RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
General information
Latest issue
Archive
Impact factor
Subscription

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Sibirsk. Mat. Zh.:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

This article is cited in 35 scientific papers (total in 35 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.

Full text: PDF file (258 kB)
References: PDF file   HTML file

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
\Bibitem{BorMam07}
\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
\mathnet{http://mi.mathnet.ru/smj4}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2304876}
\zmath{https://zbmath.org/?q=an:1164.37342}
\transl
\jour Siberian Math. J.
\yr 2007
\vol 48
\issue 1
\pages 26--36
\crossref{https://doi.org/10.1007/s11202-007-0004-6}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000244424100004}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33846585433}


Linking options:
  • http://mi.mathnet.ru/eng/smj4
  • http://mi.mathnet.ru/eng/smj/v48/i1/p33

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Koiller J., Ehlers K., “Rubber rolling over a sphere”, Regul. Chaotic Dyn., 12:2 (2007), 127–152  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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
    4. Borisov A.V., Mamaev I.S., “Conservation laws, hierarchy of dynamics and explicit integration of nonholonomic systems”, Regul. Chaotic Dyn., 13:5 (2008), 443–490  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Borisov A.V., Fedorov Yu.N., Mamaev I.S., “Chaplygin ball over a fixed sphere: an explicit integration”, Regul. Chaotic Dyn., 13:6 (2008), 557–571  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Fernandez O.E., Mestdag T., Bloch A.M., “A generalization of Chaplygin's reducibility theorem”, Regul. Chaotic Dyn., 14:6 (2009), 635–655  crossref  mathscinet  zmath  isi  elib  scopus
    7. Fedorov Yu.N., Jovanović B., “Hamiltonization of the generalized Veselova LR system”, Regul. Chaotic Dyn., 14:4-5 (2009), 495–505  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    8. Jovanović B., “LR and L+R systems”, J. Phys. A, 42:22 (2009), 225202, 18 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    9. I. S. Mamaev, “Universalnyi kompleks programm dlya issledovaniya mekhanicheskikh sistem s negolonomnymi svyazyami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2009, no. 2, 147–160  mathnet
    10. Jovanovic B., “Hamiltonization and Integrability of the Chaplygin Sphere in R-n”, J Nonlinear Sci, 20:5 (2010), 569–593  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    11. A. Yu. Moskvin, “Rezinovyi shar na ploskosti: kriticheskie resheniya”, Nelineinaya dinam., 6:2 (2010), 345–358  mathnet
    12. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Gamiltonizatsiya negolonomnykh sistem v okrestnosti invariantnykh mnogoobrazii”, Nelineinaya dinam., 6:4 (2010), 829–854  mathnet  elib
    13. Fernandez O.E., Bloch A.M., “The Weitzenbock connection and time reparameterization in nonholonomic mechanics”, J Math Phys, 52:1 (2011), 012901  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Bolsinov A.V., Borisov A.V., Mamaev I.S., “Hamiltonization of non-holonomic systems in the neighborhood of invariant manifolds”, Regul. Chaotic Dyn., 16:5 (2011), 443–464  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Kachenie bez vercheniya shara po ploskosti: otsutstvie invariantnoi mery v sisteme s polnym naborom integralov”, Nelineinaya dinam., 8:3 (2012), 605–616  mathnet
    16. Bolsinov A.V., Borisov A.V., Mamaev I.S., “Rolling of a Ball Without Spinning on a Plane: the Absence of an Invariant Measure in a System with a Complete Set of Integrals”, Regul. Chaotic Dyn., 17:6 (2012), 571–579  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    17. Alexey V. Borisov, Ivan S. Mamaev, Dmitrii V. Treschev, “Rolling of a rigid body without slipping and spinning: kinematics and dynamics”, J. Appl. Nonlinear Dyn., 2:2 (2013), 161–173  mathnet  crossref
    18. 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
    19. 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  mathnet  crossref  mathscinet  zmath
    20. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    21. Alexander A. Kilin, Elena N. Pivovarova, Tatyana B. Ivanova, “Spherical Robot of Combined Type: Dynamics and Control”, Regul. Chaotic Dyn., 20:6 (2015), 716–728  mathnet  crossref  mathscinet  adsnasa
    22. Balseiro P., Fernandez O.E., “Reduction of Nonholonomic Systems in Two Stages and Hamiltonization”, Nonlinearity, 28:8 (2015), 2873–2912  crossref  mathscinet  zmath  isi  elib  scopus
    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
    25. Fernandez O.E., “Poincaré Transformations in Nonholonomic Mechanics”, Appl. Math. Lett., 43 (2015), 96–100  crossref  mathscinet  zmath  isi  elib  scopus
    26. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “On the Hadamard–Hamel Problem and the Dynamics of Wheeled Vehicles”, Regul. Chaotic Dyn., 20:6 (2015), 752–766  mathnet  crossref  mathscinet  adsnasa
    27. 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
    28. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248  mathnet  crossref  mathscinet  zmath  elib
    29. Alexander P. Ivanov, “On Final Motions of a Chaplygin Ball on a Rough Plane”, Regul. Chaotic Dyn., 21:7-8 (2016), 804–810  mathnet  crossref
    30. Alexey V. Borisov, Alexey O. Kazakov, Elena N. Pivovarova, “Regular and Chaotic Dynamics in the Rubber Model of a Chaplygin Top”, Regul. Chaotic Dyn., 21:7-8 (2016), 885–901  mathnet  crossref
    31. 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
    32. Krakowski K.A., Leite F.S., “Geometry of the Rolling Ellipsoid”, Kybernetika, 52:2 (2016), 209–223  crossref  mathscinet  zmath  isi  scopus
    33. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “Dynamical systems with non-integrable constraints, vakonomic mechanics, sub-Riemannian geometry, and non-holonomic mechanics”, Russian Math. Surveys, 72:5 (2017), 783–840  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    34. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane”, Regul. Chaotic Dyn., 23:6 (2018), 665–684  mathnet  crossref  mathscinet
    35. Alexander A. Kilin, Elena N. Pivovarova, “Integrable Nonsmooth Nonholonomic Dynamics of a Rubber Wheel with Sharp Edges”, Regul. Chaotic Dyn., 23:7-8 (2018), 887–907  mathnet  crossref
  • Сибирский математический журнал Siberian Mathematical Journal
    Number of views:
    This page:459
    Full text:62
    References:37

     
    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2019