  Geometric optimal control utilizes tools from differential geometry to
analyze the structure of a problem to determine the control and state
trajectories to reach a desired outcome while minimizing some cost function.
For a controlled mechanical system, the control usually manifests as an
external force which, if conservative, can be added to the Hamiltonian. In this
work, we focus on mechanical systems with controls added to the symplectic form
rather than the Hamiltonian. In practice, this translates to controlling the
magnetic field for an electrically charged system. We develop a basic theory
deriving necessary conditions for optimality of such a system subjected to
nonholonomic constraints. We consider the representative example of a
magnetically charged Chaplygin Sleigh, whose resulting optimal control problem
is completely integrable.
