1.9. Ergodic system Consider a classical dynamical system with a phase space (0 < q < 1; 0 < p < 1) and equation of motion given by q(t) = q0 +t ; p(t) = p0 +α t.

a. Discuss the trajectories in the phase space when α is a rational and irrational number.

b. Show that the system is ergodic when α is irrational, i.e. the time averages of all functions f (q,p) coincide with their average on the phase space.

Hint: Use the fact that the volume of the phase space is finite to expand any function of the coordinate and momentum in a Fourier series.