Perturbations of degenerate coisotropic invariant tori of Hamiltonian systems

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

“…Assume that this system is close, in a certain sense, to a B-integrable locally Hamiltonian system [15,25], and the invariant tori of the latter are coisotropic submanifolds diffeomorphic to the standard torus T r := R r /2πZ r . Then, in the neighborhood of every invariant torus, there exist "action-angle" coordinates (y, ϕ), where y = (y 1 , .…”

“…The proof is substantially based on the reasonings used in [21,23], and, therefore, we omit technical details. We define functions…”

“…that smoothly depend on ϑ so that this torus is defined by the equations u = 0 and z = 0, the variables φ and ψ are angle coordinates on the tori T m μ (ϑ) and T n μ (ϑ), respectively, the Poisson bracket is defined by equalities (21), and ω μ has the form…”

“…The investigation of the nonclassical case where the phase space of an unperturbed Hamiltonian system is stratified by coisotropic invariant tori was originated in [14][15][16]. Despite several considerable advances in this direction [17][18][19][20][21][22][23][24], the theory of coisotropic invariant tori still contains gaps.…”

“…Despite several considerable advances in this direction [17][18][19][20][21][22][23][24], the theory of coisotropic invariant tori still contains gaps.…”