Talk:Center-of-momentum frame

A comment

"If we take two particles, one of mass m1 moving at velocity V1 and a second of mass m2" - this implies that the second particle is not moving (or the first particle's speed is relative to the second's)? Or the speed just does not matter for the calculation? It would imo be more clear if this was mentioned in a few words... Mverleg (talk) 20:12, 27 April 2008 (UTC)

It's a terrible example. Someone should make a new one. --Armaetin (talk) 23:22, 4 April 2009 (UTC)

"This relativity-related article is a stub. You can help Wikipedia by expanding it."
Although there are a couple of lines about relativity in the article, isn't this article much closer to classical mechanics than it is to relativity? --Armaetin (talk) 23:22, 4 April 2009 (UTC)

Error in article

It says: "In the COM frame, the total energy of the system is the "rest energy", and this quantity (when divided by the factor c2) therefore gives the rest mass or invariant mass of the system."

This can be readily shown to be incorrect for two particles.
Let p₁ and p₁ be the four-momenta of the particles. Then in the COM by definition:
$p_{1}+p_{2}=(E_{1}+E_{2},0)$
The articles claims that this means that,
$p_{1}+p_{2}=(m_{1}+m_{2},0)$
Taking the norm of the two vectors,
$p_{1}^{2}+p_{2}^{2}+2p_{1}^{\mu }p_{2\mu }=m_{1}^{2}+2m_{1}m_{2}+m_{2}^{2}$
Since ${\vec {p}}_{1}=-{\vec {p}}_{2}={\vec {p}}$ in the COM,
$m_{1}^{2}+m_{2}^{2}+2(E_{1}E_{2}+{\vec {p}}^{2})=m_{1}^{2}+2m_{1}m_{2}+m_{2}^{2}$
$E_{1}E_{2}+{\vec {p}}^{2}=m_{1}m_{2}$
Again using the claim of the quote,
$m_{1}m_{2}+{\vec {p}}^{2}=m_{1}m_{2}$

Finally we have,
${\vec {p}}^{2}=0$

This is obviously untrue for two particles with the same energies and masses moving toward each other.

So, I'm going to delete this. —Preceding unsigned comment added by BunnyBaby (talk • contribs) 12:50, 24 October 2009 (UTC)

Answer

The articles does NOT claim that,
$p_{1}+p_{2}=(m_{1}+m_{2},0)$

This would make total energy equal to sum of rest masses, which nobody ever claimed. The rest mass of a system is obviously not the sum of the rest masses of the particles in it, and you've merely proved that, above. Straw man. It's the rest mass (invariant mass) of the SYSTEM as a whole (the mass of a container of hot gas, say, as weighed on a scale) which is equal to its total energy (in natural units). This is surely not the sum of rest masses of the hot gas molecules, since it includes their kinetic energies as well.

When the momentum terms all cancel, as they do in the COM frame by definition, the total system energy is equal to its (the total system's) rest mass (otherwise known as the system's invariant mass). The rest mass (or invariant mass) W of a system of particles is:

$\left(Wc^{2}\right)^{2}=\left(\sum E\right)^{2}-\left\|\sum \mathbf {p} c\right\|^{2}$

In the COM frame, the last term is zero. This is all analogous to the case with single particles, where only in the rest frame is the energy of the particle equal to its rest/invariant mass.

SBHarris 20:09, 24 October 2009 (UTC)

Oh, ok my bad. I've never heard of this definition before, and I'm a physics grad student, but I see that you linked to it which is good as it will help avoid further confusion. I apologize for being so trigger happy with the delete key. BunnyBaby (talk) 22:31, 24 October 2009 (UTC)

COM frame, as defined in the article, exists only for non-accelerating systems

Should that not be explicitly stated in the lead? Before e.g. "In relativity, COM frame exists for a massive system".Ilevanat (talk) 22:42, 10 March 2013 (UTC)

This is implied by the fact that the COM frame is defined as a type of inertial frame. Inertial frames (Galilean frames) are those where the laws of physics are as simple as possible, Newton's first law holds, and (by definition) the frame isn't accelerating. We can mention specifically that this implies no acceleration if you like, but it's already there in the idea that COM frames are all inertial frames. SBHarris 01:45, 11 March 2013 (UTC)

Yes, it is implied, but I dislike implications within a definition, even when they are correct. And I believe a general reader may share the sentiment, or even get confused.

Decades ago, before computers, it was important to make simple equations even simpler. To that aim, static equilibrium was used in some accelerating system problems, of course with the inertial forces included (the approach was credited to dAlembert). A simple example would be to determine maximum acceleration of a motorbike for a given friction coefficient (no limitation on engine power). When solved in the accelerating frame in which the center of mass was at rest, one could use torques about any point (as both static equilibrium conditions apply) and achieve minor simplifications in calculation. As I recall, one would casually refer to that accelerating frame as "the motorbike CM frame". Why not, the meaning was perfectly clear, and what authority might there be to forbid such usage?

That being said, I am aware of the fact that relativistic particle collisions are best described in inertial COM frames. Some authors call them "COM systems", as e.g. in the Sears and Zeemansky College Physics. (They even claim that "COM system" is a relativistic generalization of "CM system" - people will use various conventions.)--Ilevanat (talk) 23:48, 12 March 2013 (UTC)

The main issue remains unresolved (and more)

You inserted "non-accelerated" to explain inertial frames, which was not the issue (and is hardly necessary, but does no harm). The issue is whether you want to tell the readers (explicitly, at the very outset) that COM frame is defined only for a system of masses whose center of mass is not accelerating (with respect to inertial frames). So that COM frame is not defined for e.g. the system of masses in free fall.

The first lead paragraph does not clarify that. Hypothetically, a COM frame could be an inertial frame from which a momentary "snapshot" of an accelerating system of masses is taken (for whatever purpose) at the moment when velocities of the frame and of the system coincide. Only after reading the second lead paragraph, one can indirectly deduce that the system cannot be accelerating.

I believe a reasonably good article should clarify that directly and immediately.

And, after clarifying that, I think the second and the third sentence of the present first lead paragraph should be deleted, or entirely rephrased. Consider the second sentence "Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame (a velocity and a direction)." There is no definition of what "the center of momentum of a system" might be (only COM frame is defined in the preceding sentence), so how can anybody "note" anything about an undefined concept? And what "(a velocity and a direction)" might mean? How is that "direction" different from the direction of the velocity vector?--Ilevanat (talk) 00:04, 15 March 2013 (UTC)

Clearly the “velocity and direction” should have been “speed and direction” (or just velocity).

But even that has problems, as doing it this way implicitly defines the “center of momentum” as a velocity that will transform you from some frame velocity (as an galilean inertial observer) to the COM frame for the system. That v vector is v= sumP/sumM or “sum of momentum divided by total mass”. Doing that pperation gives the velocity of the COM frame for the system with regard to your observational frame (which may not have a total momentum of zero from your inertial viewpoint, in which case you are not in the system’s COM frame). The COM frame for the system S is the one where you see the total momentum of the system S = zero, which some people have said means you are somehow “at” the center of momentum for the system. But that sounds like being at a location and it is not. Being “at” the center of momentum (p=0) doesn’t mean you’re at a place in space. Rather you’re on some single particular velocity vector which has a value with regard to each of the objects in the system. It is zero with regard to the velocity of the center of mass of the system.

Some texts just define the “center of momentum” as the “center of momentum frame” and perhaps what we really wanted to say is that either the term “center of momentum” is not well-defined, or else that when it’s used, it’s interchangeable with “center of momentum frame” and “center of momentum” refers to a particular frame and does NOT refer to the total momentum of your system. The total momentum of a system is called usually just called “the total momentum of the system.”

Any reference frame can be “galilean inertial," ala Newton’s first law, or it can be somehow accelerated. "Inertial" may be a bad term for these constant-velocity reference systems, as systems falling or moving in uniform G fields are said colloquially to be moving “inertially” (they have an inertial trajectory like the space shuttle in orbit) but are certainly not galilean, but are rather coordinate-accelerated.

But as to your objection (if I understand it). A snapshot at time t of a system in its COM frame means total momentum = 0 at that time t. A co-moving inertial observer (matched for v at that instant) at that time also gets p=0. In theory, the system CAN be accelerating with regard to the inertial observer, and in that case, its instantaneous COM frame will change to some other frame (with regard to inertial) at later t.

Now, such a system is indeed not the kind for which we define as a COM frame in this article. But that’s our fault. It’s not an impossible idea. Such an accelerating system will have an accelerating COM frame for which p is always zero (just calculate the frame for which system p = 0 at any time). A system in free fall is a good example. It is only the additional fact that we (apriori) limited ourselves to calculating “inertial” COM frames plus the fact that momentum is conserved (i.e., there are no outside forces acting on on the system over time, such as gravity or anything else) that guarantees that a system momentum = 0 at shapshot NOW will be the same system momentum = 0 at snapshot later at time t (where P is still zero, since its value is conserved over time and was zero before). In that case, the COM frame of the system must move along inertially with constant velocity, or else system p would change with regard to our initial. But otherwise, if some force acts on the system, it doesn’t.

So the COM frame for a system (defined completely generally) isn’t necessarily a galilean frame moving at constant V. Without this additional constraint (momentum conserved and no momentum transferred in or out of the system, which means no outside forces), it might not be. But conversely, if we start out by saying we’re only going to consider “inertial” COM frames that move in a galilean way (with constant V) then that implies no outside forces all by itself, and thus no gravity or other “forces.” And so of course the system center of mass is not accelerating.

Perhaps the mistake was not defining COM frames very generally at first, allowing for accelerating COM frames (with respect to an inertial frame) for accelerating systems that might indeed accelerate with regard to some initial inertial frame in which the system's momentum was once null. Instead, we immediately specified that we are only talking about the subset of systems with COM frames that are themselves galilean (as many texts do), or the subset of galilean COM frames, if you will. Maybe not necessary. SBHarris 02:25, 17 March 2013 (UTC)

Thank you for such elaborate answer. I believe it shows that we are essentially in agreement on the subject.

As for the option of non-intertial COM frames, I am not aware that they are extensively used today, but I am not a "COM frame" expert (and do remember the above mentioned "dAlembert" accelerating COM frame from some old textbook decades ago).

Therefore, I think that a reasonable (and cautious) approach would be to start the article with e.g.:

"In physics, a center-of-momentum frame (zero-momentum frame, or COM frame) of a system is any frame in which the center of mass of the system is at rest (has zero velocity) and hence the system has zero total linear momentum. It is generally used to simplify analysis of a system whose center of mass is not accelerating with respect to inertial frames, so its COM frame is any inertial frame having the same velocity as its center of mass. Such usage is implied in the rest of this article."

I think this might be enough for the first paragraph, and the second paragraph could be left as is. Of course, you are welcome to polish my English and make any other modifications. As for that "center of momentum itself" discussion, I do not feel that it is necessary. But if you feel differently, consider the formulation: The phrase "center of momentum" itself is frequently interpreted to have the meaning of "COM frame".

I am here assuming that you will do editing of the article. I would rather not do it, because of some rather controversial experience with other editors in some of my previous edits and/or suggestions. I have almost developed a phobia that some people are searching for my edits in English Wiki just in order to change them. I like to keep things precise, explicite, up-to-the-point and yet as simple and understandable as possible, but some prefer complicated and "sophisticated"... Actually, I came here from the Center of mass article, where you can see differences in approach (my suggestions remained only at the talk page). And if you care to see what really pissed me off, you can visit the Net force talk page.--Ilevanat (talk) 23:40, 17 March 2013 (UTC)

Relativistic 3-surface for summing momentum

In the section on special relativity, a subtlety might be worth pointing out: that the 3-surface in 4-space over which the momentum is summed can vary. Conservation of momentum actually makes the total momentum independent of the choice of this surface, and it can have any shape as long as its extremities are spacelike infinity. Without conservation of momentum, this would not hold. For the purposes of the article, it would be sufficient to mention that any spacelike 3-plane suffices for the summation, and that conservation of momentum is necessary for the COM to be well-defined. — Quondum 18:17, 27 September 2013 (UTC)

I've tweaked §§ Special relativity to mention that the system must be isolated, without mentioning integration surfaces. This pretty much addresses my concern without getting complicated . —Quondum 00:28, 13 July 2014 (UTC)

Relativistic Zero-Momentum frame

In the section on special relativity should treat a system of more than one particle, in relative motion. At least the difficulties that one may encounter should be addressed.

Aoosten (talk) 21:08, 22 January 2021 (UTC)