A nash-moser approach to KAM theory

Abstract

Any finite dimensional embedded invariant torus of a Hamiltonian system, densely filled by quasi-periodic solutions, is isotropic. This property allows us to construct a set of symplectic coordinates in a neighborhood of the torus in which the Hamiltonian is in a generalized KAM normal form with angledependent coefficients. Based on this observation we develop an approach to KAM theory via a Nash-Moser implicit function iterative theorem. The key point is to construct an approximate right inverse of the differential operator associated to the linearized Hamiltonian system at each approximate quasi-periodic solution. In the above symplectic coordinates the linearized dynamics on the tangential and normal directions to the approximate torus are approximately decoupled. The construction of an approximate inverse is thus reduced to solving a quasi-periodically forced linear differential equation in the normal variables. Applications of this procedure allow to prove the existence of finite dimensional Diophantine invariant tori of autonomous PDEs.