Oscillation centres and mode coupling in non-uniform Vlasov plasma

The general coupling coefficient for three electromagnetic linear modes of an inhomogeneous and relativistic plasma is derived from the oscillation-centre viewpoint. A concise and manifestly symmetric formula is obtained; it is cast in terms of Poisson brackets of the single-particle perturbation Hamiltonian and its convective time-integral along unperturbed orbits. The simplicity of the compact expression obtained is shown to lead to a new insight into the essence of three-wave coupling and of the Manley-Rowe relations governing such interactions. Thus, the interaction Hamiltonian of the three waves is identified as simply the trilinear contribution to the single-particle (new) Hamiltonian, summed over all non-resonant particles. The relation between this work and the Lie-transform approach to Hamiltonian perturbation theory is discussed.