Homogenization in Time II: Mechanical Systems Subject to Friction and Gyroscopic Forces

In our previous work 1] we have studied natural mechanical systems on Riemannian manifolds with a strong constraining potential. These systems establish fast nonlinear oscillations around some equilibrium manifold. Important in applications, the problem of elimination of the fast degrees of freedom, or homogeniza-tion in time, leads to determine the singular limit of innnite strength of the constraining potential. In the present paper we extend this study to systems which are subject to external forces that are non-potential, depending in a mixed way on positions and velocities. We will argue that the method of weak convergence used in 1] covers such forces if and only if they result from viscous friction and gy-roscopic terms. All the results of 1] directly extend if there is no friction transversal to the equilibrium manifold; elsewise we show that instructive modiications apply.