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Geometric optimal control utilizes tools from differential geometry toanalyze the structure of a problem to determine the control and statetrajectories to reach a desired outcome while minimizing some cost function.For a controlled mechanical system, the control usually manifests as anexternal force which, if conservative, can be added to the Hamiltonian. In thiswork, we focus on mechanical systems with controls added to the symplectic formrather than the Hamiltonian. In practice, this translates to controlling themagnetic field for an electrically charged system. We develop a basic theoryderiving necessary conditions for optimality of such a system subjected tononholonomic constraints. We consider the representative example of amagnetically charged Chaplygin Sleigh, whose resulting optimal control problemis completely integrable.

Optimal Control of Nonholonomic Systems via Magnetic Fields