Gevrey-smoothness Of Elliptic Lower Dimensional Invariant Tori In Hamiltonian Systems

In this paper we study Gevrey-smoothness of elliptic lower dimensional invariant tori in Hamiltonian systems under Riissmann's non-degeneracy condition. We will prove the result also holds in the case of Riissmann's non-degeneracy condition, it has four chapters: Introduction, Main results, Proof of main results and Appendix.In chapter 1 the introduction gives some basic concepts about Hamiltonian system and lays out research background. Here we introduced the following concepts: Hamiltonian system, conservative Hamiltonian system, first integral, Poisson bracket, symplectic matrix, symplectic map, integrable system and action-angle variable. These concepts are all very useful in this article, so it is necessary to explain them in advance. In this part we also have a look on progres of reseach work in relation to smoothnees of invariont tori. Then we give the main idea of the article in brief.Chapter 2 has two sections. In the first section, we give some definitions and notations. In the second section the main theorem are listed. The theorem mainly concerns with KAM theorems under the Melnikove conditions and Riissmann non-degeneracy condition , which is the main results of the article.In Chapter 3, we use a modified KAM iteration to prove the theorem. The KAM method is divided into KAM step, Iterative lemma, convergence of iteration and measure estimates. The KAM step is divided into the following subsection: truncation, extension of small divisor estimate, construction symplectic map, solving homological equation, estimating symplectic, estimating new non-resonant conditions and estimating new perturbation . Iterative lemma can be viewed as a summary of KAM step. By deliberate choice of parameters, the whole set-up of Iterative holds for all steps. So we can run iteration infinitely until the perturbation vanishes. In the convergence of iteration , we proved that the sequences of smplectic mappings {Φ~j} is convergent to a symplectic map Φ and Φ({T~n} × {0} × {0} × {0}) is an invariant tori of the Hamiltonian system. Futhermore we prove the invariant tori has Gevrey-smoothness. In the...