Invariant tori for nearly integrable Hamiltonian systems with degeneracy

Abstract

does not fall into a hyperplanethrough the origin. The condition (1.3) is the sharpest one, we have not seenits proof yet. In this paper, we will give a nondegeneracy condition by meansof the derivatives of h(p), which is equivalent to the condition (1.3) in analyticcase, and under this nondegeneracy condition we obtain the Russmann’s results¨for analytic case. Furthermore, the arguments of this paper are available to thenonanalytic case since our nondegeneracy conditions only involve the finite orderderivatives of h(p).Main resultsTheorem A. Suppose that H = h(p)+"f(p;q)is analytic in Ω¯ T