A Hamiltonian system is considered in the 2n-dimensional space and the Hamiltonian is supposed to be periodic in the first n coordinates, i.e. the system is defined on \(T^ n\times R^ n\) where \(T^ n\) is the n- dimensional torus. The problem is transformed into a variational problem whose Euler-Lagrange equation gives rise to a nonlinear partial differential equation. The latter's solution is a diffeomorphism on \(T^ n\). This solution gives rise to invariant tori of prescribed periods. The proofs are intricate. The existence proof is based upon a Newton type iteration technique. For given prescribed periods the uniqueness of the corresponding invariant torus is also proved.