# Solving the Liouville equation for classic harmonic oscillator via method of characteristics

I'm interested in solving Liouville's equation

$$\frac{\partial W}{\partial t} + \{ H,W\}=0$$

with $$H=\frac{p^2}{2m} + \frac{1}{2}m\omega^2x^2$$

using the method of characteristics. However I can't seem to find any resources on how to do it. Would appreciate any pointers.

You can start by reading the wikipedia article on the method of characteristics. You will see that in our case the tangent of the characteristic is $(1,-\partial H/\partial q,\partial H/ \partial p)$ where the components are in order $t,p,q$. When you formulate the equation of the characteristic, you will actually find out you get equations of motion of a single particle with this Hamiltonian.
Once you solve these equations for $t(s),p(s),q(s)$ with a parameter $s$, you will know that a solution will satisfy $W(t(s),p(s),q(s))=const.$. This is actually very intuitive - the probability distribution $W$ flows with the particles evolving according to their equations of motion.
The solutions will be determined by two constants corresponding to e.g. phase $\phi$ and amplitude $A$ of the oscillation, you will have $t^{\phi, A}(s),p^{\phi,A}(s),q^{\phi,A}$. Depending on the kind of initial data or problem you are solving, you should be able to get the solution by integrating over the family of characteristics parametrized by $A,\phi$ with a certain weight function $f(A,\phi)$ at a certain $s$ and then get the solution on the rest of the $t,p,q$ space by going through all values of $s$.
Say you are given an initial $W_{0}(p,q)$ and want to evolve it through time. Then you have to find the map $p,q \to \phi,A$ substitute it into $W_0$ and this will be your weight function over the $\phi, A$.