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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 [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

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

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