  The Hamiltonian description of the self-consistent interaction between an
electromagnetic plane-wave and a co-propagating beam of charged particles is
considered. We show how the motion can be reduced to a one-dimensional
Hamiltonian model (in a canonical setting) from the Vlasov-Maxwell Poisson
brackets. The reduction to this paradigmatic Hamiltonian model is performed
using a Lie algebraic formalism which allows us to remain Hamiltonian at each
step of the derivation.
