Question about energy in C-O-M frame and Lab frame.

[sukho]

Hi, I just register to this site.
I'm reading a famous classical mechanics textbook. and it state that
'Since the spatial momentum in the C-O-m frame is zero, there is clearly more energy, p⁰, in this frame than in the laboratory frame.'

I think the energy in the center of momentum(C-O-M) frame should be less than in all other frame.
However, I've still not sure that it's right or wrong because this textbook is very famous.

Thanks.
PS. Should I states the name of the text?

 

[tiny-tim]

welcome to pf!

hi sukho! welcome to pf!

that certainly looks wrong, but i wonder what the context is

can you provide a link?​

 

[Bill_K]

sukho, you are correct, it should be less. It's a well-known theorem in mechanics that the kinetic energy of a system of particles can be written as a sum of two terms: T = ½ MV² + ∑ ½ m_iv_i², where M is the total mass, V is the velocity of the center of mass, and v_i are the velocities of the individual particles with respect to the center of mass. Clearly this is a minimum when V = 0.

 

[Bill_K]

Oops, just noticed this is the relativity group. Ok, even easier. The total 4-momentum of the system of particles is P = (E/c, p). This has norm P·P = (E/c)² - p.p. Since the norm must be the same in all reference frames, E will be minimum in the center of mass frame where p = 0.

 

[Meir Achuz]

It would help, if you can give the exact reference, to see the context of that quote.
The book could be talking about available COM energy, which is larger in a colliding beam accelerator.

 

[Meir Achuz]

It would help, if you can give the exact reference, to see the context of that quote.
The book could be talking about available COM energy, which is larger in a colliding beam accelerator.

 

[sukho]

Hi, everyone.
The book I read is 'Classical Mechanics' 3ed by 'Goldstein Poole & Safko' on page 302 line 16
and if u see the footnote on this page, it's kind of support the context. I also confuse about the footnote too.
Thanks.

 

[tiny-tim]

hi sukho!

i don't think that's available online …

can you post a picture? ​

 

[sukho]

Hi. I attached pdf file of the context.
The attached file is a page 302, Goldstein. and the context is on line 16.
Thanks.

 

[tiny-tim]

hmm … it refers to equations (7.79) and (7.80) which are on the previous page

 

[Meir Achuz]

Hi. I attached pdf file of the context.
The attached file is a page 302, Goldstein. and the context is on line 16.
Thanks.

Reading it in context, it does not make too much sense. I am not sure what their p^0 refers to.
Maybe just ignore that sentence.

 

[sukho]

Reading it in context, I see (as I thought) that G is referring to the sum of the energies of the two particles. Since E^2-p^2 is invariant, this is largest when p-0.

Why it's not minimum energy when p=0, as Bill_K said? it's minus sign before p^2.
and the I've attached the previous pages.
Thanks.

 

[sukho]

I think I'll just ignore that context.
Thank you guys.