Nonholonomic constraint

A nonholonomic constraint is a nonintegrable restriction on allowed velocities: it cannot locally be replaced by relations among generalized coordinates and time. In contrast, a velocity equation $r\dot{\varphi }-R\dot{\theta }=0$ with constant radii integrates to a holonomic constraint $r\varphi -R\theta =C$. Rolling without slipping is therefore not automatically nonholonomic: planar rolling on a prescribed curve can be holonomic, whereas rolling with additional directional freedoms can impose nonintegrable velocity constraints. Unwrapped angles may be needed to record the winding of a rolling trajectory.