Generalizations of Gordon Theorem

In this article we present some generalizations of the Gordon theorem for Hamiltonian systems that are integrable on submanifolds of the phase space. The main result of the article is a generalization of the Gordon theorem. More precisely we consider a system that is integrable in "Hamiltonian" sense on such a submanifold and prove that for the conditionally periodic motion on invariant isotropic $k$-dimensional tori foliating this submanifold, the frequencies of this motion depend only on the values of $k$ "central" integrals of the system on a torus. Here, the central integrals are $k$ functions defined on the whole phase space that are integrals of the system on a submanifold on which the system is integrable. They are in involution on this submanifold and determine the foliation of the submanifold. That means that the Hamiltonian vector fields corresponding to these functions are tangent to the invariant tori foliating the submanifold of integrability. In addition, on some weaker assumptions (e.g. we do not postulate the existence of any Hamiltonian system at all), we have proved the following. Consider $k$ circular functions that correspond to the foliation of the submanifold into isotropic tori. The foliation is determined by $k$ functions in involution. Then, these $k$ circular functions are determined by the $k$ functions restricted to the submanifold.