Birkhoff normal form for some quasilinear Hamiltonian PDEs.

Consider a Hamiltonian PDE having an elliptic equilibrium at zero. Assuming a suitable condition on the frequencies we will construct a canonical transformation putting the system in Birkhoff normal form up to a small reminder. In the case of quasilinear systems the normal form will be used to describe the dynamics of smooth solutions of small amplitude. Applications to the water wave problem and to quasilinear wave equations will be given.