Analytic solution to the pendulum equation for a given initial conditions

Abstract

In this paper we give the analytical solution for the undamped pendulum equation for a given arbitrary initial conditions. This solution is expressed in terms of the Jacobian elliptic functions. Approximated trigonometric solution is also provided. Three practical formulas for the period of oscillations are given. The results are illustrated with examples. Keywords: Undamped pendulum, Duffing equation, Period of oscillations, Jacobian elliptic functions