Quasiperiodic Version of Gordon’s Theorem

We consider Hamiltonian systems possessing families of nonresonant invariant tori whose frequencies are all collinear. Then under certain conditions the frequencies depend on energy only. This is a generalization of the well-known Gordon’s theorem about periodic solutions of Hamiltonian systems. While the proof of Gordon’s theorem uses Hamilton’s principle, our result is based on Percival’s variational principle. This work was motivated by the problem of isochronicity in Hamiltonian systems.