Nonsemisimple 1:1 resonance at an equilibrium

Consider a Hamiltonian system of two degrees of freedom at an equilibrium. Suppose that the linearized vectorfield has eigenvaluesiα,iα,−iα,−iα (α ∈ ℝ, α>0) and is not semisimple. In this paper we discuss the real normalization of the Hamiltonian function of such a system. We normalize the Hamiltonian up to 4th order and show how to compute its coefficients. For the planar restricted three body problem atL4 the coefficient that plays an important role in the investigation of the qualitative behaviour of periodic solutions near the equilibrium is explicitly calculated.