# Talk:Momentum

WikiProject Mathematics (Rated B-class, High-importance)
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Field: Mathematical physics
WikiProject Physics (Rated B-class, Top-importance)
This article is within the scope of WikiProject Physics, a collaborative effort to improve the coverage of Physics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
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To-do list for Momentum:
 Here are some tasks you can do: Expand: Electromagnetism: Add preamble about relativity and violation of action-reaction law Momentum in media Manifestly covariant formulation Conjugate momentum Quantum mechanics: Relation to canonical momentum Commutation relations 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

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

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

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

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

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

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?

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

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?

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)