A beam of particles is moving along an accelerator pipe in the z direction. The particles are uniformly distributed in a cylindrical volume of length L o (in the z direction) and radius Ro . The particles have momenta uniformly distributed with pz in an interval po Opt and the transverse momentum pi inside a circle of radius Ap 1 . To increase the particles' spatial density, the beam is focused by electric and magnetic fields, so that the radius shrinks to a smaller value R. What does Liouville's theorem tell you about the spread in the transverse momentum p 1and the subsequent behavior of the radius R