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

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


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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
\by A.~V.~Borisov, I.~S.~Mamaev, A.~V.~Tsiganov
\paper Non-holonomic dynamics and Poisson geometry
\jour Uspekhi Mat. Nauk
\yr 2014
\vol 69
\issue 3(417)
\pages 87--144
\jour Russian Math. Surveys
\yr 2014
\vol 69
\issue 3
\pages 481--538

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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. Valery V. Kozlov, “The Dynamics of Systems with Servoconstraints. II”, Regul. Chaotic Dyn., 20:4 (2015), 401–427  mathnet  crossref  mathscinet  zmath  adsnasa  elib
    5. 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
    6. Yury A. Grigoryev, Alexey P. Sozonov, Andrey V. Tsiganov, “Integrability of Nonholonomic Heisenberg Type Systems”, SIGMA, 12 (2016), 112, 14 pp.  mathnet  crossref
    7. 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
    8. Andrey V. Tsiganov, “Bäcklund Transformations for the Nonholonomic Veselova System”, Regul. Chaotic Dyn., 22:2 (2017), 163–179  mathnet  crossref
    9. Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367  mathnet  crossref
    10. P. Balseiro, “Hamiltonization of solids of revolution through reduction”, J. Nonlinear Sci., 27:6 (2017), 2001–2035  crossref  mathscinet  zmath  isi  scopus
    11. B. Jovanovic, “Rolling balls over spheres in $\mathbb{R}^n$”, Nonlinearity, 31:9 (2018), 4006–4030  crossref  mathscinet  zmath  isi  scopus
    12. A. V. Borisov, I. S. Mamaev, I. A. Bizyaev, “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
    13. Tsiganov A.V., “Hamiltonization and Separation of Variables For a Chaplygin Ball on a Rotating Plane”, Regul. Chaotic Dyn., 24:2 (2019), 171–186  mathnet  crossref  mathscinet  isi  scopus
    14. Ehlers K.M., Koiller J., “Cartan Meets Chaplygin”, Theor. Appl. Mech., 46:1 (2019), 15–46  crossref  isi
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