We apply variational methods to converse KAM theory. These are useful for symplectomorphisms in the annulus that satisfy weaker hypotheses than those usually required. For instance, we do not need the existence of a global Lagrangian generating function. We obtain the variational principles from the primitive function of our symplectomorphism. They are introduced not only for the orbits of a symplectomorphism, but also for the so-called invariant Lagrangian graphs (ILG). Among the non-degenerate ILG we focus on the minimizing ones. Applications are also described for a broad class of examples.