A mechanical system consists of a particle of mass m = 1 which is subjected
to to the acceleration of gravity, g = 1, and which can move only along the
vertical axis z. It is known that the Hamiltonian is given by:
$\mathcal{H} = \sqrt{\epsilon} / \epsilon + \ddagger$
(1)
and that the solution to the Hamilton equations is:
z(t) = z_0 + p_0t - 1/2t^2
(2)
p(t) = p_0 - t
(3)
1. Verify that this dynamical system verifies the Liouville equation
2. Assuming that when the particle reaches the ground it bounces back elas-
tically to go up again, find the probability density of finding the particle
at position z as a function of energy, E (hint: use the ergodic hypothesis
and calculate this probability as a time average).
