Realizing holonomic constraints in classical and quantum mechanics

We consider the problem of constraining a particle to a smooth compact subm anifold Sigma of configuration space using a sequence of increasing potenti als. We compare the classical and quantum versions of this procedure. This leads to new results in both eases: an unbounded energy theorem in the clas sical case, and a quantum averaging theorem. Our two step approach, consist ing of an expansion in a dilation parameter, followed by averaging in norma l directions, emphasizes the role of the normal bundle of Sigma, and shows when the limiting phase space will be larger (or different) than expected.