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Uspekhi Mat. Nauk, 2014, Volume 69, Issue 3(417), Pages 87–144 (Mi umn9587)  

This article is cited in 24 scientific papers (total in 24 papers)

Non-holonomic dynamics and Poisson geometry

A. V. Borisova, I. S. Mamaevab, A. V. Tsiganovc

a Udmurt State University, Izhevsk
b Izhevsk State Technical University
c Saint Petersburg State University

Abstract: This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie–Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them.
Bibliography: 95 titles.

Keywords: non-holonomic systems, Poisson bracket, Chaplygin ball, Suslov system, Veselova system.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation
Russian Science Foundation 14-19-01303
The research of the first author was carried out in the framework of the state contract "Regular and Chaotic Dynamics" with Udmurt State University. The research of the second author was supported by grant no. 14-19-01303 of the Russian Science Foundation ("Dynamics and Control of Mobile Robototechnological Systems").


DOI: https://doi.org/10.4213/rm9587

Full text: PDF file (977 kB)
References: PDF file   HTML file

English version:
Russian Mathematical Surveys, 2014, 69:3, 481–538

Bibliographic databases:

UDC: 514.853+517.925+531.38+531.012
MSC: Primary 70F25, 70G45; Secondary 53D17
Received: 23.12.2013

Citation: A. V. Borisov, I. S. Mamaev, A. V. Tsiganov, “Non-holonomic dynamics and Poisson geometry”, Uspekhi Mat. Nauk, 69:3(417) (2014), 87–144; Russian Math. Surveys, 69:3 (2014), 481–538

Citation in format AMSBIB
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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. Paula Balseiro, O.E. Fernandez, “Reduction of nonholonomic systems in two stages and Hamiltonization”, Nonlinearity, 28:8 (2015), 2873–2912  crossref  mathscinet  zmath  isi  scopus
    2. Andrey V. Tsiganov, “On Integrable Perturbations of Some Nonholonomic Systems”, SIGMA, 11 (2015), 085, 19 pp.  mathnet  crossref
    3. I. A. Bizyaev, A. V. Bolsinov, A. V. Borisov, I. S. Mamaev, “Topology and bifurcations in nonholonomic mechanics”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25:10 (2015), 1530028, 21 pp.  mathnet  crossref  mathscinet  mathscinet  zmath  isi  elib  scopus
    4. V. V. Kozlov, “Dinamika sistem s servosvyazyami. II”, Nelineinaya dinam., 11:3 (2015), 579–611  mathnet
    5. Valery V. Kozlov, “The Dynamics of Systems with Servoconstraints. II”, Regul. Chaotic Dyn., 20:4 (2015), 401–427  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    6. A. V. Tsiganov, “Abel's theorem and Bäcklund transformations for the Hamilton–Jacobi equations”, Proc. Steklov Inst. Math., 295 (2016), 243–273  mathnet  crossref  crossref  mathscinet  isi  elib
    7. Yury A. Grigoryev, Alexey P. Sozonov, Andrey V. Tsiganov, “Integrability of Nonholonomic Heisenberg Type Systems”, SIGMA, 12 (2016), 112, 14 pp.  mathnet  crossref
    8. I. A. Bizyaev, A. V. Borisov, I. S. Mamaev, “Dinamika sanei Chaplygina na tsilindre”, Nelineinaya dinam., 12:4 (2016), 675–687  mathnet  crossref  mathscinet  zmath  elib
    9. Ivan A. Bizyaev, Alexey V. Borisov, Ivan S. Mamaev, “Dynamics of the Chaplygin Sleigh on a Cylinder”, Regul. Chaotic Dyn., 21:1 (2016), 136–146  mathnet  crossref  mathscinet  zmath  elib
    10. Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179  mathnet  crossref
    11. Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367  mathnet  crossref
    12. P. Balseiro, “Hamiltonization of solids of revolution through reduction”, J. Nonlinear Sci., 27:6 (2017), 2001–2035  crossref  mathscinet  zmath  isi  scopus
    13. B. Jovanovic, “Rolling balls over spheres in $\mathbb{R}^n$”, Nonlinearity, 31:9 (2018), 4006–4030  crossref  mathscinet  zmath  isi  scopus
    14. 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
    15. Andrey V. Tsiganov, “Hamiltonization and Separation of Variables for a Chaplygin Ball on a Rotating Plane”, Regul. Chaotic Dyn., 24:2 (2019), 171–186  mathnet  crossref
    16. Kurt M. Ehlers, Jair Koiller, “Cartan meets Chaplygin”, Theor. Appl. Mech., 46:1 (2019), 15–46  mathnet  crossref
    17. Vyacheslav P. Kruglov, Sergey P. Kuznetsov, “Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators”, Regul. Chaotic Dyn., 24:6 (2019), 725–738  mathnet  crossref  mathscinet
    18. Alexey V. Borisov, Andrey V. Tsiganov, “On the Chaplygin Sphere in a Magnetic Field”, Regul. Chaotic Dyn., 24:6 (2019), 739–754  mathnet  crossref  mathscinet
    19. A. V. Borisov, A. V. Tsyganov, “Vliyanie effektov Barnetta-Londona i Einshteina-de Gaaza na dvizhenie negolonomnoi sfery Rausa”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 29:4 (2019), 583–598  mathnet  crossref
    20. Alexey V. Borisov, Andrey V. Tsiganov, “On the Nonholonomic Routh Sphere in a Magnetic Field”, Regul. Chaotic Dyn., 25:1 (2020), 18–32  mathnet  crossref  mathscinet
    21. Tsiganov A.V., “On a Time-Dependent Nonholonomic Oscillator”, Russ. J. Math. Phys., 27:3 (2020), 399–409  crossref  mathscinet  zmath  isi
    22. Borisov V A., Tsiganov V A., “The Motion of a Nonholonomic Chaplygin Sphere in a Magnetic Field, the Grioli Problem, and the Barnett-London Effect”, Dokl. Phys., 65:3 (2020), 90–93  crossref  isi
    23. Vladimir Dragović, Borislav Gajić, Božidar Jovanović, “Demchenko's nonholonomic case of a gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis”, Theor. Appl. Mech., 47:2 (2020), 257–287  mathnet  crossref
    24. A. V. Borisov, A. V. Tsiganov, “Chaplygin ball in a solenoidal field”, Russian Math. Surveys, 76:3 (2021), 546–548  mathnet  crossref  crossref  isi  elib
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