Arnold's theorem on properly degenerate systems with the Russmann nondegeneracy

摘要

In this paper, we investigate the persistence of invariant tori for the nearly in-tegrable Hamiltonian system H(x,y) = h(y) + εp(x,y) + ε~2Q(x,y), where h(y) and εp(x,y) satisfy the Russmann non-degenerate condition. Mainly we overcome the difficulties that the order of the parameter ε in the perturbation ε~2Q(x,y) is not enough and that the measure estimate involves in parts of frequencies with small parameter.