A. V. Borisov, I. S. Mamaev, A. V. Tsiganov, “Non-holonomic dynamics and Poisson geometry”, Russian Math. Surveys, 69:3 (2014), 481–538  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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