Theorem on perturbation of coisotropic invariant tori of locally Hamiltonian systems and its applications

Abstract

We study the problem of perturbations of quasiperiodic motions on coisotropic invariant tori in a class of locally Hamiltonian systems. We prove a general KAM-theorem on the perturbation of coisotropic invariant tori for locally Hamiltonian systems. As applications of this theorem, we consider the motion of an electron on a two-dimensional torus under the action of an electromagnetic field and extend results concerning the bifurcation of a Cantor set of coisotropic invariant tori to the case of locally Hamiltonian systems.

Bifurcation of coisotropic invariant tori under locally Hamiltonian perturbations of integrable systems and nondegenerate deformation of symplectic structure

“…In the present paper, we continue the investigations begun in [1]. On a 2n-dimensional symplectic manifold (M 2n , ω 2 0 ) with symplectic structure ω 2 0 , we consider a Liouville-integrable system with Hamiltonian H 0 : M 2n → R. Assume that this system undergoes perturbations of the form…”

“…In this case, the vector field of the integrable Hamiltonian system is perturbed by a locally Hamiltonian vector field and, simultaneously, the symplectic structure is deformed, which can lead to a change in the class of cohomologies of the corresponding 2-form. Under these conditions, as in [1], we consider the bifurcation problem for a Cantor set of coisotropic invariant tori of the perturbed system near a so-called quasistationary position (definitions are given in Sec. 2).…”

“…2). However, unlike [1], we study the case where the deformed symplectic structure generates a nondegenerate matrix of the Poisson brackets of action variables (these variables "enumerate" invariant tori of the unperturbed system). Due to this nondegeneracy, quasistationary positions are isolated, whereas, in [1], they form a certain manifold.…”

“…However, unlike [1], we study the case where the deformed symplectic structure generates a nondegenerate matrix of the Poisson brackets of action variables (these variables "enumerate" invariant tori of the unperturbed system). Due to this nondegeneracy, quasistationary positions are isolated, whereas, in [1], they form a certain manifold. This increases the degree of degeneracy of the Hamiltonian in the sense of its dependence on external parameters of the system that enable one to control the influence of small denominators in the process of construction of perturbed invariant tori.…”

“…In Sec. 3, the perturbed system is reduced to a special form convenient for the application of the results of [1] concerning the existence of invariant tori of locally Hamiltonian systems. The main theorem is proved in Sec.…”