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 epsilon of thi s small time scale tends to zero but the velocity components remain os cillating with an amplitude variation of the order O(1). The process o f establishing an effective initial value problem for the limit positi ons will be called homogenization of the Hamiltonian system. This prob lem occurs in mechanics as the problem of realization of holonomic con straints, 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 sy stematic use of the notion of weak convergence in order to approach th is problem. This methodology helps to establish unified and short proo fs of many known results which throw light on the inherent structure o f the problem. Moreover, we give a careful and critical review of the literature.