How to Solve Hamilton-Jacobi Equation for a Particle in a Given Potential Field?

[fluidistic]

Homework Statement

Using Hamilton-Jacobi's equation, find the motion equations and the trajectory of a particle in the field $U (\vec r )=\frac{m \omega}{2} (x^2+y^2)$.

Homework Equations

$H(q_1,...q_s,\frac{\partial S _0}{\partial q_1},...,\frac{\partial S _0}{\partial q_s})=E$.
Where $S_0$ is the abbreviated action, apparently worth $\int \sum _i p_i dq_i$.

The Attempt at a Solution

I'm self studying CM for a final exam on 7th of March (I can choose not to take the exam if I feel not ready but this would be a pain for the next semester). I've searched google and this forum for similar problems but didn't find anything that could really help me. What I found was heavy abstract math rather than applied problems like this one.
Ok my attempt: I don't know if I can assume that the motion is in 3 dimensions or 2. Let's take 2 for the sake of simplicity.
I use Cartesian coordinates so that the velocity is $\dot {\vec r }=\dot x \hat i + \dot y \hat j$. Thus the total energy of the system is worth $\frac{m (\dot x ^2 + \dot y ^2 )}{2}+\frac{m \omega}{2} (x^2+y^2)$.
The Hamiltonian is then $H=E=\frac{1}{2m} (p_x ^2+ p_y ^2)+\frac{m \omega }{2}(x^2+y^2)$.
This is where I'm stuck. It looks like I must express the generalized momenta into the abbreviated actions, but I don't know how to do so. Any tip is greatly appreciated!

 

[sgd37]

use the poisson brackets

 

[fluidistic]

use the poisson brackets

Thanks for your reply.
Somehow I don't see how this can help me. I'd appreciate if you could specify a bit more.
I've checked into http://en.wikipedia.org/wiki/Hamilton–Jacobi_equation (there are several examples), in Landau's book and in Goldstein's book. None seems to involve any Poisson bracket.
I am not 100% sure, but it seems that from $E=\frac{1}{2m} (p_x ^2+ p_y ^2)+\frac{m \omega }{2}(x^2+y^2)$ I can replace $p_i$ by $\frac{\partial S _0 }{\partial q_i}$. I doubt if it shouldn't be $\frac{\partial S }{\partial q_i}$ or $\frac{\partial S _i }{\partial q_i}$ instead.


This would make $\frac{1}{2m} \left [ \left ( \frac{\partial S _0}{\partial x} \right )^2+\left ( \frac{\partial S _0}{\partial y} \right )^2 \right ] +\frac{m\omega (x^2+y^2) }{2}=E$. I don't know what to do next. Should I solve for S?

Wikipedia takes the example of a particle in spherical coordinates. Laudau threats the same problem. However Landau totally depreciated a term in the potential function "because it's not interesting physically" or something like that. Wikipedia however do not simply get rid of a term because it's of few interest. I really don't understand what to do next...