Nonlinear Pendulum: A Simple Generalization

TL;DR

This paper derives a closed-form solution for the nonlinear pendulum's motion with arbitrary initial conditions using elliptic functions, illustrating advanced mathematical concepts and generalizing previous standard case solutions.

Contribution

It provides a general solution for the nonlinear pendulum with arbitrary initial conditions using elliptic functions, extending prior work that only considered zero initial velocity.

Findings

Closed-form solution in terms of Jacobi elliptic functions

Educational application for advanced undergraduates

Generalization of previous standard case results

Abstract

In this work we solve the nonlinear second order differential equation of the simple pendulum with a general initial angular displacement ($\theta(0)=\theta_0$) and velocity ($\dot{\theta}(0)=\phi_0$), obtaining a closed-form solution in terms of the Jacobi elliptic function $\text{sn}(u,k)$, and of the the incomplete elliptical integral of the first kind $F(\varphi,k)$. Such a problem can be used to introduce concepts like elliptical integrals and functions to advanced undergraduate students, to motivate the use of Computer Algebra Systems to analyze the solutions obtained, and may serve as an exercise to show how to carry out a simple generalization, taking as a starting point the paper of Bel\'endez \emph{et al} \cite{belendez}, where they have considered the standard case $\dot{\theta}(0)=0$.