Stability of equilibria for 2-D resonant Hamiltonian systems: a geometrical approach

The stability of an equilibrium point of a 2-D Hamiltonian system, in the presence of resonances, is decided by means of a geometrical criterium, when the corresponding quadratic part is not sign defined. It is proven that this method is the geometrical counterpart of a theorem of Cabral and Meyer which constitutes an extension of the Arnold’s theorem.