Weak convergence and averaging for ODE

This mostly expository paper shows how weak convergence methods provide simple, elegant proofs of (i) the stabilization of an inverted pendulum under fast vertical oscillations, (ii) the existence of particle traps induced by rapidly varying electric fields and (iii) the adiabatic invariance of ∫Γpdx for slowing varying planar Hamiltonian dynamics. Under an appropriate, but very restrictive, unique ergodicity assumption, the proof of (iii) extends also to many degrees of freedom.