ng the system
        numerically, but those lines can be easily obtained in closed-form.
        Directly integrating the equations of motion is difficult, but at least
        for the case when $u_0 = 0$, we have some additional physical insight
        for this problem that we can take advantage of.  The kinetic energy,
        $T$, and potential energy, $U$, of the pendulum are given by $$T =
        \frac{1}{2}I\dot\theta^2, \quad U = -mgl\cos(\theta),$$ where $I=ml^2$
        and the total energy is $