Hi Moderation Team,

Thanks for the plaudit! But if we're going to make this a featured post, let's go ahead and add an animation.

Performing standard numerical time evolution along the pendulum C.E.S. allows us to compute a "phase space race":

This image shows trajectories for six energies, with ten approximations at each energy level, using the same coloring scheme where precision increases toward blue.

Especially for the outer higher-energy trajectories, the poorly-approximated green curves race ahead of schedule, finishing before their due time. The blue curves linger a while longer around the critical points, and arrive at the finish line in more reasonable time.

Since the period is defined as the time for one complete cycle around phase space, the energy dependence of the period is very obvious in the animation.