Centre of Mass-Energy: Griffiths Electrodynamics 4th Ed.

[PeroK]

This is actually from Griffiths Electrodynamics 4th edition, page 546.

He defines the centre of mass-energy of a system of particles as:
$$\vec{R} = \frac{1}{E} \sum E_i \vec{r_i}$$
And gives that the total momentum of the system is:
$$\vec{P} = \frac{E}{c^2} \frac{d \vec{R}}{dt}$$
In a footnote he says that the proof of this is non-trivial and refers to a couple of papers.
I took a look at this for two particles and got:
$$\frac{d \vec{R}}{dt} = \frac{1}{(E_1 + E_2)^2}[E_1^2 \vec{u_1} + E_2^2 \vec{u_2} + E_1E_2(\vec{u_1} + \vec {u_2}) + (E_1\frac{dE_2}{dt} - E_2\frac{dE_1}{dt})(\vec{r_2} - \vec{r_1})]$$
$$= \frac{1}{(E_1 + E_2)^2}[(E_1 + E_2)(E_1 \vec{u_1} + E_2 \vec{u_2}) + (E_1\frac{dE_2}{dt} - E_2\frac{dE_1}{dt})(\vec{r_2} - \vec{r_1})]$$
$$= \frac{c^2}{E} \vec{P} + \frac{1}{E^2}[(E_1\frac{dE_2}{dt} - E_2\frac{dE_1}{dt})(\vec{r_2} - \vec{r_1})]$$
The total momentum came out but I can't see how the cross terms in the position vectors disappear. I couldn't find any specific references to this online. Any ideas about what's wrong?

 

[PeroK]

I might be able to answer my own question. If the particles are accelerating owing to an EM field that they themselves create, then energy and momentum are stored in the fields. This relation is only true when you include the energy-momentum from the fields. Unlike the classical case, using centre of mass, where it's generally true regardless of how the particles are accelerated.

It's clear now. Especially after reading the next page.

 

[Meir Achuz]

However the ancient references are both wrong.
Example 12.13 and Eq. (12.72) are also wrong, as is the paper in Ref. 21.
The only correct statement is "this is not a very realistic model."

 

[PAllen]

Of course, in the case of pure neutral particle dynamics, with energy differentials zero except for collisions, the result is trivially true ...

 

[Meir Achuz]

The equation does not hold if electromagnetic energy and momentum are included, or if there are external forces.

 

[PAllen]

The equation does not hold if electromagnetic energy and momentum are included, or if there are external forces.

Consistent with what I said ...