Simple and exact nonlinear pendulum motion for all possible initial conditions

Abstract

Simple and formally exact solutions of nonlinear pendulum are derived for all three classes of motion: swinging, stopping, and spinning. Their unique simplicity should be useful in a theoretical development that requires trackable mathematical framework or in an introductory physics course that aims to discuss nonlinear pendulum. A simple formula for the complete elliptical integral of the first kind is also proposed. It reproduces the exact analytical forms both in the zero and asymptotic limits, while in the midrange maintains average error of 0.06% and maximum error of 0.17%. The accuracy should be sufficient for typical engineering applications.

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