Closed-form solution to the nonlinear pendulum equation

This article provides a detailed analysis of the nonlinear pendulum equation, presenting its exact solution in terms of Jacobi elliptic functions. It explains how the solution is derived from the differential equation, shows Python code using SciPy to compute the exact solution numerically, and compares the exact solution to both a numerical ODE solution and a stretched linearized solution. The article highlights the accuracy improvements over linearization and clarifies common misconceptions from previous plots. It is a technical, math-focused piece relevant to physics simulations and differential equations in computing.