# How to analyse equations of motion advice

I was just playing about at home with some hamiltonians to see how well I could analyse them without having to solve the equations of motion, I can't think of 3 constants of motion in this particular case so I'm guessing that they aren't integerable anyway.

My little system had $U(r) = x\ y^2 + y\ x^2$
The equations of motion were then;
$P_x ' = -y^2 - 2\ x\ y$
$P_y '= - x^2 - 2\ x\ y$
$p_x = m x'$
$p_y = m y'$

My first idea was to look for any straight lines which might allow for 'free motion' but alas the only stationary point in my potential is at x=0.

Next I thought about looking at large initial x and y so that $x \gg y = 0$ and arrived at
$p_x' \approx 0$
$p_y' \approx -x^2$

Which lead to $x \propto t$, $y \propto t^4$ for large x and ~0 y. I'm guessing this approximation is bad though since according to this y shoots off as $t^4$ which will quickly become >>0 which would invalidate the approximation.

I was going to try some principal axis stuff but I quickly realized that my potential has an $x$ beside the $y^2$ and so I didn't know how I could write it as $U(r) = r^T A r$

So I'm stumped, I dunno where to go from here (or if there is anywhere to go)

My main questions are;
1. Where can I go from here
2. What books can I look up for information on how to deal with these systems