Given a submersion pi : Q -> M with an Ehresmann connection H, we describe how to solve Hamiltonian systems on M by lifting our problem to Q. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on M along with a generalization of the magnetic force. This generalized force is described using the curvature of H along with a new form of parallel transport of covectors vanishing on H. Using the Pontryagin maximum principle, we apply this theory to optimal control problems M and Q to get results on normal and abnormal extremals. We give a demonstration of our theory by considering the optimal control problem of one Riemannian manifold rolling on another without twisting or slipping along curves of minimal length.