e height where
        the total energy equals the potential energy.  As an exercise, you can
        verify that differentiating this equation with respect to time indeed
        results in the equations of motion.</p>

        <p>The particular orbit defined by $E = mgl$ is special -- this is the
        orbit that visits the (unstable) equilibrium at the upright.  This is
        the <a href="https://en.wikipedia.org/wiki/Homoclinic_orbit">homoclinic
        orbit</a> of the pendulum.</p>

      </subsub