Hamiltonian description of a self-consistent interaction between charged particles and electromagnetic waves.

The Hamiltonian description of the self-consistent interaction between an electromagnetic plane wave and a copropagating 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 preserve the Hamiltonian character at each step of the derivation.