Submersions; Hamiltonian systems; Rolling manifolds
Abstract :
[en] Given a submersion $\pi:Q \to M$ with an Ehresmann connection~$\calH$, 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 $\calH$ along with a new form of parallel transport of covectors vanishing on $\calH$. 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.
Disciplines :
Mathematics
Author, co-author :
GRONG, Erlend ; University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit
External co-authors :
no
Language :
English
Title :
Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem
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