# Realization of Nonholonomic Constraints and Singular Perturbation Theory for Plane Dumbbells

## TL;DR

This paper investigates the dynamics of plane dumbbells under nonholonomic constraints realized through viscous friction, revealing complex long-term behaviors and challenges in applying singular perturbation theory.

## Contribution

It provides a detailed analysis of the relation between viscous friction and ideal constraints, highlighting the limitations of classical asymptotic methods in this context.

## Key findings

- Long-term behaviors of frictional and constrained systems can differ significantly.
- Matching asymptotics require careful choice of time scales and cannot always handle secular terms.
- A reduction procedure for plane dumbbells is developed, sometimes leading to analytic solutions.

## Abstract

We study the dynamics of pairs of connected masses in the plane, when nonholonomic (knife-edge) constraints are realized by forces of viscous friction, in particular its relation to constrained dynamics, and its approximation by the method of matching asymptotics of singular perturbation theory when the mass to friction ratio is taken as the small parameter. It turns out that long term behaviors of the frictional and constrained systems may differ dramatically no matter how small the perturbation is, and when this happens is not determined by any transparent feature of the equations of motion. The choice of effective time scales for matching asymptotics is also subtle and non-obvious, and secular terms appearing in them can not be dealt with by the classical methods. Our analysis is based on comparison to analytic solutions, and we present a reduction procedure for plane dumbbells that leads to them in some cases.