An acrobot is a nonlinear and underactuated system with two degrees of freedom. The control of its swing-up motion using only the actuated joint at the elbow of the system is a challenging problem that has been tackled using different optimal control approaches. This paper uses the Discrete Mechanics and Optimal Control (DMOC) method to solve the problem, allowing the system to start from its position pointing down and end at the unstable upward position. The formulation of the trajectory optimization problem for the acrobot is presented, followed by the full solution, including all the mathematical details. The results compare the obtained solution with a more classical one, showing the effectiveness of the DMOC approach on the control signal.