Talk:Newton's laws of motion

The Introduction

In the introduction is a list of areas which require improvements in Newton's formulation. It mis-characterizes the changes for special relativity and general relativity as being for high speeds and for very massive; but the reason for both of their discoveries are the allied reasons that the speed of transmission of information is finite and is still observer independent. And quantum mechanics should be characterized as being increased understanding of what constitutes information. In other words, special relativity is true at all speeds, general relativity is true at the level of an apple pip, and quanntum mechanics is true at the level of the universe writ large. YouRang? (talk) 16:49, 26 November 2024 (UTC)

Here is the entire sentence from the current introduction:

• Limitations to Newton's laws have also been discovered; new theories are necessary when objects move at very high speeds (special relativity), are very massive (general relativity), or are very small (quantum mechanics).

The sentence accurately summarizes part of the section "Relation to other physical theories". That is all it needs to do. The sentence is not here to characterize relativity or quantum mechanics. Johnjbarton (talk) 17:24, 26 November 2024 (UTC)

Incorrect formula for variable mass

I had to fix the formula for variable mass displayed in the article. A common misconception among physicists is that

$F={\frac {dp}{dt}}$

is the most general form of Newton's equations, valid both for constant and variable mass. That opinion is absurd, as can be readily seen by applying the derivative:

$F=m{\dot {v}}+v{\dot {m}}$

and noticing that it is not Galilean invariant! Any classical physical law must, by necessity, be invariant under Galilean transformations; otherwise, it would yield different physical results for observers moving at different speeds.

As an example, imagine that a bucket filled with sand is put on a support, and that a hole is made in the bottom, such that the sand can now fall to the ground, reducing the mass of the bucket. Since both $v_{x}$, ${\dot {v_{x}}}$ and $F_{x}$ are 0 in the bucket's frame of reference, the equation checks out. However, if an observer is moving, and looking at the same system, he would find out that

$F_{x}=mv_{x}$

since the velocity is constant, but non-null. In other words, the moving observer would find out that there is a horizontal force in a system just because the sand is flowing downwards, which is absurd. The forces computed in every frame of reference must be the same.

People who give this wrong opinion (that $F={\frac {dp}{dt}}$) frequently give examples where the mass ejection happens at the same coordinate axis as the velocity. They, then, neglect the fact that the ejected mass produces a force. Those cases are conveniently select where the force produced by the ejected mass is $-v{\dot {m}}$, yielding the former equation when you completely disregard the fact that this term should be in the "force" side of the equation, not on the other one.

It turns out that $F=ma$ is the correct equation, both for constant and variable mass, which can be easily verified if you adopt the frame of reference where the object is currently at rest, since then $v=0$ and $F=ma$ in that frame, but, from the principle of covariance, that means the same equation must be true in all frames of reference, since both sides are galilean invariants. — Preceding unsigned comment added by Jocryptowiki (talk • contribs) 16:08, 6 March 2025 (UTC)

If you have a source which backs your claim then we can add it to the article. Johnjbarton (talk) 17:13, 6 March 2025 (UTC)

There is no source for the claim that F=dp/dt for variable mass, which is a evidently wrong equation that violates Galilean invariance. If you require a source at all, then no statement should be left on the article regarding variable mass, especially a statement as shocking as "the relativity principle of Galileo does not hold for variable mass". I advise that, if a source for either F=ma or F=dp/dt is required, then any mention to variable mass should be removed from the article until sources are provided. Otherwise, you'd be acting like someone who insists that a statement like "145*26=3800" should be left in an article, unless it is provided a specific source that 145*26=3770 instead. My opinion is that, if we require sources to anything, then the obviously wrong statement "145*26=3800", while lacking a source, should be removed. Yet, any source claiming such an absurdity should also be immediately regarded as non reliable.

If, on the other hand, your worry is about whether F=ma is the correct equation (instead of any other possibility), then I should remark that it becomes immediately obvious if you realize that F=ma is valid for each classical particle (atoms of each element not changing mass), and that F = sum_i F_i = sum_i m_i a = ma, by the aditivity of these quantities, at any given time. Jocryptowiki (talk) 19:53, 6 March 2025 (UTC)

Contrary to your assertion that There is no source for the claim that F=dp/dt for variable mass, you may read Chapter 4 of

• Kleppner, Daniel; Kolenkow, Robert J. (2014). An introduction to mechanics (2nd ed.). Cambridge: Cambridge University Press. ISBN 978-0-521-19811-0. OCLC 854617117.

The later parts of the chapter include multiple examples of variable mass systems. Johnjbarton (talk) 23:34, 6 March 2025 (UTC)

This, here, is a direct reference regarding the matter.

https://articles.adsabs.harvard.edu/full/seri/CeMDA/0053//0000227.000.html

I should point out that, if Wikipedia ought to be regarded as at least internally consistent, we should either delete the entirety of the article https://en.wikipedia.org/wiki/Variable-mass_system and replace all the formulas expressed there with F=dp/dt, or address the issue more intelligently. Newton's second law, as an expression of the conversation of momentum, should in general state that the applied forces, MINUS the momentum leaving the system, equals the change in momentum in the system. Jocryptowiki (talk) 17:18, 7 March 2025 (UTC)

I see some edits were made simultaneously (so I'm assuming no bad faith on your part, we both want to address the issue). I've put more references (such as Sommerfeld's textbook), and tried to address your claims, but I think the text can be made to be more succinct, while referencing the other article.

I still think that, either nothing should be mentioned regarding variable mass (only the main article pointed to) or the correct, principled, formulas should be presented. F=dp/dt is wrong in principle. It is the exact same situation as if you stated that the heat applied to a system is the variation of internal total energy in the system, WHEN there is matter leaving the system. That cannot be true. The heat, MINUS the kinectic energy of the mass leaving the system, would be the amount of energy leaving, etc.

Many different sources point out F=dp/dt as a mistake (not a "non-careful application"), which seems to come from Kleppner's book. Indeed, you need to make careful applications for a wrong equation to yield the correct results! The total variation in the momentum of a system is the force applied PLUS the momentum entering directly via movement of mass. That momentum that is entering (or leaving) the imaginary boundaries of the "system" cannot be regarded as a force, since it is a mere product of inertia. Jocryptowiki (talk) 18:26, 7 March 2025 (UTC)

I included a description of why F=dp/dt can lead you astray. Sommerfeld makes essentially the same arguments as Kleppner. Johnjbarton (talk) 19:27, 7 March 2025 (UTC)

Singularities

After discussion "Flying off to infinity in a finite time" on the Science Reference Desk I have reworded the section. Other interested parties Malypaet, Baseball Bugs, jpgordon, Trovatore, PiusImpavidus and Wrongfilter were invited to comment here.

Previous text:

Singularities

It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time. This unphysical behavior, known as a "noncollision singularity", depends upon the masses being pointlike and able to approach one another arbitrarily closely, as well as the lack of a relativistic speed limit in Newtonian physics.

changed to:

Singularities

Mathematicians have investigated the behaviour of collections of point masses that may approach one another arbitrarily closely, possibly collide together, and move in accord with Newton's laws. In studies that assume no relatavistic speed limit, singularities of unphysical behavior are predicted. For example, a particle velocity can accumulate through successive near-collisions to the extent of theoretically departing the system to infinity in a finite time. Philvoids (talk) 14:32, 13 January 2025 (UTC)

I prefer the old version. Starts with the unnecessary personalisation ("mathematicians") and continues with the "prediction" (predicted for what? Nobody expects these things to be observable) and the assumption of no relativistic (sic!) speed limit — these considerations are fully within Newtonian theory, which quite simply has no speed limit. I see these studies as investigations of the mathematical structure of the theory, not of the physics that the theory is supposed to describe; the old version states that more clearly. --Wrongfilter (talk) 15:33, 13 January 2025 (UTC)

I think it would be helpful if you summarized the issues you have with the existing content. Johnjbarton (talk) 15:54, 13 January 2025 (UTC)

I think this paragraph belongs in the Multibody section. Here is a different suggested paragraph based on the sources:

• The stability of a system of multiple point point masses was introduced as a mathematical problem by Paul Painlevé and Henri Poincaré. Against intuition, systems with at least 5 point masses exhibit "non-collisional singularities" in which a particle may be launched to infinity in finite time. In the 5 particle case, a light particle oscillates between two pairs of heavy particles, each in a binary orbit; in every cycle the binary orbits shrink and the two pairs move further apart in a way that grows out of control. This problem is related to other mathematical singularities in physical models like Zeno's paradox, self-energy, and the center of a black hole.

Johnjbarton (talk) 16:26, 13 January 2025 (UTC)

A paragraph on this could work in either place, I suppose, but I think it's a little better in its current spot, where it follows the discussion of unpredictability in the previous subsection, and can then segue into Navier–Stokes existence and smoothness. Mentioning who introduced what and when would blend the history in with the concepts, which isn't in principle a bad thing but is different from how the article is currently structured. The phrasing "This problem is related to other mathematical singularities..." feels a little vague. What's the relation, apart from them all being examples of mathematical singularities? XOR'easter (talk) 21:07, 13 January 2025 (UTC)

The refs are about math (mathematical physics perhaps) but the two other summaries focused on physics picked out of the sources. The content sounds like it it trying to explain away the non-intuitive result in an unsourced way (Baez says "the set of initial conditions for which two or more particles come arbitrarily close to each other within a finite time has ‘measure zero’.", that is, phase space is too big). The Painleve/Poincare sentence was intended to position the issue in math. We could start with something based on Saari/Xia's concluding sentence: "the Newtonian n-body problem serves as a source of intriguing mathematical problems".

Baez groups these as "problems that arise from assuming spacetime is a continuum", so we could just say as much. Johnjbarton (talk) 22:54, 13 January 2025 (UTC)

I don't see how the previous version tried "to explain away the non-intuitive result in an unsourced way". It seems a reasonable paraphrase of Saari and Xia's phrasing: It turns out that particles must approach other distant particles infinitely often and arbitrarily closely; First, the particles must shuttle among each other infinitely often causing arbitrarily close approaches. And Baez says, Of course this isn’t possible in the real world, but Newtonian physics has no ‘speed limit’, and we’re idealizing the particles as points. So, if two or more of them get arbitrarily close to each other, the potential energy they liberate can give some particles enough kinetic energy to zip off to infinity in a finite amount of time! After that time, the solution is undefined. The idea that the set of initial conditions leading to these outcomes has measure zero is conjectured, but unproven. XOR'easter (talk) 00:18, 14 January 2025 (UTC)

The lack of a relativistic speed limit in Newtonian physics doesn't cause or prevent the behavior. As far as I can tell no one has worked on the relativistic problem. So Baez is only saying this is a math problem.

Saari and Xia say for the 5 body problem, "Initial conditions leading to a Xia type example are in a set of Lebesgue measure zero;" and Saari's previous work eg (Improbability of Collisions in Newtonian Gravitational Systems Donald Gene Saari, Vol. 162 (Dec., 1971), pp. 267-271) suggests this a characteristic. The previous content "depends upon the masses being pointlike and able to approach one another arbitrarily closely" amounts to the same thing. This is Baez's overall point about the continuum. The infinity that is at work here isn't one flown-off-to so much as the one the model work within.

But on the scale of all Wikipedia paragraphs, the existing content is fine. I was just trying to suggest a way forward. Johnjbarton (talk) 01:56, 14 January 2025 (UTC)

I prefer the old version, which avoids the "mathematicians" (too specific — are you 100% sure none of the people who have studied this are physicists?) and "predicted" (it's not a prediction about any realistic system). The new version also includes awkward phrasing, like "to the extent of theoretically departing". XOR'easter (talk) 20:58, 13 January 2025 (UTC)

This is the old first sentence that was questioned at the Science ref. desk: "It is mathematically possible for a collection of point masses, moving in accord with Newton's laws, to launch some of themselves away so forcefully that they fly off to infinity in a finite time." It is a poor lede if read isolated from the proviso "This behavior...depends..." as a general reader is liable to do. I cannot qualify whether people who have studied this are physicists or not but the study is a mathematical one. There was an objection to calling the studies "simulations" so I arrived at the term "predicted" qualified as (only) theoretical. I agree the new version phrasing is awkward. Philvoids (talk) 23:57, 13 January 2025 (UTC)

If this "general reader" is unable or unwilling to read all of two sentences, I don't know what we can do for them. XOR'easter (talk) 00:20, 14 January 2025 (UTC)

We could just reorder the content:

• It is mathematically possible for a collection of point masses, moving in accord with Newton's laws and able to approach one another arbitrarily closely, to exhibit unphysical behavior. Under certain contrived conditions they can interact to launch some of themselves away so forcefully that they fly off to infinity in a finite time. This unphysical behavior is known as a "noncollision singularity".

Johnjbarton (talk) 02:07, 14 January 2025 (UTC)

"Fly off to infinity in a finite time": Escape speed?

Or, what concludes Saari and Xia: "The resulting Cantor set of initial conditions allows r_max(t) to approach infinity in finite time without prior collisions."

I like the verb "to approach," which gives a different sense than the title: I approach infinity every day when I walk in a straight line for a finite time. Malypaet (talk) 00:04, 16 January 2025 (UTC)

References

Getting better.

• The abstract idea of a singular point-like particle that has mass but zero size has received study because of its implied ability to accelerate infinitely close to collision with similar particle(s). For example Zhihong (Jeff) Xia reports ^ref a contrived 5-body arrangement where point particles moving in accord with Newton's laws but without relativistic speed limit results in a particle being recursively accelerated to unphysical infinite speed. The example is known as a "noncollision singularity". Philvoids (talk) 16:55, 14 January 2025 (UTC)

Sorry I don't believe the sources support this version. Johnjbarton (talk) 17:48, 14 January 2025 (UTC)

The phrasing The abstract idea...has received study is indirect and awkward. Everything we talk about here has been studied; that's tautological. The term contrived might be accurate but sounds oddly judgmental. Is it really more "contrived" than motion in a perfect circle, or an oscillator that is perfectly harmonic? What distinction is being made here? Saying Newton's laws but without relativistic speed limit is redundant. XOR'easter (talk) 00:08, 15 January 2025 (UTC)

Using the passive voice in has been studied avoids the objection by Wrongfilter to unnecessary personalisation. No one claims that everything here has been studied for the same reason. When properly read in whole the sentence reveals the singular unphysical speculation that motivates the study. There's no tautology here. We introduce these facts without fearing a sceptic who might judge the work a doomed GIGO effort. That judgement can be his/her opinion that we neither rebut, encourage nor try to dodge. The term contrived is well chosen. I am sorry that XOR'easter's examples of an idealised perfect circle or harmonic oscillator that would be useful in other contexts are only irrelevant strawman objections to the text considered here. Saying Newton's laws but without relativistic speed limit may seem redundant but is quite lucid to a modern reader who needs to be told that an arbitrary simplification that isn’t possible in the real world (Baez) is used. Philvoids (talk) 14:21, 15 January 2025 (UTC)

The version that starts "The abstract idea of a singular point-like... " mischaracterizes the sources. The grad student Xia should not be named if Poincare does not make the list. The vast majority of the content on this problem is mathematical, but the content focuses on the lack of physics. The effects of relativity do not invalidate this work in any way, by design. In fact Xia/Saari say this outright:

• With special relativity, for instance, all velocities are bounded by the speed of light, so f (t) = ct. But Newton’s universe fails to respect Einstein’s formulation; once n ≥ 4, no such f (t) exists for Newtonian n-body systems!

If we have a limited the content budget we can't spend all of it explaining why this can't happen in "reality". We should either explain the work or not, we should not be characterizing it in ways inconsistent with the sources. This is a mathematical result illustrating unexpected outcomes within Newtonian mechanics and it fails to occur in Newtonian systems because the continuum of phase space is infinitely dense. There is no evidence that 5 body systems can be caused to reach 5% of the speed of light in experimental conditions so we don't need to explain why this non-thing does not happen. Johnjbarton (talk) 16:29, 15 January 2025 (UTC)

Yes, the passive-voice construction here avoids saying "mathematicians", but it's overlong and confusing in its own way. The phrasing accelerate infinitely close to collision is hard to parse; does "infinitely" modify "accelerate" or "close"? Is that implied ability the only reason why people study the abstract idea of a singular point-like particle that has mass but zero size? Moreover, this phrasing implies that the motivation is that implied ability to accelerate infinitely. But is that so? Is that what motivates people to study these scenarios, or are they really interested in something more fundamental, like what it means to assume that space is a continuum, and the implied ability to accelerate infinitely is just how that concern manifests? XOR'easter (talk) 18:25, 15 January 2025 (UTC)

I remind that the title of the section is Singularities. The point-like particle with its implied ability to accelerate infinitely (adverb) is the first singularity that we identify. Do you require a footnote to explain the rationale of linking Newton's law of universal gravitation when separation r = zero with Newton's 2nd Law of Acceleration when F is infinite? I think No. After agreeing that personalisation is unnecessary I don't think XOR'easter helps by posing strawman questions probing what "really interests" or exclusively motivates anyone. The motivation of our 3 concise sentences is only to report the published singular mathematical result. Philvoids (talk) 14:57, 16 January 2025 (UTC)

It's not a "strawman"; it's a legitimate question about whether your suggested text implies something that the sources cannot support. XOR'easter (talk) 18:45, 16 January 2025 (UTC)

My post stands with my word "strawman" struck out to assure you that I don't think you have posted anything illegitimate. Thank you for pointing that out. Philvoids (talk) 00:41, 17 January 2025 (UTC)

The evolved text:

Singularities

• A singular point-like particle that has mass but zero size has implicit ability to accelerate arbitrarily close to collision with similar particle(s). In mathematical analysis of systems of point particles obeying Newton's laws Xia reports ^ref the example of a 5-body arrangement contrived to allow a particle that shuttles between two other particle pairs, each in binary gravitational orbit, to accelerate recursively towards infinite speed in a "noncollision singularity" that cannot be further analysed. This unphysical result reflects the lack in Newtonian physics of Einstein's special theory of relativity that implies that only particles with zero rest mass may travel at a finite speed limit, and that nothing may travel faster. Philvoids (talk) 00:30, 23 January 2025 (UTC)

  only particles with zero rest mass may travel at a finite speed limit

  This phrase, at least, is poorly worded. My car travels at a finite speed limit every day, and it has nonzero rest mass. CodeTalker (talk) 22:11, 23 January 2025 (UTC)

  Rewording: only a photon with zero rest mass may travel at a finite speed limit, and that nothing may travel faster. Philvoids (talk) 16:15, 24 January 2025 (UTC)

  I'm afraid that's even worse. Now it's not just unclear but actually incorrect. The incorrect part is the claim that "only a photon" travels at c. Any massless particle travels at c. The unclear part is using "finite speed limit" to mean c. To me, "finite speed limit" may refer to any speed limit, such as the regulatory speed limit of 25 mph on the road outside my house. CodeTalker (talk) 18:24, 24 January 2025 (UTC)

  Einstein's special theory of relativity refers to photon travel. When you read "a finite speed limit" and see that it is a wikilink to speed of light it is time to pull your mind away from the dirt track outside your house. Philvoids (talk) 10:08, 25 January 2025 (UTC)

  I would agree, but we're trying to write for a general audience, who might love their dirt track more than life itself. Remsense ‥ 论 10:14, 25 January 2025 (UTC)

I oppose this version as well. That does "implicit ability to accelerate arbitrarily close to collision" mean? These particle cannot collide. They don't have "abilities", they are abstractions. Also Xia is not notable. The concept has not been studied with special relativity. Both infinite and special relativity are very non-intuitive. Does special relativity have non-collisional singularities? We do not know and should not speculate. We could cite the professional speculation. Note that the Xia ref talks about the particle going to infinity not about infinite speed. Johnjbarton (talk) 18:26, 25 January 2025 (UTC)

I sense the cadence of a Gish gallop. Mathematics Professor Zhihong (Jeff) Xia is notable as the originator of "Xia's five-body construction" drawn in Figure 3 on page 543 and reported in the 2nd lede sentence of the AMS peer-reviewed reference that you yourself refer to as "the Xia reference". Any general reader understands that allowing two masses to occupy exactly the same location is a recipe for a collision. A mass characterised as point-like is easier for a mathematician to understand than for a non-specialist. We help our general reader by naming this object's special ability or, if you prefer, its special property or attribute. I see that you have quoted approvingly the phrasing "being...able to approach one another arbitrarily closely". Regardless of anyone "talking about" Xia's particle going to infinity, that speculation is neither proven nor possible nor should Wikipedia venture it so. Your own reasons for not attempting to apply Special or "infinite" - you mean General surely - Relativity to the reported singularity are good. This is not the occasion to study "Does special relativity have non-collisional singularities?" Philvoids (talk) 16:50, 27 January 2025 (UTC)

Please do not characterize my replies in a derogatory way. You posted a new version with the same issues as the previous ones which you did not address.

With no disrespect to Xia, Xia is not a notable historical figure or well-known expert. Xia isn't even the first author. There is no reason to include the name in the article.

No I do not mean general relativity.

The article should summarize the published content in a way consistent with the character of the content. The published content is a mathematical study of abstractions used in Newtonian mechanics. It is not physics. Your claim that "This unphysical result reflects the lack in Newtonian physics of Einstein's special theory of relativity" is original research. The profession conjecture by Saari is that the very special initial conditions needed to trigger the singularity are unrealizable. (The analog would be an infinitely sharp pencil balanced on its point.)

If we want to clarify the issue for general readers, we should do so in the opening sentence. We should explain that these singularities occur with infinitely small masses governed by classical physics and placed in special positions. We should not speculate on the other cases. Johnjbarton (talk) 19:02, 27 January 2025 (UTC)

This is awkwardly written (has implicit ability, In mathematical analysis [...] Xia reports, etc.), and it just makes the concepts harder to understand. XOR'easter (talk) 19:39, 25 January 2025 (UTC)

Split proposal

Greetings! When I came across this page, I'm rather surprised that there are no individual articles for all three laws. This notion is simply due to the fact that they are so important, and should deserve individual articles for more detailed description. However, I'm not very sure, so I'm asking for opinions here. Pygos (talk) 11:03, 28 January 2025 (UTC)

This page, of course, shall remain, as it describes the underlying assumptions of the laws and their historical background. Pygos (talk) 11:07, 28 January 2025 (UTC)

The article is really an introduction to Newtonian mechanics, which happens to be described in three parts. Each individual law is not, by itself, useful or interesting. I believe three articles would end up being mostly repetition in order to explain how each law contributes to what is a single thing. Johnjbarton (talk) 16:38, 28 January 2025 (UTC)

The three laws together form a conceptual unit. Each one really needs to be understood in the context of the others. Splitting them into separate articles would make for needless repetition. XOR'easter (talk) 20:07, 28 January 2025 (UTC)

@Pygos It seems that your proposal did not get traction. Would you consider removing the tag in the article? Johnjbarton (talk) 02:03, 7 March 2025 (UTC)

Done Pygos (talk) 04:32, 7 March 2025 (UTC)

Any linguist is the room?

Neither English nor Latin are my first language. However I can understand the meaning of «unless» and «except insofar as» sounds quite cryptic to me, but it seems as a pretentious way to say unless.

What Newton wrote was

Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressis cogitur statum illum mutare.

• nisi means unless, except
• quatenus means in the extent that

Example:

Eadem est asini et cuiusvis imperatoris post modicum tempus gloria, nisi quatenus memoria alterutrius scriptorum beneficio prorogatur.

— John of Salisbury, Policraticus, Prologus,

The reputation of the fool and the emperor is the same after a moderate period of time except in the extent that the memory of either is prolonged by the beneficence of writers.

— Translation from Latin

I do not get why there was the change made to that more obscure translation, I've not been able to find a single Latin dictionary translating quatenus as «insofar as» and I do not think that is the traditional translation into English of the first law. So, why???

--77.75.179.1 (talk) 00:28, 20 February 2025 (UTC)

The current phrasing "except insofar as it is acted upon by a force." is just utter nonsense. No body speaks or thinks that way. It appears as a brute force attempt of word-by-word translation. kbrose (talk) 19:48, 21 February 2025 (UTC)