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Simple but accurate periodic solutions for the nonlinear pendulum equation

Abstract Despite its elementary structure, the simple pendulum oscillations are described by a nonlinear differential equation whose exact solution for the angular displacement from vertical as a function of time cannot be expressed in terms of an elementary function, so either a numerical treatment or some analytical approximation is ultimately demanded. Such solutions have been thoroughly investigated due to the abundance of distinct pendular systems in nature and, more recently, due to the availability of automatic data acquisition systems in undergraduate laboratories. However, it is well-known that numerical solutions to differential equations usually loose accuracy (due to accumulation of roundoff errors) and polynomial approximations diverge after long time intervals. In this work, I take a few terms of the Fourier series expansion of the elliptic function sn (u;k) as a source of accurate periodic solutions for the pendulum equation. Interestingly, these approximations remain accurate for arbitrarily long time intervals, even for large amplitudes, which shows its adequacy for the analysis of experimental data gathered in classical mechanics classes.