Abstract: An analytic method to study the periodic motion of double pendulums is presented.Considering enough first integrals,such as conservation of energy and angular momentum,the system is integrable by Liouville’s theorem.The dynamical equations are reduced based on Whittaker’s theorem.And rotating number is achieved to decide that the motion of the system is periodic or quasi-periodic.Then with some special parameters selected,the equation of motion in the phase plane is obtained.Because of existence of the elliptic integral in the equation of motion on time domain,the closed form solution is transcendental.The numerical examples and the simulation curves are given to verify the validity of the theoretical conclusion.

Key words: double pendulum；analytic method；Whittaker&rsquo, s theorem；elliptic function；integrable system