Conservation of multidimensional invariant tori of Hamiltonian systems

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

“…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.…”

Persistence of Hyperbolic Invariant Tori for Hamiltonian Systems

“…Such systems are usually called nonstandard Hamiltonian systems. Motivated by the works of Hermann [7], Graff [6], and Parasyuk [10], we consider the persistence of hyperbolic invariant tori for nonstandard systems in the present paper.…”

“…In recent years, there has been increasing interest in the persistence problem for Hamiltonian systems with distinct numbers of action-angle variables [4,7,10]. Such systems are usually called nonstandard Hamiltonian systems.…”

The Classical KAM Theory at the Dawn of the Twenty-First Century

“…Moreover, it turns out that the periods of the symplectic structure (its integrals over the twodimensional cycles within the tori in question) should satisfy certain Diophantinelike conditions: all the theorems on coisotropic tori proven by now include such Diophantine hypotheses. Coisotropic Hamiltonian KAM theory was founded by Parasyuk [105] in 1984; see also the subsequent papers [73], [74], [106]- [108] by Parasyuk and his co-worker Kubichka. Coisotropic invariant n-tori of Hamiltonian systems with N < n degrees of freedom were also studied by Herman [56], [57] (see also [96], [157], [158]) and by Cong and Li [42].…”

“…Example 2. Let a symplectic manifold (Π 0 , ω 0 ) and a function H 0 : Π 0 → R determine an unperturbed Hamiltonian system in the Parasyuk theory [105]. In other words, suppose that Π 0 is smoothly foliated into coisotropic invariant n-tori (of codimension p < n) of the Hamiltonian system with Hamiltonian function H 0 and that any close Hamiltonian system on (Π 0 , ω 0 ) admits many invariant n-tori close to unperturbed ones.…”