

    <subsection><h1>The damped pendulum</h1>

      <p>Now let's add damping back.  You can still add torque to move the
      fixed points (in the same way).</p>

      <figure>
        <img width="90%" src="figures/pend_damped_phase.svg"/>
        <figcaption>Phase diagram for the damped pendulum</figcaption>
      </figure>

      <p>With damping, the downright fixed point of the pendulum now becomes
      asymptotically stable (in addition to stable i.s.L.). Is it also
      exponentially stable?  How