Realizing holonomic constraints in classical and quantum mechanics

A motion of a particle constrained to lie on a submanifold of configuration space is governed by well-known differential equations. For example, if the particle is allowed to move freely on the submanifold, the classical orbits are geodesics, while the quantum motion is governed by the Laplace-Beltrami operator.

One may try to model such a constraint by allowing the particle to move in the whole configuration space, subject to a very large force that pushes the particle onto the submanifold. As this constraining force becomes larger and larger, do the orbits converge to solutions of the well-known equations on the submanifold?

Surprisingly, the answer is often no. Sometimes the particles obey an equation with extra potential terms, and sometimes the observed motion doesn't seem to obey an equation at all. Understanding the limiting motion involves quantum and classical versions of averaging, as well as the geometry of the normal bundle.