On the trajectory of the nonlinear pendulum: Exact analytic solutions via power series

TL;DR

This paper derives an exact power series solution for the nonlinear pendulum's trajectory, analyzes its convergence properties, and introduces resummation techniques to improve its applicability across all motion phases.

Contribution

It provides a comprehensive analysis of the pendulum's series solution, including optimal expansion points and a novel resummation method for better convergence and extension of the solution.

Findings

Series converges from top to bottom of the trajectory when expanded at the top.

Resummation accelerates convergence uniformly across the entire motion.

Analytic extension of the period using elliptic integral resummation.

Abstract

We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair\'en, V. L\'opez, and L. Conde, Am. J. Phys 56 (1), (1988), pp. 57-61], the series itself -- as well as the optimal location about which an expansion should be chosen to assure series convergence and maximize the domain of convergence -- was not examined, and is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the physical problem in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. In constructing the series solution, we re-derive the coefficients using an alternative approach that generalizes to other nonlinear problems of mathematical physics. Additionally, we provide an exact resummation of the pendulum series -- Motivated by the asymptotic approximant method given in [Barlow et al., Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] -- that accelerates the series' convergence uniformly from the top to the bottom of the trajectory. We also provide an accelerated exact resummation of the infinite series representation for the elliptic integral used in calculating the period of a pendulum's trajectory. This allows one to preserve analyticity in the use of the period to extend the pendulum series for all time via symmetry.