section>

<!--  these are correct, but take away from the flow.  move to an appendix?

<subsubsection><h1>Trajectory calculations</h1>

<p> For completeness, I'll include what it would take to solve for $\theta(t)$,
even thought it cannot be accomplished using elementary functions.  Feel free to
skip this subsection.  We begin the integration with

\begin{gather*} \frac{d\theta}{dt} = \sqrt{\frac{2}{I}\left[E +
mgl\cos\left(\theta(t)\right)\right]} \\ \int_{\theta(0)}^{\theta(t)}
\frac{d\theta}{\sqrt{\frac{