KAM Methods For Invariant Tori Of Hamiltonian Systems Via Smooth Perturbations

This article investigated the KAM theorem of the finite dimensional near-integrable Hamiltonian systems.Specifically,considering Hamiltonian function:H = e +(ω,y)+ 1/2<A(ω)u,u>+P(x,y,u,ω),where ω ∈ Ω(?)Rn is a parameter,P(x,y,u,w)∈ Cβ(Tn × Rn × R2m × Ω).We proved that:There exists a β*,23(n + 4m.2 + τ + 1),giving a γ>0,τ>4M2(n-1),when β>β*,perturbations |P|Cβ fully small,there is almost all of the Cantor subset Ωγ(?)Ω,the Hamiltonian systems H has a low dimensional invariant torus when ω∈Ω,and meas(Ω-Ωγ)= O(γ1/4m2).The method of proving this article:As the perturbations considered in this paper is a finite order smooth,we apply Moser-Jackson-Zehnder lemma[18,2],the smooth perturbations P is approximated by a column of analytic functions in a complex neigh-borhood.At the same time we borrowed the method from the[2].modified KAM the iteration,considered an approximate analytic Hamiltonian in each step of iteration.In the solution to the homology equation,we mainly adopt the kill of You in[21]to deal with the small denominators.