How to analyse equations of motion advice

[genericusrnme]

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 don't know 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

Thanks in advance

 

[mark726]

Thank you for sharing your findings and questions about your Hamiltonian system. As a fellow scientist, I would like to offer some insights and suggestions.

Firstly, it is important to note that not all Hamiltonian systems are integrable. In fact, it is quite rare for a system to have three constants of motion, which are necessary for integrability. So, it is possible that your system might not be integrable, as you suspected.

However, you could still try to find other constants of motion. One approach could be using Noether's theorem, which states that for every continuous symmetry in a system, there exists a corresponding constant of motion. In your case, you could look for symmetries in your potential function U(r) and see if they lead to any constants of motion.

Another approach could be to use the Hamilton-Jacobi method, which allows for the separation of variables in the Hamiltonian equations of motion. This method can be useful in finding constants of motion for non-integrable systems.

In terms of resources, there are many books on Hamiltonian systems and their analysis. Some recommended titles are "Classical Mechanics" by Herbert Goldstein, "Hamiltonian Dynamical Systems" by Richard Cushman and John Bates, and "Chaos and Integrability in Nonlinear Dynamics" by Michael Tabor.

I hope these suggestions will help you in your further explorations of your Hamiltonian system. Keep up the good work and happy analyzing!