A general formalism is developed for the analysis of the transverse coherent motion of beams in a strong focusing storage ring. Both coherent and incoherent forces arising from either direct or wake fields are included in the analysis. The beams, in the absence of coherent motion, may move along different orbits which have both time and/ or space crossings. The analysis applies to beams with arbitrary longitudinal density variations.nWe recover the well known results for a single coasting or bunched beam, but-now generalized to include the relatively minor influence of an A.G. structure. Our pre¬vious results for two bunched beams which were strongly dependent on the assumed azimuthal symmetry of the machine, are now significantly modified. The analysis leads to simple ways of evaluating the coherent motion, for any given structure;and for bunched (electron) and unbundled (proton) beams.
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