Minimum-time trajectory tracking of an underactuated mechanical system called the Acrobot is presented. The success of the controller is demonstrated by the fact that the tracking error is reduced by more than an order of magnitude when compared to the open-loop system response. The control law is obtained by linearizing the system about the nominal trajectory and applying differential dynamic programming to the resulting linear time-varying system, while using a weighted sum of the state-deviation and input-deviation as the cost function.