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

TL;DR

This paper develops a method to solve Hamiltonian systems on a base manifold by lifting to a total space with an Ehresmann connection, and applies it to optimal control problems like rolling manifolds without slipping.

Contribution

It introduces a novel approach to lift Hamiltonian systems via Ehresmann connections and describes solutions using a generalized magnetic force, extending optimal control theory.

Findings

Solutions of lifted Hamiltonian systems relate to original systems through a generalized magnetic force.

Application of Pontryagin maximum principle yields insights into normal and abnormal extremals.

Demonstration on the optimal control problem of one Riemannian manifold rolling on another.

Abstract

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 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.