  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.
