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 Regul. Chaotic Dyn., 2016, Volume 21, Issue 4, Pages 390–409 (Mi rcd85)

Realizing Nonholonomic Dynamics as Limit of Friction Forces

Jaap Eldering

Universidade de São Paulo — ICMC, Avenida Trabalhador Sao-carlense 400, CEP 13566-590, Sao Carlos, SP, Brazil

Abstract: The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carathéodory. It is known that, indeed, when friction forces are scaled to infinity, then nonholonomic dynamics is obtained as a singular limit.
Our results are twofold. First, we formulate the problem in a differential geometric context. Using modern geometric singular perturbation theory in our proof, we then obtain a sharp statement on the convergence of solutions on infinite time intervals. Secondly, we set up an explicit scheme to approximate systems with large friction by a perturbation of the nonholonomic dynamics. The theory is illustrated in detail by studying analytically and numerically the Chaplygin sleigh as an example. This approximation scheme offers a reduction in dimension and has potential use in applications.

Keywords: nonholonomic dynamics, friction, constraint realization, singular perturbation theory, Lagrange mechanics

 Funding Agency Grant Number Coordenaҫão de Aperfeiҫoamento de Pessoal de Nível Superior PVE11-2012 This research was supported by the Capes grant PVE11-2012.

DOI: https://doi.org/10.1134/S156035471604002X

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Bibliographic databases:

Document Type: Article
MSC: 37J60, 70F40, 37D10, 70H09
Language: English

Citation: Jaap Eldering, “Realizing Nonholonomic Dynamics as Limit of Friction Forces”, Regul. Chaotic Dyn., 21:4 (2016), 390–409

Citation in format AMSBIB
\Bibitem{Eld16} \by Jaap~Eldering \paper Realizing Nonholonomic Dynamics as Limit of Friction Forces \jour Regul. Chaotic Dyn. \yr 2016 \vol 21 \issue 4 \pages 390--409 \mathnet{http://mi.mathnet.ru/rcd85} \crossref{https://doi.org/10.1134/S156035471604002X} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000380679700002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84980377955} 

• http://mi.mathnet.ru/eng/rcd85
• http://mi.mathnet.ru/eng/rcd/v21/i4/p390

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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. Alexander P. Ivanov, “On Final Motions of a Chaplygin Ball on a Rough Plane”, Regul. Chaotic Dyn., 21:7-8 (2016), 804–810
2. S. Koshkin, V. Jovanovic, “Realization of non-holonomic constraints and singular perturbation theory for plane dumbbells”, J. Eng. Math., 106:1 (2017), 123–141
3. A. Kobrin, V. Sobolev, “Integral manifolds of fast-slow systems in nonholonomic mechanics”, 3rd International Conference Information Technology and Nanotechnology (ITNT-2017), Procedia Engineering, 201, eds. V. Soifer, N. Kazanskiy, O. Korotkova, S. Sazhin, Elsevier Science BV, 2017, 556–560
4. Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin, “Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation”, Regul. Chaotic Dyn., 23:4 (2018), 480–502
5. Alexey V. Borisov, Sergey P. Kuznetsov, “Comparing Dynamics Initiated by an Attached Oscillating Particle for the Nonholonomic Model of a Chaplygin Sleigh and for a Model with Strong Transverse and Weak Longitudinal Viscous Friction Applied at a Fixed Point on the Body”, Regul. Chaotic Dyn., 23:7-8 (2018), 803–820