Persistence of Invariant Tori on Submanifolds in Hamiltonian Systems

Abstract

Generalizing the degenerate KAM theorem under the Rüssmann non-degeneracy and the isoenergetic KAM theorem, we employ a quasi-linear iterative scheme to study the persistence and frequency preservation of invariant tori on a smooth sub-manifold for a real analytic, nearly integrable Hamiltonian system. Under a nondegenerate condition of Rüssmann type on the sub-manifold, we shall show the following: a) the majority of the unperturbed tori on the sub-manifold will persist; b) the perturbed toral frequencies can be partially preserved according to the maximal degeneracy of the Hessian of the unperturbed system and be fully preserved if the Hessian is nondegenerate; c) the Hamiltonian admits normal forms near the perturbed tori of arbitrarily prescribed high order. Under a sub-isoenergetic nondegenerate condition on an energy surface, we shall show that the majority of unperturbed tori give rise to invariant tori of the perturbed system of the same energy which preserve the ratio of certain components of the respective frequencies.

On Invariant Tori of Nearly Integrable Hamiltonian Systems with Quasiperiodic Perturbation

“…However, in the case of Rüssmann's nondegeneracy condition, we can only get the existence of a family of invariant tori while there is no information on the persistence of frequency of any torus. Recently, Chow et al 10 and Sevryuk 11 consider perturbations of moderately degenerate integrable Hamiltonian system and prove that the first d frequencies d < n, n denotes the freedom of Hamiltonian system of unperturbed invariant n-tori can persist. Xu and You 12 prove that if some frequency satisfies certain nonresonant condition and topological degree condition, the perturbed system still has an invariant torus with this frequency under Rüssmann's nondegeneracy condition.…”

Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy

“…In particular, it was shown in [30] (see also [29]) that the Rüssmann nondegenerate condition is equivalent to the condition A) above with respect to the present frequency map ω. We refer the readers to [8,12,16,17,18,24,27] for more KAM type of results under Rüssmann non-degenerate conditions. Unfortunately, these results as well as their proofs do not apply to the properly degenerate Hamiltonian system (1.1) directly, simply because the order of its non-integrable perturbation is not high enough for the usual KAM iterations to carry over.…”

“…1) Using arguments in [8], the above theorem also holds on a submanifold M of R n if the condition A) is only assumed for ξ ∈ M (e.g., M is a fixed energy surface). This in particular leads to an iso-energetic version of the theorem (see [8] for detail). One can further show the partial preservation of frequency components for the perturbed tori in the above theorem.…”

Persistence of lower dimensional invariant tori on sub-manifolds in Hamiltonian systems

“…They adopted the Fourier series expansion for normal form N, which is a new technique. Recently, Chow, Li and Yi [2] proved that the majority of the unperturbed tori on sub-manifolds will persist under a non-degenerate condition of Rüssmann type for standard Hamiltonian systems. Motivated by their work, in this paper, we shall show that lower dimensional tori also survive small perturbations on sub-manifolds under some assumptions.…”

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AbstractChow, Li and Yi in [2] proved that the majority of the unperturbed tori on submanifolds will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems.

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