Nonlinear stability in resonant cases: A geometrical approach

In systems with two degrees of freedom, Arnold's theorem is used for studyi ng nonlinear stability of the origin when the quadratic part of the Hamilto nian is a nondefinite form. In that case, a previous normalization of the h igher orders is needed, which reduces the Hamiltonian to homogeneous polyno mials in the actions. However, in the case of resonances, it could not be p ossible to bring the Hamiltonian to the normal form required by Arnold's th eorem. In these cases, we determine the stability from analysis of the norm alized phase flow. Normalization up to an arbitrary order by Lie-Deprit tra nsformation is carried out using a generalization of the Lissajous variable s.