Effective Hamiltonians and averaging for Hamiltonian dynamics II

We extend to time-dependent Hamiltonians some of the PDE methods from our previous paper [E-G1], and in particular the theory of "effective Hamiltonians" introduced by Lions, Papanicolaou & Varadhan [L-P-V]. These PDE techniques augment the variational approach of Mather [Mt1,Mt2,Mt3,Mt4,M-F] and the weak KAM methods of Fathi [F1,F2,F3,F4,F5]. We also provide a weak interpretation of adiabatic invariance of the action and suggest a formula for the Berry-Hannay geometric phase in terms of an effective Hamiltonian.[PUBLICATION ABSTRACT]