Integration when transform to center of mass frame

[babylonia]

Hi,

I am having some difficulty doing the integral
∫d^{3}v1d^{3}v2 | \overline{v1}-\overline{v2}|, where u1\leq|v1|,|v2|\lequ2, and \overline{v1} means vectors.

It seems better to evaluate it in the center of mass frame, by substitution \overline{v1}+\overline{v2}=\overline{V}, and \overline{v1}-\overline{v2}=2\overline{v},
However, I'm not sure what are the correct integral limits for |V| and |v|.

Can anybody give me some help? Really appreciate deeply.

Thanks.

 

[haruspex]

I tried fixing a point at a from the origin, letting another point range over a shell radius r < a, integrating the distance between them. Wasn't too difficult.

 

[babylonia]

Hi,

Thanks for your reply, but I'm not sure you are replying to my post? What you mentioned does not seem to be the thing I was asking? I'm more interested to know the limits of integral variables in CM frame instead of working out this particular integral. Could you tell more details even if you think it very easy?

Thanks.

 

[haruspex]

I'm more interested to know the limits of integral variables in CM frame instead of working out this particular integral.

Ah, yes. I thought about that for two seconds and decided it was so nasty it couldn't be the right approach. Imagine picking a midpoint near one of the shell boundaries. The range for the separation then depends in a very awkward way on the orientation of the line.
If I've convinced you of that, try my way and let me know if you need more help.

 

[babylonia]

Thanks for you reply.

I have no difficulty working out this particular integral, since I actually picked an easy form just to present my question about those limits. My problem is to know the full expression of those limits.

Thanks any way.

Ah, yes. I thought about that for two seconds and decided it was so nasty it couldn't be the right approach. Imagine picking a midpoint near one of the shell boundaries. The range for the separation then depends in a very awkward way on the orientation of the line.
If I've convinced you of that, try my way and let me know if you need more help.