  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 derive a one-dimensional Hamiltonian model
which self-consistently couples the electromagnetic field to a plasma which
experiences the ponderomotive force. We derive a family of structure preserving
discretizations of the model of varying order in space and time using
conforming and broken finite element exterior calculus spectral element
methods. In all variants of our discretization framework, 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.
