Nonlinear Stability in Resonant Cases: A Geometrical Approach

In systems with two degrees of freedom Arnold's theorem is used for studying non linear stability of the origin when the quadratic part of the Hamiltonian is a non definite form. In that case, a previous normalization of the higher orders is needed, which reduces the Hamiltonian to homogeneous polyno-mials in the actions. However, in the case of resonances it could not be possible to bring the Hamiltonian to the normal form required by Arnold's theorem. In these cases we determine the stability from the analysis of the normalized phase flow. Normalization up to an arbitrary order by Lie-Deprit transformation is carried out using a generalization of the Lissajous variables.