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

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.G34.31.003902.740.11.019514.740.11.0876MK-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
Accepted:19.11.2011
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} 

• http://mi.mathnet.ru/eng/rcd338
• http://mi.mathnet.ru/eng/rcd/v17/i2/p170

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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. A. V. Borisov, I. S. Mamaev, “Topologicheskii analiz odnoi integriruemoi sistemy, svyazannoi s kacheniem shara po sfere”, Nelineinaya dinam., 8:5 (2012), 957–975
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
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
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
5. A. V. Bolsinov, A. A. Kilin, A. O. Kazakov, “Topologicheskaya monodromiya v negolonomnykh sistemakh”, Nelineinaya dinam., 9:2 (2013), 203–227
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
7. Valery V. Kozlov, “The Euler–Jacobi–Lie Integrability Theorem”, Regul. Chaotic Dyn., 18:4 (2013), 329–343
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
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
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
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
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
13. Yu. L. Karavaev, A. A. Kilin, “Dinamika sferorobota s vnutrennei omnikolesnoi platformoi”, Nelineinaya dinam., 11:1 (2015), 187–204
14. Alexey V. Borisov, Ivan S. Mamaev, “Adiabatic Invariants, Diffusion and Acceleration in Rigid Body Dynamics”, Regul. Chaotic Dyn., 21:2 (2016), 232–248
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
16. Andrey V. Tsiganov, “Integrable Discretization and Deformation of the Nonholonomic Chaplygin Ball”, Regul. Chaotic Dyn., 22:4 (2017), 353–367
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
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