Integration when transform to center of mass frame

In summary, the conversation is about solving an integral with limits involving vector quantities in the center of mass frame. The suggested approach is to use substitution and evaluate the integral in the center of mass frame, but there is confusion about the correct limits for the variables. One person suggests a different approach, which the other person agrees is a better way to solve the integral. However, the original person is still unsure about the limits in the center of mass frame.
  • #1
babylonia
11
0
Hi,

I am having some difficulty doing the integral
∫d[itex]^{3}[/itex]v1d[itex]^{3}[/itex]v2 | [itex]\overline{v1}[/itex]-[itex]\overline{v2}[/itex]|, where u1[itex]\leq[/itex]|v1|,|v2|[itex]\leq[/itex]u2, and [itex]\overline{v1}[/itex] means vectors.

It seems better to evaluate it in the center of mass frame, by substitution [itex]\overline{v1}[/itex]+[itex]\overline{v2}[/itex]=[itex]\overline{V}[/itex], and [itex]\overline{v1}[/itex]-[itex]\overline{v2}[/itex]=2[itex]\overline{v}[/itex],
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.
 
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  • #2
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.
 
  • #3
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.
 
Last edited:
  • #4
babylonia said:
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.
 
  • #5
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.


haruspex said:
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.
 

What is the center of mass frame?

The center of mass frame is a reference frame in which the total momentum of a system is zero. In this frame, the center of mass of the system remains at rest or moves with a constant velocity.

Why is it important to transform to the center of mass frame when integrating?

Transforming to the center of mass frame simplifies the integration process as it removes the contribution of the overall motion of the system. This allows for easier calculations and a clearer understanding of the underlying physics.

How is the center of mass frame determined for a system?

The center of mass frame is determined by finding the point at which the total momentum of the system is zero. This can be calculated using the positions and velocities of each individual particle in the system.

Can the center of mass frame change during an integration?

No, the center of mass frame remains constant throughout the integration process. This is because the center of mass is a property of the system and does not depend on the reference frame being used.

What are the advantages of using the center of mass frame in integration?

Using the center of mass frame in integration can simplify calculations and make it easier to analyze the motion of a system. It also allows for a clearer understanding of how different forces and interactions affect the overall motion of the system.

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