 if you tell me $\theta$ I can determine one of two possible
        values for $\dot\theta$, and the solution has all of the richness of the
        black contour lines from the plot.  This equation has a real solution
        when $\cos(\theta) > \cos(\theta_{max})$, where $$\theta_{max} =
        \begin{cases} \cos^{-1}\left( -\frac{E_0}{mgl} \right), & E_0 < mgl \\
        \pi, & \text{otherwise}. \end{cases}$$ Of course this is just the
        intuitive notion that the pendulum will not swing above th