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

Abstract

The Hamiltonian description of the self-consistent interaction between an electromagnetic planewave 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-MaxwellPoisson 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.

Hamiltonian derivation of the Charney–Hasegawa–Mima equation

“…Given this verification, the reduced model is naturally equipped with a Hamiltonian structure since the Poisson bracket and the Hamiltonian function are provided by the derivation process (for an example of this derivation process, see Ref. [22]).…”

Variational formulation of classical and quantum models for intense laser pulse propagation

“…3.2.6. One-dimensional electron dynamics (HP 1-5 and HF [1][2][3][4] As in the Lagrangian case, a reduced electron phase space model (HP 5) is straightforward to implement. One simply considers a distribution function on a lower-dimensional phase space.…”

“…In fact, frequently the results may be explained in terms of simple physical mechanisms which are not substantially affected by the fine details contained in the complete description of the field-particle interaction. See, for example, the single-wave model for the free electron laser [2,4] and the beam-plasma instability [5]. Therefore, we are constantly motivated to seek reduced descriptions which are both numerically tractable and simple enough to permit theoretical analysis of the results and novel experimental predictions.…”

“…[8,9]. Then, simplifying hypotheses for a given problem are incorporated directly into the variational formulation, whether by applying the hypotheses to the action, the Hamiltonian and Poisson bracket [4], or some combination of the two, as in gyrokinetic theory [10]. Employing a variational formulation poses several advantages over a reduction performed directly on the equations of motion.…”