Kinematic constraints are classically
divided into two classes: constraints
that can be integrated to yield constraints
on the position variables,
called holonomic constraints, and constraints
for which this integration is
not possible, called nonholonomic constraints.
A typical example of a nonholonomic
constraint is a wheel rolling
vertically without slipping on a surface.
The constraint on the allowable velocity
(the point of contact of the wheel
with the surface cannot slip in all
directions) cannot be integrated to
yield a constraint on the position of
the wheel. This nonintegrability is
intuitively clear, as illustrated by the
fact that an automobile can go anywhere
it pleases by suitable maneuvering.
(This example hints at the
intimate relationship between nonholonomic
constraints and controllability.)
Loosely speaking, mechanical
systems with holonomic constraints
can be reduced to lower dimensional
mechanical systems without constraints;
for systems with nonholonomic
constraints, this reduction is not
possible, and as a result some distinguishing
features arise