The complex geometry of the spherical pendulum

In this paper we describe the geometry of the energy momentum mapping of the complexified spherical pendulum. For background on the classical spherical pendulum we refer the reader to [4, chpt. IV]. We show that this complex Hamiltonian system is Mumford-Jacobi completely integrable [9, p.3.51–3.53] and admits an analogue of the period lattice [5] and action angle coordinates [4, Apend. D]. Following the variation of the period lattice around a closed loop in the complement of the discriminant locus of a family of elliptic curves (see equation (9) below), we show that the complexified spherical pendulum has monodromy [4, p.383].