Elementary analysis of phase space painting

Analytical expressions are given to help understand phase space painting used in H^- injection. Under simple assumptions, relations between the real space density and the phase space density are obtained for both one- and two-dimensional transverse betatron motions. The phase space painting for a beam injected with a small emittance is studied, thus allowing the time trajectories of the injection point for various real space densities to be obtained. Simple analyses of the longitudinal painting are presented. The results are of interest in connection with the use of H^- injection and phase space painting in the Compressor/Stretcher Ring of the JHP (Japanese Hadron Project)