Mathematica to help for an Hamiltonian problem

Question

I have an Hamiltonian problem whose 2D phase space exhibit islands of stability (elliptic fixed points).

I can calculate the area of these islands in some cases, but for other cases I would like to use Mathematica (or anything else) to compute it numerically.

The phase space looks like that :

This is a contour plot make with Mathematica. Could anyone with some knowledge of Mathematica provide a way to achieve this ?

Answer 1

There is rather nice function in Mathematica 7, which allows one to integrate over an arbitrary complicated region. It is Boole:[True,False]$\to${1,0}. Below is just an example taken from Mathematica Documentation Center. If you have a 2D area defined by the inequality $4 x^4-4 x^2+y^2\leq 0$,

you can integrate any function $f(x,y)$ over this domain as follows:

Integrate[f[x,y] Boole[y^2 - 4 x^2 + 4 x^4 <= 0], {x, -Infinity,Infinity}, 
{y, -Infinity,Infinity}]

For example, if $f(x,y)$ is unity then it gives you the total volume of the integration domain:

In[1]:= Integrate[Boole[y^2 - 4 x^2 + 4 x^4 <= 0], {x, -Infinity,Infinity}, 
{y, -Infinity,Infinity}]
Out[1]= 8/3

In fact, you can use any condition you want, including that is determining your islands of stability. Numerical integration is also possible:

In[1]:= NIntegrate[Boole[y^2 - 4 x^2 + 4 x^4 <= 0], {x, -Infinity,Infinity}, 
{y, -Infinity,Infinity}]
Out[1]= 2.66667