Bambusi D. An Averaging Theorem for Quasilinear Hamiltonian PDEs (81K, TeX) ABSTRACT. We study the dynamics of Hamiltonian quasilinear PDEs close to elliptic equilibria. Under a suitable nonresonance condition we prove an averaging theorem according to which any solution corresponding to small amplitude smooth initial data remain very close to a torus up to long times. An application to quasilinear wave equations in an $n$ dimensional paralleliped is given.