  We consider the problem of constraining a particle to a submanifold Sigma of
configuration space using a sequence of increasing potentials. We compare the
classical and quantum versions of this procedure. This leads to new results in
both cases: an unbounded energy theorem in the classical case, and a quantum
averaging theorem. Our two step approach, consisting of an expansion in a
dilation parameter, followed by averaging in normal directions, emphasizes the
role of the normal bundle of Sigma, and shows when the limiting phase space
will be larger (or different) than expected.
