At time \(t=0\) a large number of particles, each of mass \(m\), is strung out along the \(x\) axis from \(x=0\) to \(x=\Delta x\), with momenta \(p_{x}\) varying from \(p=p_{0}\) to \(p=p_{0}+\Delta p\). No forces act on the particles and they do not collide.

(a) Show that the points representing these particles fill a rectangle in the \(x, p_{x}\) phase plane, and sketch it identifying the four points at the corners of the rectangle with their positions and momenta.

(b) Sketch the locations of the same particles in the phase plane some time \(t_{1}\) later, where \(t_{1}>m x_{0} / p_{0}\).

(c) What then is the shape of the area on the phase plane occupied by all of these particles?

(d) Prove that the area of the occupied region at \(t_{1}\) is the same as it was at \(t=0\). (Note that if the number of points and the area are both unchanged, then the average density of points is also unchanged, in accord with the Liouville theorem.)