In this talk we discuss recent work on a geometric approach to certain optimal control problems and the relationship of the solutions of these problems to some classical integrable dynamical systems. These systems include the rigid body equations, geodesic flows on the ellipsoid and the Toda lattice flows. We discuss the Hamiltonian structure of these systems and relate our work to some classical work of Moser. We also discuss the link to discrete dynamics and symplectic integration. The work is joint with Francois Gay-Balmaz and Tudor Ratiu.