Types of integrability on a submanifold and generalizations of Gordon’s theorem

Abstract:

At the beginning of the paper the concept of Liouville integrability is analysed for systems of general form, that is, ones that are not necessarily Hamiltonian. On this simple basis Hamiltonian systems are studied that are integrable only on submanifolds $N$ of the phase space, which is the main subject of the paper. The study is carried out in terms of $k$-dimensional foliations and fibrations defined on $N$ by the Hamiltonian vector fields corresponding to $k$ integrals in involution. These integrals are said to be central and may include the Hamiltonian function of the system. The parallel language of sets of functions is also used, in particular, sets of functions whose common level surfaces are the fibres of fibrations.

Relations between different types of integrability on submanifolds of the phase space are established. The main result of the paper is a generalization of Gordon’s theorem stating that in a Hamiltonian system all of whose trajectories are closed the period of the solutions depends only on the value of the Hamiltonian. Our generalization asserts that in the case of the strongest “Hamiltonian” integrability the frequencies of a conditionally periodic motion on the invariant isotropic tori that form a fibration of an integrability submanifold depend only on the values of the central integrals. Under essentially weaker assumptions on the fibration of the submanifold into such tori it is proved that the circular action functions also have the same property. In addition, certain general recipes for finding the integrability submanifolds are given.