This article is cited in 5 scientific papers (total in 5 papers)
Realizing Nonholonomic Dynamics as Limit of Friction Forces
Universidade de São Paulo — ICMC, Avenida Trabalhador Sao-carlense 400, CEP 13566-590, Sao Carlos, SP, Brazil
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.
nonholonomic dynamics, friction, constraint realization, singular perturbation theory, Lagrange mechanics
MSC: 37J60, 70F40, 37D10, 70H09
Jaap Eldering, “Realizing Nonholonomic Dynamics as Limit of Friction Forces”, Regul. Chaotic Dyn., 21:4 (2016), 390–409
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\paper Realizing Nonholonomic Dynamics as Limit of Friction Forces
\jour Regul. Chaotic Dyn.
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Alexander P. Ivanov, “On Final Motions of a Chaplygin Ball on a Rough Plane”, Regul. Chaotic Dyn., 21:7-8 (2016), 804–810
S. Koshkin, V. Jovanovic, “Realization of non-holonomic constraints and singular perturbation theory for plane dumbbells”, J. Eng. Math., 106:1 (2017), 123–141
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
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
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
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