Jacobi zeta function and action-angle coordinates for the pendulum

The Jacobi elliptic functions and integrals play a defining role in analytically describing the motion of the planar pendulum. In the present paper, the Jacobi zeta function is given the physical interpretation as the generating function of the canonical transformation from the pendulum coordinates ϑ and p ≡ ∂ ϑ / ∂ t to the action-angle coordinates ( J , ζ ) for both the librating pendulum and the rotating pendulum.