Moments method in charged particle ring beam dynamics

Using the method of distribution function total moments equations have been derived for root-mean square sizes of ring charged beams with regard to betatron oscillations and energy spread of particles moving in linear electromagnetic fields. Beam steady states and their stabilities are studied. The influence of radiation effects and scattering on accumulated ions on sizes and lifetime of electron rings is considered. Adiabatic invariants have been obtained for the case of a slow variation of external electromagnetic fields, basing on which beam sizes are determined at arbitrary instants. Obtained equatiors for changes in small sizes for adiabatic variations of external fields can be used to calculate the compression of electron rings in collective ion accelerators when the sizes are determined both by the energy spread and betatron oscillattions. A conclusion has been drawn that in case of adiabatic changes in system parameters and at small number of particles the ''betatron'' and ''synchrotron'' sizes do not depend on one another and vary with time according to their invariants. As in the stationary case the square of the radial size is the sum of squares of ''synchrotron'' and ''betatron'' sizes