A beam of protons with a circular cross-section of radius \(r_{0}\) moves within a linear accelerator oriented in the \(x\) direction. Suppose that the transverse momentum components \(\left(p_{y}, p_{z}\right)\) of the beam are distributed uniformly in momentum space, in a circle of radius \(p_{0}\). If a magnetic lens system at the end of the accelerator focusses the beam into a small circular spot of radius \(r_{1}\), find, using Liouville's theorem, the corresponding distribution of the beam in momentum space. Here what may be a desirable focussing of the beam in position-space has the often unfortunate consequence of broadening the momentum distribution.