I allowed NDSolve to automatically pick its solution method for this Euler-Lagrange equation. I have explored symplectic methods in notebooks I'm preparing for the future on the Kepler problem (following "Moving Planets Around") and for the Inverted Spherical Pendulum but don't yet have anything to report about them in general.

In my current series of notebooks (upcoming soon: the symmetric top), I'm focusing on discrete Lagrangians and using NDSolve only as a "source of truth" to compare against my discrete solutions. While numerical integration of the Euler-Lagrange equations is a deep and interesting topic, I am "taking the tangent" of integrating by discretizing the Lagrangians and computing algebraic recurrences. Such methods are guaranteed to be "on the manifold." I'm exploring the promise that they can be applied generally without generating a large "choice space" of integration methods, as does the standard numerical approach.