Double pendulum and θ-divisor

The equations of motion of integrable systems involving hyperelliptic Riemann surfaces of genus 2 and one relevant degree of freedom are integrated in the framework of the Jacobi inversion problem, using a reduction to the theta-divisor on the Jacobi variety, i.e., to the set of zeros of the theta-function. Explicit solutions are given in terms of Kleinian sigma-functions and their derivatives. The procedure is applied to the planar double pendulum without gravity, but it is worked out for any Abelian integral of first or second kind.