Lower dimensional tori in Hamiltonian systems. Variational approach

In this report the problem of persistence of invariant tori of codimension $k\geq 1$ with a fixed frequency vector under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between weakly hyperbolic invariant tori and critical points of the Parceval potential defined on $\mathbb T^k\times [0,1]^k$ is established. We give sufficient conditions which being imposed on the unperturbed Hamiltonian provide the existence at least one weakly hyperbolic invariant torus. We also give an example of a convex unperturbed Hamiltonian for which all perturbed low dimensional invariant tori are elliptic.