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.(adsbygoogle = window.adsbygoogle || []).push({});

My little system had [itex]U(r) = x\ y^2 + y\ x^2[/itex]

The equations of motion were then;

[itex]P_x ' = -y^2 - 2\ x\ y[/itex]

[itex]P_y '= - x^2 - 2\ x\ y[/itex]

[itex]p_x = m x'[/itex]

[itex]p_y = m y'[/itex]

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 [itex]x \gg y = 0[/itex] and arrived at

[itex]p_x' \approx 0[/itex]

[itex]p_y' \approx -x^2[/itex]

Which lead to [itex]x \propto t[/itex], [itex]y \propto t^4 [/itex] for large x and ~0 y. I'm guessing this approximation is bad though since according to this y shoots off as [itex]t^4[/itex] 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 [itex]x[/itex] beside the [itex]y^2[/itex] and so I didn't know how I could write it as [itex]U(r) = r^T A r[/itex]

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

Thanks in advance

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: How to analyse equations of motion advice

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**