A Hamiltonian structure preserving discretization of Maxwell's equations driven by a ponderomotive current

In the presence of an inhomogeneous oscillatory electric field, charged particles experience a net force, averaged over the oscillatory timescale, known as the ponderomotive force. We first derive a simple one-dimensional Hamiltonian model which self-consistently couples the electromagnetic field to a plasma which experiences the ponderomotive force. We then derive a Hamiltonian structure preserving discretization of this system using a finite element exterior calculus spectral element method. The method is found to conserve the Casimir invariants of the continuous model to machine precision and the energy to the order of the splitting method used.