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 +(?,y)+ 1/2<A(?)u,u>+P(x,y,u,?),where ? ? ?(?)Rn is a parameter,P(x,y,u,w)? C?(Tn × Rn × R2m × ?).We proved that:There exists a ?*,23(n + 4m.2 + ? + 1),giving a ?>0,?>4M2(n-1),when ?>?*,perturbations |P|C? fully small,there is almost all of the Cantor subset ??(?)?,the Hamiltonian systems H has a low dimensional invariant torus when ???,and meas(?-??)= O(?1/4m2).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.