  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 example. This approximation scheme offers a reduction in
dimension and has potential use in applications.
