Talk:Norton's dome

Untitled

Note: this article was started from text from the page User:Danski14/prep, created by User:Danski14. -- The Anome (talk) 16:59, 5 August 2014 (UTC)

Heh, thanks for the attribution, although I only wrote one sentence . My original thinking was to have a page on "non deterministic Netwonian mechanics" because I didn't think the topic of Norton's dome alone was enough for a page. The page you put together looks good though. Great work Danski14^(talk) 20:38, 12 November 2014 (UTC)

No consensus?

The page states that there is no consensus on how to reconcile Newtonian mechanics with Norton's dome, but reference 6 (http://www.pitt.edu/~jdnorton/Goodies/Dome/) gives a perfectly valid explanation. To that explanation I would only add that Norton's dome has infinite curvature at r=0, and so it is quite obvious that applying the rules valid for masses moving on infinitely smooth surfaces to this dome will give funny results. 144.173.208.87 (talk) 17:14, 13 December 2017 (UTC)

Lipschitz Continuity

This sentence:

"While many criticisms have been made of Norton's thought experiment, such as it being a violation of the principle of Lipschitz continuity[citation needed],"

should be expanded and modified. I don't know what the "principle of Lipschitz continuity" refers to, but the dynamical system in Norton's dome is markedly not Lipchitz continuous and this is precisely the mathematical reason that it fails to have unique solutions. And it should be perhaps noted that this isn't the first case of nonphysical uniqueness being dealt with by considering physical constraints. For example, the entropy condition used to control shockwave solutions in systems of conservation laws. I think that the phenomenon occurring due to nonuniqueness of solutions and that this is dealt with through physical constraints in other parts of mathematical physics is worth a mention.

For references, any graduate book on ODE will suffice, for example, Jack Hale's Ordinary Differential Equations, or Miller and Michel's Ordinary Differential Equations. For entropy solutions, Evans Partial Differential Equations works. — Preceding unsigned comment added by 23.92.134.163 (talk) 23:02, 21 March 2018 (UTC)

Constant

@Tercer: I just undid your edit, reference 8 provides all equations using "b". The idea is that when you write the energy to derive the equations of motion, "mgh", g disappears. But to me (and some other authors) the equation looks very unphysical (or a list dimensionally weird) without a proportionality constant b to fix the units. See also this thread in stackexchange [1].--ReyHahn (talk) 15:09, 21 January 2021 (UTC)

But neither Ref 1 nor Ref 8 uses this b. Introducing it doesn't fix anything, because I can simply define another constant $g'=g/b^{2}$ and rewrite the equations in terms of g'. It's just an arbitrary constant. Making dimensional analysis here is missing the point. We're not studying a law of Nature here, but a specific system. It's like how you can solve a problem for a particle of mass 1 using the "wrong" equation $F=a$. The interesting part is how the force depends on r, introducing more constants just makes the reader lose focus. Tercer (talk) 19:52, 21 January 2021 (UTC)

Do you mean ref 1 and 9? If we decide to keep just g, when somebody will try to derive the force they would get F=m (g'/g) r^(1/2), where g' is the acceleration due to gravity, that they would either take it as g'=g or not, but it is still clunky don't you think?. Also in Norton's original derivation he makes it clear that g is the acceleration due to gravity for that same purpose to make that ratio disappear, we cannot say it is just a constant. I am just thinking of somebody that just discovers the equations, has some experience in Newton mechanics, and wants to do the calculation on its own, I think leaving g as standard gravity and adding a constant to correct all dimensional analysis is more natural.--ReyHahn (talk) 20:45, 21 January 2021 (UTC)

What we do is keep just g, and do state that it is acceleration due to gravity. Indeed, it would be clunky otherwise, that's why Norton does it. It does not matter that dimensional analysis fails. Just imagine that we are using a constant b that happens to be numerically equal to 1. I noticed that you also introduced an extraneous mass parameter m. This is really not pedagogical.

Also, I looked a bit more carefully at the references. Refs 1,2,5,7 use the version without the b. Ref 3 is not about Norton's dome specifically, but also use equations of motion that don't make dimensional sense. I couldn't access Ref 4, and the others are repetitions, so it's really only Ref 6 that introduces this constant. The consensus is clear. Tercer (talk) 22:06, 21 January 2021 (UTC)

I did not add an extraneous m (it appears in some of the refs, not only the one with b), m has to appear again, to cancel m in F=ma. Sure we can set everything to 1 (why don't we set g=1?), but I still think that it misses the point. I could add more references, there is for example Luboš Motl version [2], which does use a constant and argues that the equations remain dimensionally weird even if we set m=1. Yet, I am open to recognize that most references not have b. Generally because the problem is studied from a philosophical or mathematical point of view. But shouldn't we try to be a least pedagogical? If somebody fell into this article a few week ago, they would have stumbled a lot (as we did), there was no picture to have an idea (thanks again for the picture btw), they would have to notice after a while that r is bounded, that the units are weird, that g is there to cancel g from gravity (as well as the 2/3 cancels factors from the derivative). Sure they could have read it up, but most people who fall into this problem are not doing research, they are just searching for curious stuff. We can have it either way, but we should explain the units issue one way or another. If you want we can ask for a second opinion in the physics wiki project.--ReyHahn (talk) 12:46, 22 January 2021 (UTC)

In the original paper m is set to 1. If I had written the paper I would have indeed set g to 1 as well, but Norton didn't, and none of the references do it, so we don't really have a choice here. Luboš Motl is definitely not an expert on this subject, and his blog is not a reliable source. Moreover, I'm personally glad to see that he disagrees with me here. I had the misfortune of talking to him a couple of times about unrelated subjects, and argh, what an aggressive pig-headed ignoramus.

Of course I think we should be pedagogical, what we disagree about is what is the most pedagogical exposition. I think going on about dimensional analysis is drawing the attention of the reader to an irrelevant point. Moreover, it makes the equations harder to read. I would merely note that units indeed do not match, and that it doesn't matter. Sure, we can ask in the WikiProject Physics. And I'm glad you liked the picture. Tercer (talk) 13:22, 22 January 2021 (UTC)

If I had to choose between either just g (without b) and g=1, I will go for the second one, some references we cite do it like that too. I still prefer the constant though. I will open a Wikiproject discussion.--ReyHahn (talk) 13:52, 22 January 2021 (UTC)

Which reference uses g=1? Tercer (talk) 14:09, 22 January 2021 (UTC)

I stand corrected, I checked none do that. I hope you can agree that is unusual. Bhat&Bernstein inspiring paper was constantless. --ReyHahn (talk) 14:28, 22 January 2021 (UTC)

Yeah, Bhat&Bernstein did it best, in my opinion. It's not about Norton's dome specifically, though, and since g is used everywhere readers would be confused by not seeing it here. Tercer (talk) 14:34, 22 January 2021 (UTC)

Initiated readers can just set b=1 to retrieve the same results. Is there not a guideline for this in Wikipedia? It is not like I am not using a source or something. I agree with consensus, but this is just dimensional manipulation (supported by at least one source).--ReyHahn (talk) 14:38, 22 January 2021 (UTC)

I'm not saying you're making a mistake or WP:OR, I'm just saying that it's more pedagogical to present it without b. The initiated can easily insert b if they care about the units; but the innocent readers will get scared away by this messy equation with this mysterious constant. Let me try to convince you with a different example. Suppose a dome is described by the equation $h=1-x^{2}$, for $x\in [-1,1]$, which gives us $h\in [0,1]$. Dimensional analysis clearly fails, as both x and h should have dimension of length, but h is the sum of something with dimension length squared and a suspiciously dimensionless "1". Now, what is the problem with this failure? If I interpret x as metres or centimetres but keep the domain unchanged, the equation still makes sense. But if I say that x really is measured in metres, and try to convert it as 100 centimetres, making $x\in [-100,100]$, the equation clearly doesn't describe the same shape anymore. For unit conversion to work we need to make the units consistent; we can for example redefine the equation as $h=L-x^{2}/L$ for some length L. Now we can define L as one metre, and if we convert it consistently to 100 centimetres now the changed equation does describe the same shape still.

Is it really worth it? Why would anybody want to change units in this problem? I mean, that's in general nice, but at the cost of introducing an irrelevant constant and making the equation more complicated? Tercer (talk) 18:24, 22 January 2021 (UTC)

I would concur with you if g was named k or if it was not there at all. But as it is bothering because it makes you stumble in the calculation, you either worry of the units and relation to gravity or you wonder why is there in the first place. I do not see anything in WP:SCICITE that would settle this. It is written that For reasons of notation, clarity, consistency, or simplicity it is often necessary to state things in a slightly different way than they are stated in the references, to provide a different derivation, or to provide an example., do you think we can take off g entirely or change its name? --ReyHahn (talk) 18:46, 22 January 2021 (UTC)

I don't think Wikipedia policy forces us to do it one way or the other, we just have to decide among ourselves what is the best presentation. If g was not there at all the units wouldn't make sense anyway, so removing it is not an improvement. Just don't worry about the units and accept that it is gravity, it is there to make your life easier. Tercer (talk) 21:23, 22 January 2021 (UTC)

Physics equations should be dimensionally consistent.--Srleffler (talk) 16:35, 24 January 2021 (UTC)

Can the least action principle resolve this non-determinism?

The situation like the Norton's dome is non-deterministic from evolution perspective: Euler-Lagrange formulation of classical mechanics.

But there is also alternative formulation: the least action principle ( https://en.wikipedia.org/wiki/Stationary-action_principle ), which at least for Norton's dome seems to lead to unique solution: fixing position in two points in time: earlier at the top of the dome and later away from it, should determine the moment to start falling (?).

Is it universal solution to non-determinism problem?--Jarek Duda (talk) 11:07, 9 November 2021 (UTC)

Remove "See also: Folk Science"

To quote the first line of the Folk Science page, "Folk science describes ways of understanding and predicting the natural and social world, without the use of rigorous methodologies (see Scientific method). One could label all understanding of nature predating the Greeks as "folk science"." Since Newtonian Mechanics is very well based in scientific method (obviously), why reference Folk Science here in this page? There is hardly any mention or description of any philosophical discussion on Norton's Dome, and so I see no reason for Folk Science to be listed as a related article. It just seems completely out of place for it to be on a rather mathematically dense page.

I removed it. I agree that it didn't seem appropriate. If anything, Norton's Dome is the opposite of folk science—an unanticipated effect seen only through the use of mathematical analysis of a scientific model. --Srleffler (talk) 18:51, 30 January 2022 (UTC)

Error in image?

The y label in the Dome's section is probably wrong. It should be "y", not "h" (or height, elevation, ...).--Maillage (talk) 01:02, 2 February 2022 (UTC)

Why? The function plotted is h(x).--Srleffler (talk) 02:59, 2 February 2022 (UTC)

Because h is used in the article and means something else, "where h is the vertical displacement from the top of the dome to a point on the dome".--Maillage (talk) 04:11, 2 February 2022 (UTC)

Just to be clear, y should be replaced by 1-h(x)?--ReyHahn (talk) 13:08, 2 February 2022 (UTC)

Yes, or change the ylabel in the Python code to "H" (capital H), and write something like $h=1-H$. The French version (just translated) had the same issue, and is being corrected.--Maillage (talk) 13:24, 2 February 2022 (UTC)

Indeed, I plotted 1-h instead of h. Just plot it again with the correct label, the python source code is in the file description page. Tercer (talk) 13:27, 2 February 2022 (UTC)

I have uploaded the function's plot : "Dôme de Norton.svg".--Maillage (talk) 06:04, 3 February 2022 (UTC)

Why didn't you just use my code? I was careful to show the only aspect of the dome that matters: the second derivative at the apex is undefined. One does not see the apex in your version. Tercer (talk) 09:12, 3 February 2022 (UTC)

I guess x is also not defined in the article? --ReyHahn (talk) 13:50, 3 February 2022 (UTC)