Talk:Momentum

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)

I studied College Physics for a year in the 1980s, and a lot of the introduction was opaque to me. I can only imagine the casual reader is not well served by this. Seems the technical experts never seem to care about the readability of introductions. I agree the main body of the article can be more technical. 2601:281:CB00:2A0:FD0B:480B:BCA7:85F8 (talk) 04:56, 30 December 2017 (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)

Why isn't there a section somewhere explaining the core idea itself simply in plain text? — Preceding unsigned comment added by Bearsca (talk • contribs) 01:24, 1 August 2015 (UTC)

The first couple of paragraphs in the lede give a simple explanation in plain language. Remember Wikipedia is an encyclopedia, not a text book. I think the lede is appropriately introductory for an encyclopedia. Dolphin (t) 06:11, 1 August 2015 (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):

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

which fits in with the linked definition by Quondum:

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

(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:

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

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

$U_{\rm {EM}}={\frac {1}{2}}\int _{V}(\mathbf {E} \cdot \mathbf {D} +\mathbf {B} \cdot \mathbf {H} )dV\,\!$

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:

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

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"

$F=m{\frac {dv}{dt}}=ma,$

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)

Is momentum a vector or covector?

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

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)

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)

Article clarity

Where is the definition, I don't know what it is, but thought it was something like the tendency of an object in motion to continue in motion, Something like that would be helpful. This article is chaotic, with reams of references all over the place. There is also an assumption that beyond a certain level this concept is the property of science. but it has many every day application, not just in academics, whether grade school or beyond. Those are actually the places where if you have a reason to know this stuff you already do. — Preceding unsigned comment added by 70.27.118.120 (talk • contribs) 2015-04-11T08:39:38

Generator of translation?

The article on the quantum momentum operator states "As it is known from classical mechanics, the momentum is the generator of translation," and "momentum" in that sentence is a link to this article. However, this momentum article does not cover the concept of "generator of translations". It would be good if someone could add this. Tpellman (talk) 01:55, 18 May 2015 (UTC)

At this time the article mentioned that momentum conservation is connected to the symmetry of translation, but I agree that concisely describing momentum as the generator of translations would be helpful (it is not only true in QM, but classical mechanics also). M∧Ŝc²ħεИ_τlk 07:06, 10 June 2016 (UTC)

Angular momentum

There is a link to angular momentum but do we need a hat note to it since this article is specifically linear momentum? RJFJR (talk) 20:04, 7 June 2016 (UTC)

I think a hat note would be a good means to clarify the distinction between the two concepts. Dolphin (t) 20:43, 7 June 2016 (UTC)

I agree. RockMagnetist(talk) 21:13, 7 June 2016 (UTC)

I have inserted a hatnote. See my diff. Dolphin (t) 04:22, 10 June 2016 (UTC)

"conservation of linear momentum is implied by Newton's laws."?

The lead has this sentence. It is not helpful, it's like saying charge conservation is implied by Maxwell's equations. But momentum conservation and charge conservation are more fundamental than the laws (which are empirical/postulate equations), and separate statements. It would be better to say Newton's laws are consistent with momentum conservation, but even then this leads to awkward issues with variable mass systems (in this article, section Momentum#Objects of variable mass), in which case you must use momentum conservation to set up the equation of motion correctly.

A stronger way to infer momentum conservation is by translation symmetries (see previous section on this talk page). Any thoughts? M∧Ŝc²ħεИ_τlk 07:17, 10 June 2016 (UTC)

Afaict, the derivation is routinely taught in first year classical mechanics courses. Yes, perhaps the conservation laws are more fundamental in light of today's theories, but the fact remains that the laws can indeed be derived from Newton's laws, as for instance, Euler already knew. To me that statement sounds pretty OK, but perhaps we need to source it. I propose this one:

References

- DVdm (talk) 08:44, 10 June 2016 (UTC)

Newton's laws of motion can be expressed in several different ways. For example, F = dp/dt is a legitimate expression of the second law, but F = ma is equally legitimate because it is derived from the former. Similarly, the law of conservation of momentum (LCM) is a legitimate derivation from Newton's Laws. LCM was not discovered independently of Newton and his Laws of Motion - it is implied by those laws of motion and credit should be given to Newton. Dolphin (t) 12:00, 10 June 2016 (UTC)

Yes, I know Newton's 2nd law can be expressed in different ways, what I mean is that some readers could think that momentum conservation follows from Newton's 2nd law. It doesn't. Yes, as viewed today, Newton's laws should respect momentum conservation, not the other way round. By all means add sources to support the statement already there, but I still maintain the lead could be misleading. M∧Ŝc²ħεИ_τlk 07:13, 11 June 2016 (UTC)

But of course our statement does not say that LCM can be derived form the 2nd law. It follows from the combination of the laws (see, for instance Ohanian, Physics I, section 10.1). I think that your remark about generality is well covered by the remainder of the paragraph. But I have added the source. Certaily won't do any harm . - DVdm (talk) 08:28, 11 June 2016 (UTC)

Formatting an equation

One of the equations in this article has been revised multiple times recently. The main issue is whether v/c should be represented by a slash,

$\gamma ={\frac {1}{\sqrt {1-\left(v/c\right)^{2}}}},$

or as a vertical fraction:

$\gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

or

$\gamma ={\frac {1}{\sqrt {1-{\frac {\textstyle v^{2}}{\textstyle c^{2}}}}}}$

Those in favor of the first version are trying to maximize the size of the characters; an IP editor claims this is arbitrary and has expressed a distaste for slashes. I have two comments:

1. The Physical Review style guide recommends using slashes in the numerator and denominator of fractions "except where excessive bracketing would obscure your meaning or slashing would interfere with continuance of notation."
2. I think a non-arbitrary criterion for the size of characters is that, if possible, they should all be the same size (and here I am including numbers).

Thus, I favor the first version. The third isn't bad, but requires rather more TeX commands than a simple equation like this should need. What is your preference ? RockMagnetist(talk) 15:28, 29 June 2016 (UTC)

A fourth option would get rid of the parentheses in the first version:

$\gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}.$

RockMagnetist(talk) 15:33, 29 June 2016 (UTC)

Fourth version preferred, immediately followed by third version. First and second versions rather not. - DVdm (talk) 15:36, 29 June 2016 (UTC)

With the addition of \textstyle, the size of the text/variables is as before any changes were made. 87.254.76.130 (talk) 15:39, 29 June 2016 (UTC)

I too prefer the fourth version narrowly over the first. RockMagnetist(talk) 15:40, 29 June 2016 (UTC)

"The third isn't bad, but requires rather more TeX commands than a simple equation like this should need." Please could you elaborate on what the correct number of LaTeX commands that an equation should need? 87.254.76.130 (talk) 15:55, 29 June 2016 (UTC)

My point is that a given article should have a consistent style, but many people contribute to this article, so conforming to that style should be as easy as possible. And there is no question that the fourth equation is easier to format than the third. RockMagnetist(talk) 16:31, 29 June 2016 (UTC)

Could you be more specific about the difficulties in the two different forms of LaTeX? Would you like me to prepare further options for making things as easy as possible, as I can see other possibilities if this is an issue? I note that I am asking such a question. 87.254.76.130 (talk) 16:49, 29 June 2016 (UTC)

Mainly I object to using \textstyle inside an expression. That is quite technical and unorthodox; it shouldn't ever be necessary to mix textstyle with displaystyle in the same equation. Please do feel free to suggest other ways of formatting this equation. RockMagnetist(talk) 16:59, 29 June 2016 (UTC)

English variety

I scanned for British spellings "our" and "ise", and came up empty. "Generalize" is used frequently, therefore I conclude the article is already in American English. JustinTime55 (talk) 14:39, 11 August 2016 (UTC)

Listed alternate units

In the "other units" section of the information box, the article lists "ft/s" as a unit of momentum. This seems like an error; ft/s stands for feet per second, which is a unit of speed and velocity (not momentum, at least as far as I am aware). Furthermore, the text links to the page for feet per second, which makes no mention whatsoever of momentum. Should this be changed, or is there something I'm missing here? 108.20.171.228 (talk) 22:51, 17 October 2016 (UTC)Gus Wiedey

You are indeed missing something. The ft/s is multiplied by a slug, which is a unit of mass, so the product is in units of momentum. RockMagnetist(talk) 01:18, 18 October 2016 (UTC)

Should be multiplied by a pound, the US Customary unit of mass. The slug is a derived unit that no one ever actually uses. John G Hasler (talk) 23:18, 21 May 2022 (UTC)

Transfer of momentum

Don’t declaration and acceleration produced during the infinitesimal contact period of time in collision affects the transfer of momentum (final velocities) of objects after collision2001:56A:7399:1200:3078:21D0:B447:EDC6 (talk) 05:24, 31 March 2017 (UTC)eek

Newton's Cradle does *not* demonstrate the conservation of momentum well

After all, the balls constantly change direction (as they generally do in pendulums) which actually appears as a *violation* of the conservation law. The conservation is only true for the entire earth-pendulum system: The momentum differences are equalized by corresponding changes in earth's momentum, but since we are actually applying tangential forces to a sphere (earth) spinning with the speed of sound around its axis ... you get the picture. A good demonstration would be to put the cradle on a low-friction vehicle which will oscillate, moving back and forth, compensating the ball's momentum at each point in time.

One should **remove Newton's cradle.** 80.146.228.85 (talk) 14:16, 24 May 2018 (UTC)

You're right - it's not a clear demonstration of conservation of momentum. In the article, Newton's cradle, it points out that any function of mass and velocity would be conserved. I'll remove it. RockMagnetist(talk) 15:39, 24 May 2018 (UTC)

OK not clear demonstration, but it is one of the few easily viewed demonstrations. OK, it more demonstrates conservation of energy. If one didn't know anything above physics and collisions, one might expect when one ball hit, motion was transferred to all the other balls equally. That also satisfies momentum conservation, but not energy conservation (in the case of elastic collisions). If you build one with sticky glue balls, that is what will happen. Newton's cradle more demonstrates elastic collisions, which are not common in many ordinary situations. Not understanding elastic collisions complicates understanding momentum conservation. Gah4 (talk) 12:15, 1 November 2020 (UTC)

Section: History of the concept

This section is biased and inaccurate on at least three points. (1) It interprets and rubbishes Descartes' contribution "This should not be read as a statement of the modern law of momentum since ... he believed that it is speed rather than velocity that is conserved." No reference is given for this interpretation and the original clearly refers to head-on collisions (as seen in the subsequent rules). See also his second law of motion in The World (Descartes), and in Descartes' Principia Philosophiae Book 2 Section 37 and 40 [1] (page 57). (2) There is no reference at all to Huygens, even though he is often quoted as the originator of the principle of conservation of momentum. This is probably due to an anglo-centric reporting of the Royal Society Proceedings of 1668. See [2] and [3] (page 229). (3) It gives all the credit to Wallis for this principle: "The first correct statement of the law of conservation of momentum was by English mathematician John Wallis in his 1670 work". No reference to this claim by a historian is given except for the original work, and then the quote that is supposed to support this claim, namely "the initial state of the body, either of rest or of motion, will persist", makes no reference to conservation and indeed already appears in works of Descartes and Galileo. In fact Wallis, Wren, and Huygens all presented the same principles in 1668 and, according to the textbook above by David C. Cassidy, Gerald Holton, and F. James Rutherford, Huygens was more complete than them. This section should be rewritten by a historian of physics. — Preceding unsigned comment added by Chrystomath2 (talk • contribs) 15:06, 30 November 2019 (UTC)

@Chrystomath2: I suspect that if we wait for a historian of science, we will be waiting a long time. So let's see what we can do with the sources that historians have already written:

(1) There are actually three sources supporting the statement about speed, but they are after the next sentence. That makes sense, since the statement "if a moving object ... no change in its quantity of motion" clarifies the meaning. Rothman, on the top of p. 86, states clearly that Descartes did not take into account the direction of motion. (Unfortunately, Rothman is a bit sloppy, and uses "velocity" in a few places where he means "speed", but the intent is clear.) Rothman also states on p. 85 that "Descartes himself wrote down, as part of his laws of motion, seven rules governing the collisions between objects of various masses and velocities. All but one of these rules were later proved incorrect ..." The Stanford Encyclopedia states: 'In the Principles, Descartes conservation law only recognizes a body’s degree of motion, which correlates to the scalar quantity “speed”, rather than the vectorial notion “velocity” (which is speed in a given direction).' Unfortunately, I can't access all of the relevant pages in Garber, but he is clearly saying that Descartes' program fell short in many ways. In particular, "Descartes and his contemporaries did not have a notion of mass independent of size." And as an example of the shortcomings of using speed instead of velocity, the Stanford Encyclopedia states, "Astonishingly, Descartes claims that a smaller body, regardless of its speed, can never move a larger stationary body."

(2) and (3) Rothman states "In November of 1668 John Wallis submitted a memoir that is perhaps the first correct statement of the Law of Conservation of Momentum." But I agree, the contribution of Huygens probably warrants an entire paragraph. Rothman discusses his contributions on pages 85-88, and also mentions Wren. (Neither Rothman nor the textbook you have linked goes into any detail about Wren. Apparently his contribution was less significant.)

In summary, I think the assessment of the work by Descartes is correct and well supported by sources, although it could be expanded. You are welcome to add coverage of Huygens and Wren - but, in accordance with the policy on original research, please use secondary sources wherever possible, and don't try to interpret the primary sources yourself.

RockMagnetist(talk) 18:51, 30 November 2019 (UTC)

@RockMagnetist: It seems there is agreement about points (2) and (3). I don't know what Rothman is referring to exactly, but it seems that Wallis did not give an explicit statement of the Law of Conservation of Momentum in 1668 but in 1670 in a separate paper. Before that Huygens had already written a paper (not published by the Royal Society, hence the controversy) explicitly mentioning conservation of momentum with direction (see for example "The Mechanical Philosophy 1660-1675" by E.B. Davies of King's College, London). As for (1), the issue is not whether Descartes was correct or not, but the importance that should be attached to the fact that it was the first statement of its kind. Note that Rothman is not being sloppy about speed and velocity: they were used interchangeably not only in the 17th but also part of the 18th century. This is part of the controversy - one can interpret Descartes harshly or favorably. Was he referring to head-on collisions as his subsequent rules make clear? Were they what we now call elastic collisions? Descartes certainly knew the difference between motion and motion in a certain direction, as evidenced by his arguments in the refraction of light. The importance of Descartes' law stands out when one considers that Wallis and Huygens were reacting to, and correcting, his notion. With the same reasoning, one should write that "Wallis' statement should not be read as a statement of the modern law of momentum since he believed that it is weight times velocity rather than mass times velocity that is conserved" (and as you know, this makes a big difference if it is applied to the Earth-Moon system, as he does). Indeed, what is a correct statement? even $mv$ is corrected in "modern" formulations. The article picks on Descartes' faults and ignores Wallis'. Edward Slowik, who is quoted as a reference ([57]), himself states in Stanford Encyclopedia "Not only did Descartes provide the first distinctly modern formulation of laws of nature and a conservation principle of motion,..." (my emphasis). My suggestion would be to rewrite this section, emphasizing Descartes' contribution more (while retaining that it is not totally correct), and amending the Wallis paragraph to include Huygens as joint originator. Otherwise it misleads the reader; meanwhile, no harm done in waiting for a historian of physics. — Preceding unsigned comment added by Chrystomath2 (talk • contribs) 07:59, 2 December 2019 (UTC)

Please sign all your talk page messages with four tildes (~~~~) and indent them as outlined in wp:THREAD and wp:INDENT — See Help:Using talk pages. Thanks. - DVdm (talk) 10:25, 2 December 2019 (UTC)

As I said above, you're welcome to rewrite the coverage of Descartes, Wallis and Huygens. I could do it in a couple of weeks, but I'm very busy now. There is no point in waiting for an expert. I have been editing Wikipedia for a decade and I don't know of a single example where an expert actually responded to an expert tag. RockMagnetist(talk) 16:43, 2 December 2019 (UTC)

Section: Kinetic momentum

Please, add "kinetic momentum" to the synonyms in first line. — Preceding unsigned comment added by 81.40.235.248 (talk) 18:51, 31 August 2020 (UTC)

I think that would be confusing because it is a type of momentum, not momentum in the broad sense. It is already mentioned in the third paragraph of the lead. I think that changing that mention to boldface would be justified, though, as there is a redirect here. RockMagnetist(talk) 14:35, 1 September 2020 (UTC)

Semi-protected edit request on 27 October 2020

Change "In Hamiltonian mechanics, the Lagrangian (a function of generalized coordinates and their derivatives)" to "In Hamiltonian mechanics, the Lagrangian (a function of time, generalized coordinates and their derivatives)" 178.23.234.209 (talk) 20:38, 27 October 2020 (UTC)

 Not done: please provide reliable sources that support the change you want to be made. Goldsztajn (talk) 22:16, 27 October 2020 (UTC)

Hamiltionian

There is discussion in Talk:Hamiltonian on where Hamiltonian should be directed. It seems now to direct to the classical version of Hamiltonian mechanics. Yet more often now, it is likely that the quantum mechanics version is meant. It then occurred to me that most likely momentum would direct to the classical explanation and not the quantum version, and it does seem that we have Momentum operator for the quantum version. (I might wonder why it isn't Momentum (quantum mechanics), but otherwise.) Without actually suggesting it, should all the quantum version be in its own article, with only classical here? Gah4 (talk) 12:20, 1 November 2020 (UTC)

My feeling is that a discussion of the quantum version belongs in this article, with perhaps a more extensive discussion elsewhere (I agree with you about the name, but that's best discussed at Talk:Momentum operator). We already have Newton, relativistic and generalized momentum, and it's useful to have the different approaches beside each other for comparison. RockMagnetist(talk) 23:09, 1 November 2020 (UTC)

Relativistic

"Momentum is also conserved in special relativity (with a modified formula)".

This phrase misleading, it is not the momentum that has been defined in previous paragraph the one that is conserved, but the spatial components of the four-momentum.

To be clarified. — Preceding unsigned comment added by 81.40.239.244 (talk) 12:43, 28 November 2020 (UTC)

Momentum as a political usage

When I type in momentum it comes to this article which is about physics but that is not why I was looking for momentum.

Whenever I hear the word momentum it has something to do with politics not physics.

This article makes no mention about the word being used in politics.

I asked that this be rectified or at least create an article around the political usage of momentum. 49.3.72.79 (talk) 04:46, 13 February 2021 (UTC)

The same thing could be said about momentum in economics and momentum in entertainment. That's why we have that line at the top of the article, saying "This article is about momentum in physics. For other uses, see Momentum (disambiguation)." We don't have an article about momentum in politics, and if we had one, we could (and should) have an entry on the disambiguation page, solving your problem. Hope this helps. - DVdm (talk) 12:11, 13 February 2021 (UTC)

I suspect that all the non-physics momenta are related, and could have a single article, with redirects. There is also sports momentum (defying statistics). Gah4 (talk) 17:59, 13 February 2021 (UTC)

when bodies come in contact there is always some dissipation

The article says: when bodies come in contact there is always some dissipation. Classically, it is hard to avoid some dissipation, but in quantum systems, there can only be dissipation if there are states available for energy to go into. When atoms in a gas collide, there is not normally enough energy to get one to the next available energy state. Or, one might say that atoms never collide, but instead there is repulsion from the electron cloud. But it is Fermi exclusion more than electron repulsion that keeps atoms apart. In any case, in quantum systems there is not always some dissipation. Gah4 (talk) 18:35, 16 June 2021 (UTC)

Format Introduction for iPhone

On my iPhone 11, the introduction says "then the object's momentum p is", then shows the billiard balls with caption, then the formula floating all by itself below. If it's possible, if someone could get the formula to stick with the paragraph it would be easier to follow. Thanks. Pbackstrom (talk) 13:53, 30 December 2021 (UTC)

Novel Equation for Momentum - Reg;

following is for editing "Momentum" Page on Wikipedia.org:

NOTE : Recent research findings reported a novel equation for Momentum, which is NOT based upon mass and velocity of the body under motion but, it is based upon the mass and "frequency" of the particle under motion. You could find that research work at the DOI given below. This work is available online, for your kind information.

Research Article Title: "Kinetic Energy of a Particle Independent of its Mass & Velocity"; DOI:10.9790/4861-1501013441

So, I request you to kindly update this novel equation and derivation on Wikipedia for the benefit of the wide readers of Wikipedia.

Thank you

PS: I am suggesting the following edits for addition [from START to END, present below] into the "Momentum" Page of Wikipedia; you could add more if you wish. Related novel equation for kinetic energy is also discovered in that research work at the above DOI document. You could update it as well on the Kinetic energy page of Wikipedia.

---START of EDIT---

Novel equation for "Momentum" recently discovered by research and reported in the scientific literature is given below. This discovery was done by the Indian researcher Mr. Bhargava R. Kotur, as recorded in the above DOI document.

p = sqrt (h. n . m)

In the above equation, p - stands for the Momentum, f - is the frequency of the body or particle under motion, m - is the mass of the body, and h - is the famous Planck's constant, sqrt - stands for the mathematical function "square root";

The derivation of the above work could be found at the DOI already mentioned earlier.

--- END of EDIT --- Ganesh Siva75 (talk) 19:28, 10 February 2023 (UTC)