Regular and Chaotic Dynamics

, Volume 21, Issue 4, pp 390–409 | Cite as

Realizing nonholonomic dynamics as limit of friction forces

  • Jaap Eldering


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 

MSC2010 numbers

37J60 70F40 37D10 70H09 


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Copyright information

© Pleiades Publishing, Ltd. 2016

Authors and Affiliations

  1. 1.Universidade de São Paulo — ICMCSao CarlosBrazil

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