Homogenization of Hamiltonian systems with a strong constraining potential

The paper studies Hamiltonian systems with a strong potential forcing the solutions to oscillate on a very small time scale. In particular, we are interested in the limit situation where the size ∈ of this small time scale tends to zero but the velocity components remain oscillating with an amplitude variation of the order O(1). The process of establishing an effective initial value problem for the limit positions will be called homogenization of the Hamiltonian system. This problem occurs in mechanics as the problem of realization of holonomic constraints, as various singular limits in fluid flow problems, in plasma physics as the problem of guiding center motion and in the simulation of biomolecules as the so-called smoothing problem. We suggest the systematic use of the notion of weak convergence in order to approach this problem. This methodology helps to establish unified and short proofs of many known results which throw light on the inherent structure of the problem. Moreover, we give a careful and critical review of the literature.