Re: A difficult hamiltonian problem


"GLari" <turbolenceATgmailDOTcom> wrote in message
news:42e8e24c$1_2@xxxxxxxxxxxxxxxxxx
> Hi,
>
> Consider the following functions in coordinates (p1,p2,x,y):
>
> h1 = 1/2 ( p12 + p22) + 1/2 A (x2 + y2) ? 1/32 y4 -3/16 x2 y2 ? 1/32 x4
> ? B/(2 y2)
>
> h2 = (p1 p2 + x y (-1/8 x2 - 1/8 y2 +A))2 ? B p12/y2 + 1/4 B x2.
>
> It is easy to see that the Poisson parenthesis between h1 and h2 is 0:
>
> {h1,h2} = 0.
>
> My problem is to find two (real or complex) functions r and s (in coord
> p1, p2, x, y) verifying these two equations:
>
> {s,h1} = {r,h2}
>
> {s,h2} = r {r,h2} ? s {r,h1}.

Interesting (and probably difficult) problem. The only thing I could see is
that {s,h2} = r^2 {s/r, h1}, but you probably already knew that.

Where did this problem come from? It looks like two interacting oscillators,
except that there is a negative singularity at y=0, and of course at
x=infinity and y=infinity.

Are there any stable fix points?

Have you tried plugging in a specific Ansatz, i.e. something like r = p1 +
c1 x + c2 x y + c3 y + c4 / y, and similar for s. Perhaps the Ansatz needs
to contain more terms, but using Mathematica or Maple this is a very quick
task.

-Michael.


.

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