  We present solutions to the classical Liouville equation for ergodic and
completely integrable systems - systems that are known to attain equilibrium.
Ergodic systems are known to thermal equilibrate with a Maxwell-Boltzmann
distribution and we show a simple derivation of this distribution that also
leads to a derivation of the distribution at any time t. For illustrative
purposes, we apply the method to the problem of a one-dimensional gravitational
gas even though its ergodicity is debatable. For completely integrable systems,
the Liouville equation in the original phase space is rather involved because
of the group structure of the integral invariants, which hints of a gauge
symmetry. We use Dirac's constrained formalism to show the change in the
Liouville equation, which necessitates the introduction of gauge-fixing
conditions. We then show that the solution of the Liouville equation is
independent of the choice of gauge, which it must be because physical
quantities are derived from the distribution. Instead, we derive the solution
to the classical Liouville equation in the phase space where the dynamics
involve ignorable coordinates, a technique that is akin to the use of the
unitarity gauge in spontaneously broken gauge theories to expose the physical
degrees of freedom. It turns out the distribution is time-independent and
precisely given by the generalized Gibbs ensemble (GGE), which was solved by
Jaynes using the method of constrained optimization. As an example, we apply
the method to the problem of two particles in 3D interacting via a central
potential.
