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arXiv · 1212.3651

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

Erlend Grong
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Abstract

Given a submersion $π:Q \to M$ 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 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 $\mathcal{H}$ along with a new form of parallel transport of covectors vanishing on $\mathcal{H}$. 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.

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Erlend Grong. 2016-07-05. Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem. https://doi.org/10.1137/15m1008919

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