Realizing nonholonomic dynamics as limit of friction forces

TL;DR

This paper rigorously demonstrates that nonholonomic dynamics can be approximated as a limit of friction forces using geometric singular perturbation theory, providing explicit schemes and applications like the Chaplygin sleigh.

Contribution

It formulates the problem in a geometric framework and develops an explicit approximation scheme for systems with large friction forces.

Findings

Proves convergence of solutions to nonholonomic dynamics as friction tends to infinity.

Provides an explicit approximation scheme for large friction systems.

Illustrates the theory with analytical and numerical analysis of the Chaplygin sleigh.

Abstract

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.