Explicit reduction of N-body dynamics to self-consistent particle–wave interaction

The one-dimensional (1-D) spatially periodic system of N classical particles, interacting via a Coulomb-like repulsive long-range force, is studied using classical mechanics. The usual Bohm–Gross dispersion relation for the collective modes is obtained in the absence of quasiresonant particles. In the presence of R quasiresonant particles, the evolution equations for M long-wavelength modes are coupled to the particles’ motion through a self-consistent wave–particle Hamiltonian. The wave–particle Lagrangian is derived from the full N-body Lagrangian. The derivation relies on an explicit scale separation argument and avoids the use of kinetic theory and continuous medium formalism.