The large amplitude simple pendulum and Fourier series; an analysis for undergraduates

A pedagogical and theoretical study of the behavior of a simple pendulum is presented. Fourier series and perturbation analysis at levels appropriate for undergraduate physics students are used to accurately derive the motion of a pendulum. Instead of using the elliptic integral of the first kind, appropriate higher order terms of the Taylor-expanded differential equation are considered, leading to increasingly accurate corrections to the period. It is found that the relation between the order parameter and the initial conditions is not fixed, allowing many solutions to the motion in terms of the order parameter but a unique solution in terms of the initial conditions. The error in the presented corrections is compared to published approximations.