Discuss the implications of Liouville's theorem on the focusing of beams of charged particles by considering the following simple case. An electron beam of circular cross section (radius R0) is directed along the z-axis. The density of electrons across the beam is constant, but the momentum components transverse to the beam (px and py) are distributed uniformly over a circle of radius p0 in momentum space. If some focusing system reduces the beam radius from R0 to R1, find the resulting distribution of the transverse momentum components. What is the physical meaning of this result? (Consider the angular divergence of the beam.)