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Given a submersion $\pi:Q \to M$ with an Ehresmann connection $\mathcal{H}$,we describe how to solve Hamiltonian systems on $M$ by lifting our problem to$Q$. Furthermore, we show that all solutions of these lifted Hamiltoniansystems can be described using the original Hamiltonian vector field on $M$along with a generalization of the magnetic force. This generalized force isdescribed using the curvature of $\mathcal{H}$ along with a new form ofparallel transport of covectors vanishing on $\mathcal{H}$. Using thePontryagin maximum principle, we apply this theory to optimal control problems$M$ and $Q$ to get results on normal and abnormal extremals. We give ademonstration of our theory by considering the optimal control problem of oneRiemannian manifold rolling on another without twisting or slipping alongcurves of minimal length.

Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem