Realizing Nonholonomic Dynamics as Limit of Friction Forces

The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carath´eodory. 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.