The underactuated system under consideration is a magnetically-suspended, differential drive robot utilizing a winch system to articulate a suspended mass. A dynamic model of the system is first constructed, and then a nonlinear, infinite-dimensional optimization algorithm is presented. The system model uses the principles of kinematic reduction to produce a mixed kinematic-dynamic model that isolates the modeling of the system actuators from the modeling of the rest of the system. In this framework, the inputs become generalized velocities instead of generalized forces facilitating real-world implementation with an embedded system. The optimization algorithm automatically deals with the complexities introduced by the nonlinear dynamics and underactuation to synthesize dynamically feasible system trajectories for a wide array of trajectory generation problems. Applying this algorithm to the mixed kinematic-dynamic model, several example problems are solved and the results are tested experimentally. The experimental results agree quite well with the theoretical showing promise in extending the capabilities of the system to utilize more advanced feedback techniques and to handle more complex, three-dimensional problems.