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Sibirsk. Mat. Zh., 2007, Volume 48, Number 1, Pages 33–45 (Mi smj4)  

This article is cited in 44 scientific papers (total in 44 papers)

Isomorphism and Hamilton representation of some nonholonomic systems

A. V. Borisovab, I. S. Mamaevab

a Udmurt State University
b Institute of Computer Science

Abstract: We consider some questions connected with the Hamiltonian form of the two problems of nonholonomic mechanics: the Chaplygin ball problem and the Veselova problem. For these problems we find representations in the form of the generalized Chaplygin systems that can be integrated by the reducing multiplier method. We give a concrete algebraic form of the Poisson brackets which, together with an appropriate change of time, enable us to write down the equations of motion of the problems under study. Some generalization of these problems are considered and new ways of implementation of nonholonomic constraints are proposed. We list a series of nonholonomic systems possessing an invariant measure and sufficiently many first integrals for which the question about the Hamiltonian form remains open even after change of time. We prove a theorem on isomorphism of the dynamics of the Chaplygin ball and the motion of a body in a fluid in the Clebsch case.

Keywords: nonholonomic system, reducing multiplier, Hamiltonization, isomorphism.

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English version:
Siberian Mathematical Journal, 2007, 48:1, 26–36

Bibliographic databases:

UDC: 531.38
Received: 04.07.2005

Citation: A. V. Borisov, I. S. Mamaev, “Isomorphism and Hamilton representation of some nonholonomic systems”, Sibirsk. Mat. Zh., 48:1 (2007), 33–45; Siberian Math. J., 48:1 (2007), 26–36

Citation in format AMSBIB
\by A.~V.~Borisov, I.~S.~Mamaev
\paper Isomorphism and Hamilton representation of some nonholonomic systems
\jour Sibirsk. Mat. Zh.
\yr 2007
\vol 48
\issue 1
\pages 33--45
\jour Siberian Math. J.
\yr 2007
\vol 48
\issue 1
\pages 26--36

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    This publication is cited in the following articles:
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    2. Borisov A.V., Mamaev I.S., “Rolling of a non-homogeneous ball over a sphere without slipping and twisting”, Regul. Chaotic Dyn., 12:2 (2007), 153–159  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. Borisov A.V., Mamaev I.S., Marikhin V.G., “Explicit Integration of One Problem in Nonholonomic Mechanics”, Dokl. Phys., 53:10 (2008), 525–528  mathnet  crossref  mathscinet  zmath  zmath  adsnasa  isi  elib  elib  scopus
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    20. 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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    23. Alexander P. Ivanov, “On the Control of a Robot Ball Using Two Omniwheels”, Regul. Chaotic Dyn., 20:4 (2015), 441–448  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    24. Valery V. Kozlov, “The Dynamics of Systems with Servoconstraints. I”, Regul. Chaotic Dyn., 20:3 (2015), 205–224  mathnet  crossref  mathscinet  zmath  adsnasa  elib
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    30. Grebenev V.N., Oberlack M., Megrabov A.G., Grishkov A.N., “Symmetry Transformations of An Ideal Steady Fluid Flow Determined By a Potential Function”, J. Math. Phys., 57:10 (2016), 103506  crossref  mathscinet  zmath  isi  scopus
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    35. A. V. Borisov, A. O. Kazakov, E. N. Pivovarova, “Regulyarnaya i khaoticheskaya dinamika v rezinovoi modeli volchka Chaplygina”, Nelineinaya dinam., 13:2 (2017), 277–297  mathnet  crossref  elib
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    38. A. A. Kilin, T. B. Ivanova, E. N. Pivovarova, “Control of the rolling motion of a spherical robot on an inclined plane”, Dokl. Phys., 63:10 (2018), 435–440  mathnet  crossref  crossref  isi  elib  scopus
    39. Garcia-Naranjo L.C., Montaldi J., “Gauge Momenta as Casimir Functions of Nonholonomic Systems”, Arch. Ration. Mech. Anal., 228:2 (2018), 563–602  crossref  mathscinet  zmath  isi  scopus
    40. Kurt M. Ehlers, Jair Koiller, “Cartan meets Chaplygin”, Theor. Appl. Mech., 46:1 (2019), 15–46  mathnet  crossref
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