Semi-global symplectic invariants of the spherical pendulum

We compute the semi-global symplectic invariants near the focus–focus point of the spherical pendulum. A modified Birkhoff normal form procedure is presented to compute the expansion of the Hamiltonian near the focus–focus point in Eliasson-variables. Combining this with explicit formulas for the action we find the semi-global symplectic invariants near the focus–focus point introduced by Vũ Ngọc (2003) [32]. We show that the Birkhoff normal form at the focus–focus point is the inverse of a complete elliptic integral over a vanishing cycle. We close with some remarks about the pendulum, for which the invariants can be related to theta functions in a beautiful way.