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The question What's wrong with this argument that Newton's second law implies all potentials are quadratic? has been closed as being “non-mainstream”. But one could argue that this would come under the clause:

Any post that attempts to work within the bounds of what we have determined to be "mainstream physics" is considered on topic for this site barring any other issues.

The question arose due to a misunderstanding of notation (a surprisingly common issue) and was clarified in the answers, but is still working within the bounds of “mainstream physics” and I don’t see any other issues.

So for future reference, are questions of this kind considered non-mainstream by the community?

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    $\begingroup$ For the record and since it is invisible to non-moderators afaik, only three close votes are due to "non-mainstream". The other two were "needs details or clarity". $\endgroup$ – ACuriousMind May 12 at 16:06
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    $\begingroup$ And since I am one of the person who voted to close the question, I confirm what ACM said above. I voted to close it because it lacked clarity. Though I am a bit surprised to see it getting closed as non mainstream... $\endgroup$ – FakeMod May 12 at 16:34
  • $\begingroup$ It is hard to say whether it is mainstream or not, because it lacks clarity! For one thing, what is this talking about potentials? There are no potentials in Newton's second law. $\endgroup$ – Norbert Schuch May 12 at 17:13
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    $\begingroup$ @NorbertSchuch I think it is fairly obvious that the OP has (incorrectly) concluded that all forces are linear, so then all potentials are quadratic since $F=-\nabla U$. No one said Newton's laws explicitly talk about potentials. How is that unclear? $\endgroup$ – BioPhysicist May 12 at 17:43
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    $\begingroup$ @AaronStevens Seriously, I have no idea what the OP is talking about. All of a sudden, they talk about potentials. Why should the force relate to a potential? This is either completely ad hoc or non-mainstream. Somewhere, implicitly, and without any clear mention, must be the claim that the force comes from a potential. Then, indeed, this seems to suggest that the force is linear, i.e. the potential quadratic. But why make this assumption? It is completely unjustified, not being explained in the question, and against the laws of Newtonian mechanics. Why should this be re-opened? $\endgroup$ – Norbert Schuch May 12 at 18:04
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    $\begingroup$ @NorbertSchuch "It is completely unjustified...and against the laws of Newtonian mechanics." Hence the question was posted. Also any linear function can be described by a quadratic potential. This follows naturally from the definition of potential energy. It isn't some additional assumption. $\endgroup$ – BioPhysicist May 12 at 18:09
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    $\begingroup$ The question is open now $\endgroup$ – BioPhysicist May 12 at 19:41
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    $\begingroup$ @AaronStevens The issue is that without any further comment, it is used that F(x) must relate to a potential. There is no reason for that. Also, hypothesizing that "this inconsistency related to the fact the classical mechanics is not a complete description of nature" is, for the best, obscure, and basically non-mainstream physics. $\endgroup$ – Norbert Schuch May 12 at 20:03
  • $\begingroup$ @NorbertSchuch If $F(x)$ is linear in $x$, then it automatically has a quadratic potential energy associated with it. The OP incorrectly finds that all forces are linear. It directly follows (not as an additional assumption) that the potential of these forces are quadratic. I do agree that the conclusions following this that you just quoted are not valid at all. But the OP is asking about how their incorrect method arrived at this. It doesn't do much to criticize incorrect conclusions derived from incorrect work. It would be better to just address the work. $\endgroup$ – BioPhysicist May 12 at 20:08
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    $\begingroup$ @AaronStevens But why should F(x) come from a potential? That's something I don't get! This is implicitly used and there is no reason whatsoever to do so. Not every force comes from a potential! (You seem to make the same assumption.) $\endgroup$ – Norbert Schuch May 12 at 20:09
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    $\begingroup$ @NorbertSchuch Read the question again, I never say that all forces come from a potential, but on the contrary, all potentials have a force associated with them. If all forces are linear, then all potentials will be quadratic, I don't see how this statement is so controversial. $\endgroup$ – Godzilla123 May 13 at 13:30
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    $\begingroup$ @miroslav351 The very fact that you write $F(\vec x)$ assumes that F only depends on the position. Why would that be the case, if not if it comes from a potential? If at all, it would make sense to write $F\equiv F[\vec x(t)]$ in the sense of a functional. As you can see, the r.h.s. of the equation does not depend on the position, but it's second time derivative. I stick to what I said: You make a very strong implicit assumption in writing $F$ as $F(\vec x)$. Whether this is a potential or something in the same vein is hard to tell. (That's the problem with implicit assumptions.) $\endgroup$ – Norbert Schuch May 13 at 13:42
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    $\begingroup$ @NorbertSchuch Oh yes, I completely agree with this last comment, that was the problem in the question anyway and several people were kind enough to point this out very clearly. I was just confusing myself with very bad notation! $\endgroup$ – Godzilla123 May 13 at 13:44
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    $\begingroup$ @miroslav351 Let me state it differently: You say "Newton's second law states $F(\vec x)=m\ddot{\vec x}$." That's not what Newton's second law states. It states $F=m\ddot{\vec x}$. There is not $F(\vec x)$. $\endgroup$ – Norbert Schuch May 13 at 13:44
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    $\begingroup$ @miroslav351 Good to see that we agree. (Although I still feel the answers are partly too complicated and/or not the point. You simply can't write $F(\vec x)$, that's pretty much it ... ) $\endgroup$ – Norbert Schuch May 13 at 13:52
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Based on some of the comments (now deleted), it looks like the first user who voted to close did so because they were focusing on how the OP was trying to apply scaling space to Newton's second law. Perhaps the first user figured that the OP was talking about their own theory of scaling space then? Although in my (speculative) opinion, this user seemed upset about the OP's question and their approach, but instead of supplying an answer to help relieve the confusion a sort of debate in the comments was opened up. So perhaps the closure was initiated due to what happened in the comments. Then others saw this closure reason and followed along with it.

Of course, being a user who answered this question, I do not think it should have been closed (I have cast a reopen vote). I do not think that this question falls into "non-mainstream" at all. I can understand why users would vote that it needs more clarity, but I do no agree with this either. The question makes sense to me, and so I supplied an answer addressing what I believed the confusion of the OP was.

As ACM has pointed out, only three of the five users voted to close as non-mainstream. Therefore, I don't think this question is something that is a good example of this closure reason, and it should not set precedence moving forward.

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    $\begingroup$ When I saw this question, I figured the OP was just another one of those people who thought they'd disproven physics with a few lines of bad math, so I left a snarky comment and a close vote. Later I realized OP was sincere, so I added more substantial comments and voted to reopen. $\endgroup$ – knzhou May 12 at 19:08
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    $\begingroup$ That said, I am extremely disappointed that this is the top question of the day. I answered it in comments precisely because I didn't want an answer with my name associated with it... $\endgroup$ – knzhou May 12 at 19:10
  • $\begingroup$ @knzhou I agree, I don't think it deserved HNQ status. I was under the impression that comments should never be used for answers. Is my understanding too strict? $\endgroup$ – BioPhysicist May 12 at 19:24
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    $\begingroup$ See physics.meta.stackexchange.com/q/6978/50583 and its linked discussions about answering unclear questions in comments vs. giving a "guessing answer". $\endgroup$ – ACuriousMind May 12 at 20:58
  • $\begingroup$ @ACuriousMind So the answer is "don't ever post answers in comments"? :) $\endgroup$ – BioPhysicist May 12 at 21:24
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    $\begingroup$ @knzhou Even if OP was one of "those people" (amazing that you are able to tell that based off a few sentences), it is not in the spirit of this site to not answer such a question. Answering a question does not only help OP, and contributes to making this site a comprehensive FAQ "encyclopedia". Of course its an entirely different matter if it isn't clear what the question asks (it is "not even wrong"), but that was not the case here. What is gained by casting a close vote and adding a "snarky comment"? $\endgroup$ – Jannik Pitt May 22 at 11:05
  • $\begingroup$ @Jannik Pitt Because we’ve had thousands of insincere crackpot questions asked on this site, by people who are really only looking to promote their personal theories of everything. Without negative feedback, they would constitute a huge fraction of the site. I’ve seen it happen before, and it’s not pretty. $\endgroup$ – knzhou May 22 at 16:54
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I am the OP in question and I feel like I should post some clarifications, since some people seem to be jumping to even further conclusions than I did in the post. I don't think there is anything non-mainstream about my question, it was mostly a question about sloppy maths and I was well aware that I was in the wrong when I posted it but couldn't see why mathematically. I deliberately used a provocative wording to attract good replies, but had I known it would attract that much attention, I would not have posted it at all. I use physics for a living and I am well aware of the meaning of these equations (and the correctness of physics) but I was just confusing myself over something trivial and decided to ask other people instead of agonising over my bad maths.

The original post never really implied that "physics is wrong". It asked whether this silly equation reflects something deeper about classical mechanics - and we do know that classical mechanics is not a complete description of nature - again purely as speculation and I didn't think much of it. Again, had I known that this would cause so much upset, I wouldn't have included it.

Finally, I would also like to point out that unless the question is obviously anti-science, it benefits everyone to treat even the silliest-looking questions seriously. After all, if they are so obviously wrong, then they should be very easy to disprove. It seems that some people regard questions on everything that they have understood well as trolling / questions that should be censored, and this is in my view not a productive way to look at things, especially on a website made for people asking questions on things they don't understand sufficiently.

The moral of the story is that one should be very exact and careful when asking questions without being too provocative, but also that one should answer questions with an open mind, unless there is overwhelming evidence to the contrary, in which case the question should be marked as non-mainstream.

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    $\begingroup$ "It seems that some people regard questions on everything that they have understood well as trolling / questions that should be censored..." Honestly, I think this is what happened here, and I see it from time to time on this site. I apologize that this happened to you in this instance. Thanks for the post here. $\endgroup$ – BioPhysicist May 13 at 13:31
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    $\begingroup$ No, thanks for clearly explaining my mistake :) As I said, I also had it coming in a way, especially since that was my first post on the website. In any case I appreciate the community with all its upsides and shortcomings, because at the end of the day we are all here to understand the world around us better and that's all that matters. $\endgroup$ – Godzilla123 May 13 at 13:34

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