Hamiltonian description of self-consistent wave-particle dynamics in a periodic structure

TL;DR

This paper presents a Hamiltonian framework for classical electrodynamics that accurately models wave-particle interactions in periodic structures, ensuring energy conservation and excluding self-acceleration.

Contribution

It introduces a Hamiltonian formulation satisfying Maxwell's equations and the Lorentz force, incorporating eigenfunction expansion and electrostatic coupling for self-consistent dynamics.

Findings

Conserves energy and momentum in wave-particle interactions.

Excludes self-acceleration in the dynamics.

Provides a complete Hamiltonian model for radiating electrons.

Abstract

Conservation of energy and momentum in the classical theory of radiating electrons has been a challenging problem since its inception. We propose a formulation of classical electrodynamics in Hamiltonian form that satisfies the Maxwell equations and the Lorentz force. The radiated field is represented with eigenfunctions using the Gel'fand $\beta$-transform. The electron Hamiltonian is the standard one coupling the particles with the propagating fields. The dynamics conserves energy and excludes self-acceleration. A complete Hamiltonian formulation results from adding electrostatic action-at-a-distance coupling between electrons.