Nonradiating normal modes in a classical many-body model of matter-radiation interaction

We consider a classical model of matter--radiation interaction, in which the matter is represented by a system of infinitely many dipoles on a one--dimensional lattice, and the system is dealt with in the so--called dipole (i.e. linearized) approximation. We prove that there exist normal--mode solutions of the complete system, so that in particular the dipoles, though performing accelerated motions, do not radiate energy away. This comes about in virtue of an exact compensation which we prove to occur, for each dipole, between the ``radiation reaction force'' and a part of the retarded forces due to all the other dipoles. This fact corresponds to a certain identity which we name after Oseen, since it occurs that this researcher did actually propose it, already in the year 1916. We finally make a connection with a paper of Wheeler and Feynman on the foundations of electrodynamics. It turns out indeed that the Oseen identity, which we prove here in a particular model, is in fact a weak form of a general identity that such authors were assuming as an independent postulate.