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Regul. Chaotic Dyn., 2012, Volume 17, Issue 2, Pages 170–190 (Mi rcd338)  

This article is cited in 18 scientific papers (total in 18 papers)

Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support

Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev

Institute of Computer Science, Udmurt State University, ul. Universitetskaya 1, Izhevsk 426034, Russia

Abstract: We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu.N.Fedorov, deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation to the case of a rigid body in a spherical support – the problem where almost no progress has been made since Yu.N. Fedorov posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support system does not affect its integrability.

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

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation 11.G34.31.0039
02.740.11.0195
14.740.11.0876
MK-8428.2010.1
This research was supported by the Grant of the Government of the Russian Federation for state support of scientific research conducted under supervision of leading scientists in Russian educational institutions of higher professional education (contract no. 11.G34.31.0039) and the Federal target programme “Scientific and Scientific-Pedagogical Personnel of Innovative Russia”, measure 1.1. “Scientific-Educational Center Regular and Chaotic Dynamics” (project code 02.740.11.0195), measure 1.5 “Topology and Mechanics” (project code 14.740.11.0876). The work of A. A.Kilin was supported by the Grant of the President of the Russian Federation for the Support of Young Russian Scientists–Candidates of Science (MK-8428.2010.1).


DOI: https://doi.org/10.1134/S1560354712020062


Bibliographic databases:

Document Type: Article
MSC: 37J60, 37J35, 70E18, 70F25, 70H45
Received: 27.07.2011
Language: English

Citation: Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support”, Regul. Chaotic Dyn., 17:2 (2012), 170–190

Citation in format AMSBIB
\Bibitem{BorKilMam12}
\by Alexey V.~Borisov, Alexander A.~Kilin, Ivan S.~Mamaev
\paper Generalized Chaplygin’s Transformation and Explicit Integration of a System with a Spherical Support
\jour Regul. Chaotic Dyn.
\yr 2012
\vol 17
\issue 2
\pages 170--190
\mathnet{http://mi.mathnet.ru/rcd338}
\crossref{https://doi.org/10.1134/S1560354712020062}
\zmath{https://zbmath.org/?q=an:1253.37063}


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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, “Topologicheskii analiz odnoi integriruemoi sistemy, svyazannoi s kacheniem shara po sfere”, Nelineinaya dinam., 8:5 (2012), 957–975  mathnet
    2. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “The Problem of Drift and Recurrence for the Rolling Chaplygin Ball”, Regul. Chaotic Dyn., 18:6 (2013), 832–859  mathnet  crossref  mathscinet  zmath
    3. Alexey V. Borisov, Ivan S. Mamaev, “Topological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere”, Regul. Chaotic Dyn., 18:4 (2013), 356–371  mathnet  crossref  mathscinet  zmath
    4. Alexey V. Borisov, Alexander A. Kilin, Ivan S. Mamaev, “How to Control the Chaplygin Ball Using Rotors. II”, Regul. Chaotic Dyn., 18:1-2 (2013), 144–158  mathnet  crossref  mathscinet  zmath
    5. A. V. Bolsinov, A. A. Kilin, A. O. Kazakov, “Topologicheskaya monodromiya v negolonomnykh sistemakh”, Nelineinaya dinam., 9:2 (2013), 203–227  mathnet
    6. A. V. Borisov, A. A. Kilin, I. S. Mamaev, “Kak upravlyat sharom Chaplygina pri pomoschi rotorov. II”, Nelineinaya dinam., 9:1 (2013), 59–76  mathnet
    7. Valery V. Kozlov, “The Euler–Jacobi–Lie Integrability Theorem”, Regul. Chaotic Dyn., 18:4 (2013), 329–343  mathnet  crossref  mathscinet  zmath
    8. 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
    9. 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
    10. 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
    11. Bolsinov A.V., Kilin A.A., Kazakov A.O., “Topological Monodromy as An Obstruction to Hamiltonization of Nonholonomic Systems: Pro Or Contra?”, J. Geom. Phys., 87 (2015), 61–75  crossref  mathscinet  zmath  isi  scopus
    12. Rosemann S., Schoebel K., “Open Problems in the Theory of Finite-Dimensional Integrable Systems and Related Fields”, J. Geom. Phys., 87 (2015), 396–414  crossref  mathscinet  zmath  isi  scopus
    13. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204  mathnet  elib
    14. 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
    15. Sokolov S.V. Ryabov P.E., “Bifurcation Analysis of the Dynamics of Two Vortices in a Bose–Einstein Condensate. the Case of Intensities of Opposite Signs”, Regul. Chaotic Dyn., 22:8 (2017), 976–995  mathnet  crossref  mathscinet  isi  scopus
    16. Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367  mathnet  crossref
    17. 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
    18. 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
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