A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows

We study the determination of the second-order normal form for perturbed Hamiltonians $H_{\epsilon}=H_0 +\epsilon H_1 +\frac{\epsilon^2}{2} H_2$, relative to the periodic flow of the unperturbed Hamiltonian $H_0$. The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the H\'enon-Heiles and the elastic pendulum Hamiltonians.