Talk:Momentum

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edit·history·watch·refresh Stock post message.svg To-do list for Momentum:

Here are some tasks awaiting attention:
  • Cleanup: Rewrite lead to summarize contents properly
  • Expand:

    Newtonian mechanics:

    Add derivation of conservation of linear momentum from that of angular momentum

    Relativitistic mechanics:

    Add momentum of field and stress-energy tensor

    Quantum mechanics:

    Relation to canonical momentum
    Commutation relations

    Electromagnetism:

    Manifestly covariant formulation

    History of the concept:

    Clarify Galileo's contribution
    Add Huygens
    Add post-Newton physics

    Crystal momentum

    Units (SI, English, natural)
  • Verify:

    History of the concept:

    Add sources for judgements or remove

Introduction[edit]

The introductory paragraph is somewhat obscure and is not meaningful from a lay perspective; this section should give a good physical insight into momentum and why it is important, not simply give a mathematical definition and some scientific statements which happen to be true. Metsfanmax (talk) 18:59, 2 July 2009 (UTC)

I agree. The first paragraph of an article should be comprehensible to someone with with minimal experience reading descriptions of mathematical concepts. Otherwise we risk discouraging people from learning about this, an article that's very important to WikiProject Mathematics. Saprophage (talk) 17:12, 15 May 2010 (UTC)
I agree. This is a regular problem throughout the science and mathematics articles in Wikipedia. It should be possible for someone doing their high-school homework to look up a subject and get a comprehensible explanation. Although the present policy in WP is purist and technically impeccable, these articles are virtually useless for all but the cognescenti. JMcC (talk) 09:58, 22 July 2012 (UTC)
I agree too. I have noticed a tendency in articles in the physical sciences for a useful, clear and simple introduction to evolve, suitable for teenagers and the uninitiated; and then someone comes along and inserts a statement about relativity, or the speed of light, or quantum mechanics. This statement is technically correct (as far as I can tell) but arguably inappropriate in the introduction. The best defence we have against this tendency appears to be WP:Make technical articles understandable so I guess we need to wield that as effectively as possible so these articles have something useful for the uninitiated. Dolphin (t) 12:56, 22 July 2012 (UTC)

Accessibility[edit]

We had a complaint at WikiProject Physics (see Simple explanations, please) that this article is not easy for someone with no background in physics to read. There are some good efforts in this article to explain things simply, but often they come after more difficult concepts like frame dependence or Noether's theorem. I am reorganizing the article to put more difficult material later. RockMagnetist (talk) 16:27, 22 July 2012 (UTC)

There may have been a thread called Simple explanations, please at some time in the past, but it isn't there now. I suggest you find a diff or two to show Users exactly what was said in the complaint. Dolphin (t) 11:59, 23 July 2012 (UTC)
Sorry, the link should have been to the talk page. It is fixed now. RockMagnetist (talk) 05:53, 24 July 2012 (UTC)

Original research[edit]

The section Momentum#History of the concept has the appearance of original research, with many judgements stated without backing by an independent source. That is particularly true of the discussion of Newton's mechanics, with statements like this: "The extent to which Isaac Newton contributed to the concept has been much debated. The answer is apparently nothing, except to state more fully and with better mathematics what was already known. ". There is a lot of good material here, and I hope someone can find suitable third-party sources for it. RockMagnetist (talk) 16:44, 30 July 2012 (UTC)

In its current state it may be incomplete, the talkheader items above show how. However, I don't see any justification for the OR tag right now so removing it. 76.180.168.166 (talk) 18:46, 14 February 2013 (UTC)
I gave specific reasons just above your comment. You should discuss them before deciding that the tag is not justified. RockMagnetist (talk) 19:00, 14 February 2013 (UTC)
I only quickly read the section, so I don't wanna judge if it still contains OR, but the particular passage you quoted above is not in the article any more. Maybe you want to check it again. — HHHIPPO 19:18, 14 February 2013 (UTC)
True, I hadn't noticed that passage was missing. I remember finding that not all the qualitative statements were really backed up by the references, but I'm not sure which were at fault. I'll go ahead and remove the tag. RockMagnetist (talk) 19:28, 14 February 2013 (UTC)

Merger proposal[edit]

The article Kinetic momentum is based on a backwards interpretation. Being equal to mv, kinetic momentum does not contain the vector potential, while the canonical momentum does. It is also known as mechanical momentum (see the Goldstein and Jackson references in the first paragraph of Momentum#Generalized momentum). Since that is the main subject of the article momentum, this article should be merged into Momentum. RockMagnetist (talk) 21:45, 1 August 2012 (UTC)

The article Kinetic momentum confuses me. The words suggest considering the particle momentum as separable into the momentum of the classical "bare" (uncharged) particle, and the momentum of the mass–energy of the coupled EM field. The latter would related to some integral over space of E and H, and not A. The mathematical formulae, on the hand, suggests the kinetic momentum as being related to the vector potential A, which can have an arbitrary uniform vector field added to it, so the so-called kinetic momentum is arbitrary. A more sensible definition seems to be given here, not that I'd know how to translate this into maths, but would guess it matches the formulae. In short, (it seems to me at least) that that article is inconsistent, and should be corrected or deleted. — Quondum 02:02, 2 August 2012 (UTC)

As far as I know, the following is the correct terminology (copied from kinetic momentum):

  • \bold{P} \,\! is the canonical momentum,
  • \bold{\Pi} = m\bold{v} is the kinetic (mechanical ?) momentum,
  • e\bold{A} is the potential momentum (no standard symbol),

which fits in with the linked definition by Quondum:

\bold{\Pi} = m\bold{v} = \bold{P}- e\bold{A}\,\!

(difference between total/canonical and potential momenta). But clearly RockMagnetist has a point: "kinetic momentum" mv is the subject of this article. The "canonical momentum" P is the generalized momentum found from Lagrangian mechanics:

\bold{P} = \frac{\partial L}{\partial \bold{\dot{r}}}, \quad L=\frac{m}{2}\mathbf{\dot{r}}\cdot\mathbf{\dot{r}}+e\mathbf{A}\cdot\mathbf{\dot{r}}-e\phi\,\!

Also the worded description does seem to indicate the EM field energy:

U_{\rm EM} = \frac{1}{2}\int_V(\bold{E}\cdot\bold{D}+\bold{B}\cdot\bold{H})dV\,\!

which is not the same as the "potential momentum" eA, pointed out by Quondum... So a merge would be fine. Maschen (talk) 01:37, 15 August 2012 (UTC)

I think the confusion arises in kinetic momentum because the momentum operator corresponds to the canonical momentum. So in quantum mechanics it may seem that you are adding the effect of the electromagnetic field, when in fact you are removing it to get m v. I won't do the merge until I have made that clear in momentum. RockMagnetist (talk) 02:33, 16 August 2012 (UTC)
True. Thanks for clarification. Maschen (talk) 06:21, 16 August 2012 (UTC)
For now, I'll tweak Kinetic momentum to give a head start on merging. Maschen (talk) 09:33, 21 August 2012 (UTC)
Maschen, you're making substantial improvements to Kinetic momentum. Do you still support a merger? If so, how do you envision merging it into Momentum? RockMagnetist (talk) 22:23, 24 August 2012 (UTC)
It should be merged, I'd say blend Kinetic momentum, Non-relativistic dynamics with Momentum, Particle in field (I'm working on it now in the sandbox), and then just paste the Kinetic momentum, Canonical commutation relations and Kinetic momentum, Relativistic dynamics sections after it (or if no-one likes the relativistic derivation it could be scrapped or placed in a show/hide box, it can always be recovered from the edit history). Maschen (talk) 18:08, 25 August 2012 (UTC)
These are the lines I was thinking parallel to... It's a bit long, so feel free to trim any bits you like.Maschen (talk) 22:07, 25 August 2012 (UTC)

I will do this by tomorrow if no-one else does. There has been plenty of time for people to object or do the merge themselves. Aside from that I will stay out of the article, for reasons below, and leave it to others thenfrom... Maschen (talk) 22:05, 6 September 2012 (UTC)

Sounds good. My feeling, by the way, is that you should reduce the material on equations of motion because this article could easily get too bloated. You can always link to the appropriate articles. RockMagnetist (talk) 22:45, 6 September 2012 (UTC)
What do we know - today is the day. Time to merge. Maschen (talk) 00:16, 7 September 2012 (UTC)
Done - now over to you (RockMagnetist, Zueignung, anyone else)... Maschen (talk) 01:12, 7 September 2012 (UTC)
Thanks, Maschen, for putting so much effort into improving Kinetic momentum before merging it. RockMagnetist (talk) 04:15, 7 September 2012 (UTC)
You have been the driving force all along! I only deleted/rephrased/replaced symbols/terminology in kinetic momentum... Maschen (talk) 09:10, 7 September 2012 (UTC)

Analogies[edit]

I removed the section on analogies with mass and heat transfer. It is misleading, because the viscosity law of a Newtonian fluid is not an equation for momentum transfer, although it does contribute to the equation for conservation of momentum. Similarly, the relevant analogues for heat and mass are also conservation equations. RockMagnetist (talk) 16:57, 4 August 2012 (UTC)

new footnote[edit]

A general expression for Newton's law does apply for a system with variable mass by treating mass as a variable wrt time:

 \begin{align} \mathbf{F} & = \frac{{\rm d}\mathbf{p}}{{\rm d}t} = \frac{{\rm d}(m \mathbf{v})}{{\rm d}t} \\
& = \frac{{\rm d}m}{{\rm d}t}\mathbf{v} + m\mathbf{a} \\
& = \frac{{\rm d}m}{{\rm d}t}\frac{{\rm d}\mathbf{r}}{{\rm d}t} + m\frac{{\rm d}^2\mathbf{r}}{{\rm d}t^2} \\
\end{align} \,\!

So the statement:

"It is important to note that we cannot derive a general expression for Newton's second law for variable mass systems by treating the mass in F = dP/dt = d(Mv) as a variable. [...] We can use F = dP/dt to analyze variable mass systems only if we apply it to an entire system of constant mass having parts among which there is an interchange of mass.|isbn=0-471-03710-9}} [Emphasis as in the original]</ref> and so it is equivalent to write"
F = m\frac{dv}{d t} = m a,

doesn't make much sense to me. Maschen (talk) 15:56, 4 September 2012 (UTC)

For future readers - forget this. It is incorrect reasoning as pointed out below and in Kleppner & Kolenkow, p.135. Maschen (talk) 09:10, 7 September 2012 (UTC)
Yesterday I removed the assertion that F = ma is a specific case of F = dp/dt that emerges by assuming m is constant. This point of view is explicitly rejected in a number of modern textbooks, including Halliday & Resnick and Kleppner & Kolenkow. It is also explicitly rejected in the first paragraph of the variable-mass system article (with citations).
I was reverted, with the rationale that F = ma indeed follows from the special relativity form F = dp/dt by assuming relativistic mass γm is constant. First of all, we are talking about content in the section of the article titled Newtonian mechanics, so the objection is irrelevant. The objection is also false, as you can see by examining Special relativity#Force. It is possible to massage the relativistic second law into a form that looks like F = γ^3 ma_parallel + γma_perp, but you cannot recover F = γma. Zueignung (talk) 15:55, 4 September 2012 (UTC)
You are correct that if the total mass is constant independent of the dynamics of the mass within the system, then we can use F = ma where F and a are for the centre of mass.
But when you say variable mass in special relativity, what about mass-energy conversion? Particle interactions (which include creation and annihilation)? Nuclear reactions, binding energy and decay? What do you consider the system to be to have total mass, when some has transformed to energy (or vice versa)? It's the total 4-momentum that's conserved, right?
The point was that F = dP/dτ (not dt but proper time τ) is essentially the general form of Newton's 2nd law as definition of 4-force (which would contain the 3-force f = γdp/dt, and hence f = γma for m = constant) no matter what the invariant mass M is (which does not change from reference frame to ref frame but could change in value) in the 4-momentum P = MdX/dτ is. Variable mass-energy is allowed. Maschen (talk) 16:55, 4 September 2012 (UTC)
That's not quite my rationale. I don't have access to Halliday & Resnick; but based on the derivations I see in Goldstein, they are probably arguing that you cannot derive F=dp/dt for variable mass from a given set of assumptions. However, you can postulate F=dp/dt, and that works fine for variable mass in relativity. Of course, classical mechanics is a special case.
It's true that you do have to be careful about applying this equation to rocket propulsion, but that kind of subtle problem doesn't belong in the first section. The first section is about momentum of a single particle, and I am trying to make it accessible to high school students. RockMagnetist (talk) 16:29, 4 September 2012 (UTC)
I have been meaning to add a section on momentum in variable mass systems. Perhaps you could do that? RockMagnetist (talk) 16:35, 4 September 2012 (UTC)
Even for a single particle it is incorrect to apply the second law to a situation where dm/dt ≠ 0, since you must consider the behavior of the other particles which are created (or destroyed) when your particle loses (or gains) mass. High school students (and college and grad students for that matter) will see the phrase "if the mass is constant" and then assume that they can simply treat variable-mass systems by applying the product rule to mv as Maschen has done above, which will give them wrong answers. If you really want to make a statement in this section about what happens in special relativity (which I don't really see the reason for, since the section is called Newtonian mechanics), then I think you should explicitly state "X is what happens in Newtonian mechanics, and Y is what happens in special relativity".
On the issue of accessibility, I do wonder whether the stuff about relativistic mass should be scaled down a bit. As far as I can tell it's a concept that has really fallen out of favor pedagogically; it seems like most people just stick to rest mass and write out factors of γ where appropriate.
I will think about how to write an accessible section on variable-mass systems. I too am dissatisfied with the opacity of a lot of the physics articles. Zueignung (talk) 16:52, 4 September 2012 (UTC)
Above was for classical mechanics and it is correct for classical mechanics. This can be cited (e.x. P.M. Whelan, M.J. Hodgeson (1978). Essential Principles of Physics (2nd ed.). John Murray. ISBN 0-7195-3382-1. ). You are correct that there are gamma factors in special relativity. I apologize for talking at cross purposes. Maschen (talk) 16:59, 4 September 2012 (UTC)
I realize that there are textbooks (including Feynman's) that assert in passing that F = d(mv)/dt can be applied to variable mass systems in classical mechanics by product-rule differentiation. I have yet to see such a textbook actually investigate this claim in detail and use it to successfully calculate the motion of a rocket (or something similar). On the other hand, there are many sources and papers which do undertake such investigation and conclude that the claim is bunk. Zueignung (talk) 17:15, 4 September 2012 (UTC)
Actually, "X is what happens in special relativity, and Newtonian mechanics is a special case." Anyway, there is no need to discuss relativity because "If the mass is constant" is a correct statement. Coupled with a section explicitly dealing with variable-mass momentum, I don't see how anyone can be led astray. RockMagnetist (talk) 17:22, 4 September 2012 (UTC)
A major reason that so many Wikipedia articles are difficult to read is that editors try to say everything up front. RockMagnetist (talk) 17:24, 4 September 2012 (UTC)
Note also that in the variable-mass system, F=dp/dt is correct for variable-mass systems if you define the changes in momentum correctly. I think the whole business of product-rule differentiation is a red herring. RockMagnetist (talk) 17:34, 4 September 2012 (UTC)
In every derivation I've seen, defining the changes in momentum "correctly" amounts to considering the momentum of both the rocket and its ejected fuel, which means you are applying the second law to the entire, constant-mass system. You are not applying the second law to a variable-mass system. Zueignung (talk) 18:05, 4 September 2012 (UTC)

On the whole, I'm o.k. with "does not exchange matter with its surroundings". Pedagogically, I prefer a statement about variable mass that is true for relativity as well, but your statement is simple and clear. And I like the reference. So I'll leave it the way it is. RockMagnetist (talk) 18:21, 4 September 2012 (UTC)

Elastic Collisions animation not quite right?[edit]

In the animation, the top "box" covers more ground than the bottom one does. Further, shouldn't they always meet in the middle? I was lost for a second trying to figure out just what was trying to be taught to me because of this. I'd propose the v's be the same, top and bottom, and that the boxes on the bottom start at far left and far right so that they meet at the same place as the collision on the top.
Tgm1024 (talk) 13:32, 9 September 2012 (UTC)

What I'd like to see is the right box starting at rest in both frames and the other two boxes starting from the left with the same speed. However, these graphics were made by someone in Germany in 2006, so anyone wanting to change them will probably have to start from scratch. RockMagnetist (talk) 15:24, 10 September 2012 (UTC)

Variable mass[edit]

I made a first pass at writing a section on variable mass systems. Possible issues:

  • I didn't really put in much of a derivation of the variable-mass formula since this is already done at variable-mass system (and also Tsiolkovsky rocket equation, though this is not linked in the section).
  • I avoided the phrase variable-mass system since the "system" essentially gets redefined halfway through the derivation. It seems confusing to talk about "variable-mass system" this and "variable-mass system" that, and then say "just kidding, here's the real system we need to analyze". Zueignung (talk) 03:32, 11 September 2012 (UTC)
I like it! I agree that there is no need to derive the formula here. RockMagnetist (talk) 06:09, 11 September 2012 (UTC)
Yea just about to say the same. Good work! Maschen (talk) 06:10, 11 September 2012 (UTC)

Italic or bold?[edit]

We have a difference of opinion about whether to represent vectors with upright or italic bold. I was arguing the mathematical convention mentioned in Vector notation, while Dger cites the SI rules that vector physical quantities be in italic boldface. I was able to confirm that in the NIST guide. Since this is a physics article, I would be inclined to agree with Dger. The entire article should follow the convention consistently - that shouldn't be too hard. RockMagnetist (talk) 00:54, 14 November 2012 (UTC)

Italic bold is really gross, and as far as I can tell a minority typesetting choice here on Wikipedia and elsewhere. Zueignung (talk) 03:48, 14 November 2012 (UTC)
Maybe this would be worth discussing at WikiProject Physics. RockMagnetist (talk) 05:49, 14 November 2012 (UTC)
Italic-bold (\boldsymbol) has the advantage that Greek and Latin latters can be uniformly bold (instead of switching between \boldsymbol and \mathbf), and if there is a convention then it should be used, though there are many books where italic-bold isn't used...
Upright-bold looks much cleaner. It's going to take time to convert everything on WP into italic-bold, not that it's a problem...
It's perfectly fine that my reversion was reverted... Maschen (talk) 08:12, 14 November 2012 (UTC)
Convert everything on Wikipedia to italic bold? What? Physics-related Wikipedia articles should follow the conventions of physics textbooks and journal articles, which usually (but not always) means upright bold. What the SI people wish to happen is irrelevant. Notice that they also want everyone to typeset tensors in bold italic sans-serif, which I don't think I've ever seen done. Zueignung (talk) —Preceding undated comment added 15:52, 14 November 2012 (UTC)
Since this question affects all physics articles, I have started a discussion at WikiProject Physics. I am not suggesting we change all the existing articles (that would be crazy); but the project could make a recommendation one way or the other. RockMagnetist (talk) 16:01, 14 November 2012 (UTC)

Is momentum a vector or covector?[edit]

This article says that momentum is a vector. But is not it really a covector? At least generalized momentum  p_j = \frac{\partial \mathcal{L} }{\partial \dot{q}_j} looks like covector to me. --Alexei Kopylov (talk) 21:52, 23 May 2014 (UTC)

I think physicists don't generally distinguish between vectors and covectors. Certainly, it would only cause confusion in the earlier parts of this article that are (intentionally) written at a level that a high school student would understand. RockMagnetist (talk) 17:09, 24 May 2014 (UTC)
I agree that we should not scare high school students with words such as covector. But should at least mention it somewhere? I'm not sure what is appropriate place for this.Alexei Kopylov (talk) 09:08, 27 May 2014 (UTC)
The section on relativistic momentum already has a passing mention of position as contravariant and momentum as covariant. (Actually, it said momentum is contravariant, so it's just as well you drew this to my attention!) The books I have read (like Goldstein's Classical Mechanics) call both vectors (contravariant and covariant), rather than calling one a vector and the other a covector. But we could add a footnote mentioning the alternative terminology. Is there some area of physics where you see it used? RockMagnetist (talk) 16:03, 27 May 2014 (UTC)
I'm not a physicist, I'm a mathematician. I was confused, and I'm afraid someone else would be confused. But I'm not familiar with standard physics terminology, so I don't know how to change the article. Alexei Kopylov (talk) 05:47, 29 May 2014 (UTC)
Because a Minkowski space (or more generally a pseudo-Riemannian manifold) has a metric tensor defined on it, the covariant and contravariant vector spaces are dual spaces, so it's always possible to express any 4-vector as either a contravariant or covariant vector, by raising or lowering the index as needed. However, any particular 4-vector is typically more naturally expressed as one or the other, and I believe you're right, canonical momenta at least are more naturally expressed as covectors. But what trumps any discussion of naturalness is that equations have to at least be correct, and not mix contravariant and covariant vectors as if it didn't matter which of those a vector is. Elementary vector analysis can get by with just ignoring the difference between contravariant and covariant vectors, because when using a standard basis or some other orthonormal basis on a Euclidean space, the components of the metric tensor are just the identity matrix, so the components of the contravariant and covariant versions of a vector are identical. But you can't get by with ignoring the distinction on a Minkowski space. (It isn't possible to define an orthonormal basis on a Minkowski space, because the metric isn't positive definite.) Red Act (talk) 00:40, 7 August 2014 (UTC)
For Minkowski space we have metric, but what about generalized momentum? Alexei Kopylov (talk) 20:33, 13 August 2014 (UTC)
Ah. When I wrote the above, I had just edited the "Four-vector formulation" section, so I was thinking about the relativistic version of momentum. But the same thing applies with Newtonian momentum, except with the Euclidean metric instead of the Minkowski metric. Newtonian momentum can be expressed as either a contravariant or a covariant vector, by raising or lowering the index as needed with the Euclidean metric. Red Act (talk) 16:22, 14 August 2014 (UTC)

Claim about classical electromagnetism[edit]

The section Momentum#Classical_electromagnetism makes a factually incorrect claim, but cites a Goldstone source to which I have no access. I would appreciate the relevant quote being given here. The section refers to classical electromagnetism, which is is Maxwell's theory. In this theory, there is no violation of Newton's law – a particle does not apply a force on another particle without a return force, it applies a force to the field, which applies a balancing force on the particle; by no mental acrobatics can this be construed as "no return force". The field, some time later, may as a result of its perturbation apply a force to another particle, again obeying Newton's law of action and reaction. Let's get this fixed. —Quondum 14:21, 13 July 2014 (UTC)

Quondum, here is a quote from the note at the bottom of page 8:

Consider, further, two charges moving (instantaneously) so as to "cross the T," i.e., one charge moving directly at the other, which in turn is moving at right angles to the first. Then the second charge exerts a nonvanishing force on the first, without experiencing any reaction force at all.

I know this is disturbing, and I don't fully understand how this is reconciled with Newton's laws. Definitely more should be said in the article. When I have some time, I'll look into it. RockMagnetist (talk) 16:03, 13 July 2014 (UTC)
The example appears to be an illustration showing that where, if one naïvely ignores the EM field as a momentum-carrying entity, apparent violations of conservation of momentum occur (though it is too small a snippet for me to build a full picture). In this instance, the very concept that one particle exerts a force directly on the other is a misconception and a violation of Maxwell's equations and of special relativity. As such, Goldstone may be illustrating the need to switch cleanly to a new paradigm: from the instantaneous action at a distance of Newton's picture, to that of the intermediary field of Maxwell governed by its own dynamics, with energy and momentum densities of its own. I find the phrase "the second charge exerts a non-vanishing force on the first" unfortunate as it suggests the Newtonian paradigm, but maybe this is intentional to highlight the old way of thinking. Relativistic classical mechanics makes it abundantly clear that this (old) way of thinking does not apply at all. Reconciliation with Newton's law of action and reaction is straightforward simply by realizing that forces are propagated rather than instantaneous, and that the forces across any boundary between two regions of space are always balanced according to Newton's law.
In any event in special relativity, Newton's law, which amounts to conservation of momentum, is obeyed, locally and globally. We need to avoid paraphrasing anything in such a way that that it gives the impression that this does not hold, and I hope you'll let me rephrase to remove falsehoods (e.g. that Maxwell's equations violate Newton's laws of action and reaction) until such time as we can get to the bottom of what the source is really trying to say, for which I'll need to get more. —Quondum 18:34, 13 July 2014 (UTC)
Here is a more extended quote, with the location of the footnote given by an asterix:

In a system involving moving charges, the forces between the charges predicted by the Biot-Savart law indeed may violate both forms of the action and reaction law.* Equations (1-23) and (1-26), and the corresponding conservation theorems, are not applicable in such cases, at least in the form here given. Usually it is then possible to find some generalization of P or L that is conserved. Thus, in an isolated system of moving charges it is the sum of the mechanical angular momentum and the electromagnetic "angular momentum" of the field that is conserved.

Here is equation (1-23):
\mathbf{P} = \sum m_i \frac{d\mathbf{r}_i}{dt} = M\frac{d\mathbf{R}}{dt}
(the total linear momentum of the system is the total mass times the velocity of the center of mass).
And here is equation (1-26):
\frac{d\mathbf{L}}{dt} = \mathbf{N}^{(e)}
(the time derivative of the total angular momentum is equal to the moment of the external force about the point). I think the above is pretty close to what I have written in the article, but if you want to try a rewording that better represents the above material, be my guest. RockMagnetist (talk) 22:19, 13 July 2014 (UTC)
It may also help to quote the laws of action and reaction as stated in Goldstein:
  • Weak law: "Newton's third law of motion in its original form: that the forces two particles exert on each other are equal and opposite." (p. 5)
  • Strong law: the forces also lie along the line joining the particles. (p. 7)
RockMagnetist (talk) 22:23, 13 July 2014 (UTC)
Ah, "predicted by the Biot–Savart law". No wonder. This is an "instantaneous action at a distance" type of law. Unmodified, it is incompatible with Maxwell's equations and special relativity when applied to general nonstatic movement of charge. Goldstone might in effect be making the point that one needs to attribute a stress–energy field to space even in the Newtonian case – in essence that one cannot attribute all energy and momentum to the particles while remaining consistent with the conservation laws. And once one allows an energy–momentum tensor to permeate space, Newton's conservation laws are restored, though Goldstone's wording seems to use the implicit assumption that forces act only on particles. I'm not familiar with the phraseology such as "generalized momentum", but one does not need it when one allows the EM field itself to have energy and momentum (certainly in the relativistic case, and I suspect also in the Newtonian case). At the very least, we need to be clearer about when we are dealing with Newtonian and when with relativistic mechanics, or when what we are saying applies to both. I personally don't have much taste for Newtonian electromagnetism, and would be happy to omit it. —Quondum 01:23, 14 July 2014 (UTC)
Generalized momentum is an important concept in Lagrangian and Hamiltonian mechanics (see Momentum#Generalized_coordinates). But I agree that the context should be clearer. RockMagnetist (talk) 17:20, 14 July 2014 (UTC)
Agreed. And this article clearly tries to cover the different cases: nonrelativistic classical mechanics, special relativity, quantum mechanics etc., so we should cover it. Part of my problem is the juxtaposition of the terms classical mechanics and classical electromagnetism, which are both non-quantum, but only the latter is relativistic. Another (and bigger) part of what I am reacting to is that the article seems to deal with a simplification that is inaccurate. For example, even in the relativistic case, the table provides a Lagrangian and Hamiltonian that seems to ignore radiative contributions and only allows for static external fields (but keep in mind that this is not a strong area for me). This is fine in a pedagogical setting, but in a reference setting these contextual assumptions must be highlighted. In short, I'm guessing that what is presented here is wrong because it is simplified to show only certain aspects of mechanics, but is fairly common for a presentation in a textbook. I'd say this needs some careful work to clarify it. —Quondum 18:36, 14 July 2014 (UTC)
The table was cut and pasted into this article when Kinetic momentum was merged. For some time, I have been meaning to turn it into prose with a better linkage to the rest of the article. RockMagnetist (talk) 19:15, 14 July 2014 (UTC)

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Domains of validity

A thought that occurs to me is that this article is roughly built along the lines of domains of applicability – as per the diagram here – where three of the domains appear to be addressed (§Newtonian mechanics, §Relativistic mechanics and §Quantum mechanics, which I've now grouped together). The lead also introduces the topic as being "In classical mechanics ...", yet the article goes beyond this. Perhaps this should read "In physics ..."? The section §Classical electromagnetism seem to span several of the these domains, so I feel that this should be split up as subsections under the domain-of-validity headings. The phrase "classical electromagnetism" could also be avoided, as it is not clear (to me, anyway) whether it means electromagnetism in classical mechanics (hence nonrelativistic) or electromagnetism as per Maxwell (relativistic). Does this line of thought make sense? —Quondum 17:58, 16 July 2014 (UTC)

I think the main advantage of keeping an electromagnetism section is that it keeps all the relevant terminology and symbols together, making it easier to read. However, we could change it to §Electromagnetism and develop some of the general themes in other sections before applying them to electromagnetism. For example, the concept of the stress-energy tensor can be developed in the section on relativity and then applied to the momentum of the electromagnetic field. RockMagnetist (talk) 00:55, 17 July 2014 (UTC)
As for the lead, I would say it needs a thorough rewrite, as it doesn't come close to summarizing the contents (see the todo list at the top of this page). So far, I have been more interested in editing the body, naively thinking that I would soon be done with that. RockMagnetist (talk) 01:10, 17 July 2014 (UTC)
Rewriting the lead will be tricky. If we say "In physics", the first sentence is wrong, as it does not apply to many of the uses in the article. However, I think it would be a mistake to attempt a general definition in the first sentence. That just makes the lead difficult to read and would discourage a lot of people from reading the article. I think we should keep "In classical mechanics" and then discuss the generalizations in subsequent paragraphs. RockMagnetist (talk) 01:25, 17 July 2014 (UTC)
Sure, best leave the lead for later. I've changed the heading as you say, and now we can address each of the topics introduced above (separate domains of validity, plus the different formalisms etc.) separately as need be, within this section. The previously problematic statement can thus be treated under the Newtonian mechanics domain where it belongs, so that the relativistic approach does not get confused with it. —Quondum 02:49, 17 July 2014 (UTC)