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The Dome: A simple violation of determinism in Newtonian mechanics (2005)

117 points by chmaynard 3y ago

francasso 3y ago
TL;DR: Ordinary Differential Equations (ODEs) can have multiple solutions for the same initial condition, and you can imagine physical models where Newton's second law gives you such ODEs. Relevance for reality or physics: none as far as I can see.
AbrahamParangi 3y ago
This is more of a quirk of which axioms you choose and which you discard rather than anything else. For instance, implicit in this model is a continuous representation of the universe, but the universe is not actually continuous!
- H8crilA 3y ago
  The universe is continuous (both spatially and time-wise) in Newtonian mechanics.
  - azeemba 3y ago
    And in quantum mechanics, we don't even know if spacetime is quantized right? People usually talk about planck length as some quantized length but it's just a scale where quantum effects are significant.
- eximius 3y ago
  That hasn't been proven, has it?
- jakeinspace 3y ago
  There is no evidence that spacetime is discrete. It’s quite plausible, but even in a quantum gravity theory it’s not a given.
- Zamicol 3y ago
  Exactly. This makes the assumption that the universe is continuous instead of discrete. Quantum physics is all about discreteness. Newton himself suspected light was discrete, what he named "corpuscles". Even if continuous, I would still argue against this article. In a continuous universe non-determinism, randomness, is not needed. In the provided example I would expect no action to take place, or acknowledge that a continuous universe infers an infinite resolution of information for physical systems.
  - kergonath 3y ago
    There’s some confusion here. > This makes the assumption that the universe is continuous instead of discrete. Quantum physics is all about discreteness. This is irrelevant. The discussion is within the framework of classical Newtonian dynamics. The discrete-ness of the universe has no effect whatsoever, because the moment we say we’re using classical mechanics, we assume a flat Euclidean continuous spacetime. The argument is made in these terms, and in can be proven or disproven in this framework. > Newton himself suspected light was discrete, what he named "corpuscles". This has absolutely nothing to do with whether spacetime itself is quantised or not. You can have a concept of point particles in a continuous space, that’s not a problem. > In a continuous universe non-determinism, randomness, is not needed. In the provided example I would expect no action to take place That is a good intuition. > or acknowledge that a continuous universe infers an infinite resolution of information for physical systems. However, this is not. Even in a continuous universe (and, until proven otherwise, our understanding is that ours is), infinite “information” is not really a thing.
    - Zamicol 3y ago
      >infinite “information” is not really a thing. That's the point. That's why a continuous universe is absurd from an informational perspective.
      - sebzim4500 3y ago
        To my knowledge, there are very few concrete proposals for what a discrete universe would actually look like. Most physicists believe that the universe should be symmetric under lorentz transforms, which rules out all the obvious ways like splitting the universe into little cubes.
- tsimionescu 3y ago
  All current models of space-time (including quantum mechanics, QFT, and GR) have space and time as continuous quantities.
- romwell 3y ago
  Well sure. The post is about the axioms, and the paradox that follows from them.
azeemba 3y ago
The post itself doesn't seem to have much details. The linked article seems to have more info: https://sites.pitt.edu/~jdnorton/Goodies/Dome/ https://sites.pitt.edu/~jdnorton/Goodies/Dome/ Edit: Looks like the post has been updated to point here. Just for completeness, it used to point to https://johncarlosbaez.wordpress.com/2023/08/05/nortons-dome/ https://johncarlosbaez.wordpress.com/2023/08/05/nortons-dome...
- sfink 3y ago
  And the article itself is quite short and readable, and does a much better job of pointing out the weirdness. I recommend reading it.
- dang 3y ago
  Yes, a mod updated it, probably after seeing your comment (thanks!)
kergonath 3y ago
It’s frustrating as the text does not go anywhere near demonstrating any “simple violation of determinism in Newtonian mechanics”. The core issue of the discussion linked in the story is that there are mathematical solutions to Newton’s second law that are inconsistent with other bits of classical physics (in this case that a particle at rest in a given Galilean frame of reference cannot just start moving without something else applying some force to it). That is entirely true, interesting, and there is nothing wrong with this. What this does not tell us is how it somehow violates any kind of determinism. > Then there are many solutions of Newton’s law F = ma. In one the ball remains at rest on top of the dome. But in others, it starts to roll down the dome in some arbitrary direction! Moreover it can start rolling at any time. That is just not going to happen in (classical) reality, though. Because once you properly set the initial state of the ball (force=velocity=0, or any other values), then the solution becomes unique and that’s it, there is one possible trajectory. The ball starting to move without anything acting on it would violate other principles of classical mechanics. It’s not going to happen regardless of whether that trajectory is consistent with Newton’s second law.
- SlySherZ 3y ago
  The article is hard to follow for me, but if I understood it correctly, this is not true: > That is just not going to happen in (classical) reality, though. Because once you properly set the initial state of the ball (force=velocity=0, or any other values), then the solution becomes unique If you set velocity = velocity = 0, then the ball staying at the top is a valid solution, AND the ball rolling down the hill (in any direction) is also a valid solution. If this sounds confusing (it did for me), look at the example at the end, it's possible to do the reverse - send the ball rolling up the hill with perfect velocity, such that it stops at the very top after time T. And if that is possible, the opposite is also possible because NM is time reversible.
  - lisper 3y ago
    > If you set velocity = velocity = 0, then the ball staying at the top is a valid solution, AND the ball rolling down the hill (in any direction) is also a valid solution. Yes, that is exactly right. Not only in any direction, but beginning at any time. The easiest way to see this is described at the end: imagine the ball is initially in motion and the initial conditions are precisely those that bring it precisely to rest at the apex of the dome at some time T. (Making this possible is the reason the dome has to be a specific shape. Not all shapes allow this.) The time-reversal of this motion is the ball beginning to move in some arbitrary direction at some arbitrary time.
lisper 3y ago
I don't understand why anyone would think this is even the slightest bit weird. The situation described is dynamically unstable. The object remaining at rest forever is only possible in the Platonic ideal: zero friction, infinitely rigid materials, no thermal motion, no external perturbations. As soon as the state diverges from the Platonic ideal in any way, positive feedback will amplify that divergence. So the prediction for the Platonic ideal is exactly what one would expect: the object might stay still, or it might start to move at any time without any apparent cause. Of course, this situation can never actually be observed because even if you could somehow construct it (and good luck with that), you couldn't actually look at it because merely shining a light on the object would give it a nudge.
- divs1210 3y ago
  > merely shining a light on the object would give it a nudge so you're saying you would change the outcome by measuring the system?
  - bee_rider 3y ago
    Sure, but more in the “I knocked the plank off the desk by accident while measuring it, by hitting it really hard with the ruler” sense, and less in the quantum physics is weird sense.
- kp1197 3y ago
  The article is asking us to consider an idealized situation
  - lampiaio 3y ago
    In an idealized situation though, why would the ball not stay at rest?
    - tsimionescu 3y ago
      The whole point is that Newtonian mechanics doesn't uniquely predict the motion of the ideal ball on that ideal shape. The ball could stay there forever, but it could also start moving down along the shape at any point in time - both are valid possibilities in the idealized model. This is the unintuitive part.
tylerneylon 3y ago
The article is pointing out that, if we think of Newtonian mechanics as a set of math equations, then there exist simple systems in which the future of the system has multiple equally-valid solutions to those equations. Mathematically, it’s possible to have a differential equation with an initial condition, and still have multiple different solutions. That’s what the author creates as well. I don’t think this article is about continuous vs discrete physics, as some commenters suggest - nor about reality at all (not directly). Rather, this is pointing out a surprising property of the model of Newtonian mechanics.
koryk 3y ago
I am not a physicist, but the "no probabilities" problem could just be solved with a constant offset right? Or just say that whichever direction the marble rolls is the zero degree mark. I don't see how "no cause" problem is fully true either. The system has potential energy due to the marble being places at the top of the dome, so that was the cause.
- romwell 3y ago
  To see "no probabilities" easier: there's no probability distribution for when that ball starts rolling.
kp1197 3y ago
I don't buy the argument that merely because solutions exist, they can spontaneously be "chosen" by the system, even if such choosing doesn't break rules. We are talking about an idealized setup using disproven physics, so it's weird to talk about correctness. But I think the author is invoking a variety of Murphys law: Anything that can happen, might happen spontaneously. If that's true in idealized Newtonian physics world, maybe we should "fix" that? Or maybe the question of what to do about weird multiple solution sotuations is simply not in the scope of Newtons theory.
- sebzim4500 3y ago
  This is just what determinism means: that for a given intial condition there is only a single solution obeying the rules. The article demonstrates that newtonian mechanics is not deterministic, which is surprising at least to me.
  - mvaliente2001 3y ago
    I think parent comment (by @sebzim4500) is the clearest most concise summary of the argument stated in the article.
ajkjk 3y ago
This seems like a mathematical trick -- pretending like perfectly smooth functions exist in reality -- which is then extrapolated from in bizarre and unilluminating ways. The only reasonable reading of Newton's laws (and descriptions of e.g. physically-constructed curves like this dome) is that they are true up to some small epsilon length scale. No matter how small the epsilon, as long as it is not literally zero, this doesn't work. (for instance if things aren't perfectly smooth then there is some small force proportional to the discrepancy ~O(e^2) or O(sin e) or whatever, which moves the ball off the dome) If I was teaching physics from sceatch, I would state on day one: physics is the practice of building models whose low-order approximations give correct predictions about reality. There is no such thing as a perfect model to infinite decimal places. (That one 9 decimal place calculation from QFT doesn't invalidate this: given a model, the calculations may be perfectly accurate! But the model is still fuzzy because of fuzzy inputs. In that case, it has been possible to make it very very not fuzzy.)
- spekcular 3y ago
  You may be interested to know that this exact objection has been made in the philosophical literature. See "Causal Fundamentalism in Physics" by Zinkernagel (2010). Available here: https://philsci-archive.pitt.edu/4690/1/CausalFundam.pdf https://philsci-archive.pitt.edu/4690/1/CausalFundam.pdf At the end, the author notes (as you do) that if you consider a finite difference equation with small time steps, there are no pathological solutions. He also mentions that Newton takes this difference equation approach when solving problems in his Principia. See also "The Norton Dome and the Nineteenth Century Foundations of Determinism" by van Strien: >> Abstract. The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the contemporary discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome.
selimthegrim 3y ago
I’m pretty sure this is a failure to understand virtual work and virtual displacement.
- gus_massa 3y ago
  I agree, ut I think your comment needs more details. I'll try: The idea is that the Newton's Laws an be rewritten and get equivalent equations that look very different. One of the rewrites is "Lagrangian Mechanics". Instead of a differential equation, you have a calculus of variations. In this new formulation it's more clear that the solution is not unique, and you get the particle that rest on top of the dome, the one that falls down, and even some solutions where the particle waits a little and then "decides" to fall down without an eternal trigger. For this formulation, two of the man concepts are virtual work and virtual displacement. More details in: https://en.wikipedia.org/wiki/Lagrangian_mechanics#D'Alembert's_principle https://en.wikipedia.org/wiki/Lagrangian_mechanics#D'Alember...
amluto 3y ago
Off the top of my head (haven’t verified carefully): What they’re really saying is that they have an initial value problem in classical mechanics that does not have a unique solution. Fortunately, the theory of such things is very well established. From the article, the dome results in motion like this: > d2r/dt2 = sqrt(r) [reformatted as text] This function is interesting at r=0 — it’s not differentiable, and it’s not even locally Lipschitz. So one would not generally expect solutions to the initial value problem to be unique, and math and classical mechanism still work. You can read more about the theory here: https://en.m.wikipedia.org/wiki/Initial_value_problem https://en.m.wikipedia.org/wiki/Initial_value_problem (The differential equation in question is second order and is in two spatial dimensions. The standard transformation to an ordinary vector differential equation applies, and you end up with four variables (x, v_x, y, v_y or however you like to name them) plus time, and the time derivatives of v_x and v_y as functions of everything else are not Lipschitz in any open set containing the origin.) And, intuitively, what’s going on is that the acceleration of the mass is exquisitely sensitive to position around the top of the dome (it varies infinitely quickly with a small displacement), which is precisely what breaks the uniqueness of the initial value problem.
jasonhansel 3y ago
Doesn't this just prove that Newton's First Law is not, in fact, equivalent to the claim that "in the absence of a net external force, a body is unaccelerated"? (There's also the question of whether r(t) in (3) is differentiable at t=T, right?)
- mathieutd 3y ago
  There is an external force here: gravity.
  - sudosysgen 3y ago
    Yes, hence the two not being the same - there is no net force.
Cushman 3y ago
Hoping I’m not too late to head off the usual confusion: this is an interesting result in philosophy of physics, not in physics. We already know that classical mechanics is non-physical. This result (and others) show that it is not even internally consistent— that is, you shouldn’t need any empirical evidence to know that there’s something else going on. That’s interesting to philosophers and historians, but since you and I already know empirically that it’s non-physical, it shouldn’t come as much of a surprise. Anyway, if you enjoyed learning about the dome, you may also look up the lesser-known “space invaders”, in which arbitrary objects can appear at infinity, with infinite velocity, and then be brought to rest at any time T. But again, don’t look for reasons that doesn’t actually happen— it means the theory is wrong.
- kergonath 3y ago
  > We already know that classical mechanics is non-physical. This result (and others) show that it is not even internally consistent— that is, you shouldn’t need any empirical evidence to know that there’s something else going on. I don’t think this is a right approach. We already know that no theory is complete and perfect, so we can say the same thing about any theory. Even worse than that, we can make the philosophical point that any theory, being conceived within our limited brains, physically cannot be anything other than models and approximations. The logical conclusion of this argument is then that we should throw our hands in the air and stop discussing anything. It’s also wrong in this case specifically because there is absolutely no reason why this thought experiment cannot be proven or disproven within Newtonian mechanics.
  - Cushman 3y ago
    > I don’t think this is a right approach. You’re welcome not to think so! I’m just pointing out the relevant context. This isn’t an argument in a vacuum, there’s an academic discipline that has thoroughly engaged with it. Part of that is a vast body of literature that in turns agrees and disagrees with your observations— to take a side, I’d start clicking links from “theory-ladenness”. But the dome is rather boring. P1, this math describes a deterministic system; P2, here is a non-deterministic result; QED, P1 is false. Since we know that P1 doesn’t describe the universe, it should be hard to have a strong opinion unless you’re invested in a philosophical position about pre-modern scientific practice. (I’m not invested, so I don’t have an opinion other than that it isn’t trivially refutable as stated.)
nabla9 3y ago
More generally: There are first-order differential equations that don't have unique solution for a given initial condition. This is just one example of non-uniqueness condition in ordinary differential equations. Picard–Lindelöf theorem tells when there are unique solutios
deleted 3y ago
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bjornsing 3y ago
Interesting… If you take the initial conditions of the ball sitting still at the top of the dome and integrate Newton’s laws stepwise forward in time, then surely the ball will just sit there forever. But for some reason this is not certain in continuous time?
JohnDeHope 3y ago
"...with acceleration due to gravity g... A point-like unit mass slides frictionlessly over the surface under the action of gravity." If you make up funny physics problems that don't look like the real world, then the math works out funny. What about heat, light, the motion of the larger entire system? There are a lot of forces they are just hand waving away. I think it's disingenuous to hand wave away much of the complexity of the real world and still expect your algebra problem to not be overly simplistic.
aidenn0 3y ago
ELI5: why do they go to the trouble of constructing a dome when a simple cone would have the same properties? It seems to me that the motion of the object is not infinitely differentiable in either case, and the dome shape only serves to obscure this fact.
- eigenket 3y ago
  The easiest way to see it is to consider the time-reversed version. You chuck the ball up the shape so it perfectly stops balanced at the top. It turns out this is possible for the weird dome in question, but not for pretty much any other shape - including a cone. Going back to time running normally it turns out that this means that a ball balanced perfectly on a cone only has one option consistent with Newton's laws, it'll stay balanced there forever, while there are multiple trajectories consistent with Newton's laws for the dome.
  - aidenn0 3y ago
    Why is it not possible for a cone? Slide it up the (frictionless) cone with kinetic energy equal to mgh and it should stop on the point.
    - scatters 3y ago
      It's the reverse of a point mass sliding up a hemispherical dome. It takes infinite time to reach the top.
    - eigenket 3y ago
      It never reaches the top in finite time on a cone, it just keeps going up and up getting slower and slower.
      - aidenn0 3y ago
        That makes sense; thanks.
- amluto 3y ago
dang 3y ago
Related: The Dome: A Simple Violation of Determinism in Newtonian Mechanics - https://news.ycombinator.com/item?id=26507118 https://news.ycombinator.com/item?id=26507118 - March 2021 (2 comments)
MightyBuzzard 3y ago
[dead]
sudosysgen 3y ago
I'm not satisfied with the argument that this satisfies Newton's first law. The fact that it has zero acceleration before T and at the moment T is necessary but it isn't sufficient. I take issue with the interpretation of Newton's first law. What's happening here is some sleight of hand that allows the author to interpret Newton's first law in such a way that it becomes completely empty - it simply becomes a restatement of the second law. Indeed, we look at it in its instantaneous form, which means that what we actually confirm is that dv/dt=k(t)*F(t), where F(t) is force at any time T and k(T) is a nonzero function - this is the functional equivalent of the statement that "In the absence of a net external force, a body is unaccelerated.". This is of course just a more general statement for F=ma where k(T) = 1/m. So that leads me to conclude that either this instantaneous version of the law isn't sufficient, or that it's a misinterpretation of the law, or that Newton is an idiot and didn't realize that his first law is redundant. Personally I just think that this is a misreading of the first law - I read it as saying that force is causative to changes in velocity, and not the reverse, and hence that the logic in this argument suddenly stops making sense, and only perpetual rest is a correct solution, since we cannot resolve the trajectory from the forces anymore, we have to presuppose the trajectory before finding implied forces, thus breaking the causative relationship implied by the first law. I don't think that when Newton wrote that rest in the inertial frame is something that must be broken by action, he meant something that is straightforwardly implied by the second law, as of course he knew the elementary calculus necessary to see that a weaker statement is redundant.
majikaja 3y ago
I kind of doubt Newton was that concerned of cases like this when he defined his laws of mechanics and if he was given such an example he could have easily made an alteration to the laws if he wanted. This discussion is only happening because he did not get around to doing that. It's not important.

117 points by chmaynard 3y ago

TL;DR: Ordinary Differential Equations (ODEs) can have multiple solutions for the same initial condition, and you can imagine physical models where Newton's second law gives you such ODEs. Relevance for reality or physics: none as far as I can see.

AbrahamParangi 3y ago

This is more of a quirk of which axioms you choose and which you discard rather than anything else. For instance, implicit in this model is a continuous representation of the universe, but the universe is not actually continuous!

H8crilA 3y ago
The universe is continuous (both spatially and time-wise) in Newtonian mechanics.
- azeemba 3y ago
  And in quantum mechanics, we don't even know if spacetime is quantized right? People usually talk about planck length as some quantized length but it's just a scale where quantum effects are significant.
eximius 3y ago
That hasn't been proven, has it?
jakeinspace 3y ago
There is no evidence that spacetime is discrete. It’s quite plausible, but even in a quantum gravity theory it’s not a given.
Zamicol 3y ago
Exactly. This makes the assumption that the universe is continuous instead of discrete. Quantum physics is all about discreteness. Newton himself suspected light was discrete, what he named "corpuscles". Even if continuous, I would still argue against this article. In a continuous universe non-determinism, randomness, is not needed. In the provided example I would expect no action to take place, or acknowledge that a continuous universe infers an infinite resolution of information for physical systems.
- kergonath 3y ago
  There’s some confusion here. > This makes the assumption that the universe is continuous instead of discrete. Quantum physics is all about discreteness. This is irrelevant. The discussion is within the framework of classical Newtonian dynamics. The discrete-ness of the universe has no effect whatsoever, because the moment we say we’re using classical mechanics, we assume a flat Euclidean continuous spacetime. The argument is made in these terms, and in can be proven or disproven in this framework. > Newton himself suspected light was discrete, what he named "corpuscles". This has absolutely nothing to do with whether spacetime itself is quantised or not. You can have a concept of point particles in a continuous space, that’s not a problem. > In a continuous universe non-determinism, randomness, is not needed. In the provided example I would expect no action to take place That is a good intuition. > or acknowledge that a continuous universe infers an infinite resolution of information for physical systems. However, this is not. Even in a continuous universe (and, until proven otherwise, our understanding is that ours is), infinite “information” is not really a thing.
  - Zamicol 3y ago
    >infinite “information” is not really a thing. That's the point. That's why a continuous universe is absurd from an informational perspective.
    - sebzim4500 3y ago
      To my knowledge, there are very few concrete proposals for what a discrete universe would actually look like. Most physicists believe that the universe should be symmetric under lorentz transforms, which rules out all the obvious ways like splitting the universe into little cubes.
tsimionescu 3y ago
All current models of space-time (including quantum mechanics, QFT, and GR) have space and time as continuous quantities.
romwell 3y ago
Well sure. The post is about the axioms, and the paradox that follows from them.

azeemba 3y ago

The post itself doesn't seem to have much details. The linked article seems to have more info: https://sites.pitt.edu/~jdnorton/Goodies/Dome/ https://sites.pitt.edu/~jdnorton/Goodies/Dome/ Edit: Looks like the post has been updated to point here. Just for completeness, it used to point to https://johncarlosbaez.wordpress.com/2023/08/05/nortons-dome/ https://johncarlosbaez.wordpress.com/2023/08/05/nortons-dome...

sfink 3y ago
And the article itself is quite short and readable, and does a much better job of pointing out the weirdness. I recommend reading it.
dang 3y ago
Yes, a mod updated it, probably after seeing your comment (thanks!)

kergonath 3y ago

It’s frustrating as the text does not go anywhere near demonstrating any “simple violation of determinism in Newtonian mechanics”. The core issue of the discussion linked in the story is that there are mathematical solutions to Newton’s second law that are inconsistent with other bits of classical physics (in this case that a particle at rest in a given Galilean frame of reference cannot just start moving without something else applying some force to it). That is entirely true, interesting, and there is nothing wrong with this. What this does not tell us is how it somehow violates any kind of determinism. > Then there are many solutions of Newton’s law F = ma. In one the ball remains at rest on top of the dome. But in others, it starts to roll down the dome in some arbitrary direction! Moreover it can start rolling at any time. That is just not going to happen in (classical) reality, though. Because once you properly set the initial state of the ball (force=velocity=0, or any other values), then the solution becomes unique and that’s it, there is one possible trajectory. The ball starting to move without anything acting on it would violate other principles of classical mechanics. It’s not going to happen regardless of whether that trajectory is consistent with Newton’s second law.

SlySherZ 3y ago
The article is hard to follow for me, but if I understood it correctly, this is not true: > That is just not going to happen in (classical) reality, though. Because once you properly set the initial state of the ball (force=velocity=0, or any other values), then the solution becomes unique If you set velocity = velocity = 0, then the ball staying at the top is a valid solution, AND the ball rolling down the hill (in any direction) is also a valid solution. If this sounds confusing (it did for me), look at the example at the end, it's possible to do the reverse - send the ball rolling up the hill with perfect velocity, such that it stops at the very top after time T. And if that is possible, the opposite is also possible because NM is time reversible.
- lisper 3y ago
  > If you set velocity = velocity = 0, then the ball staying at the top is a valid solution, AND the ball rolling down the hill (in any direction) is also a valid solution. Yes, that is exactly right. Not only in any direction, but beginning at any time. The easiest way to see this is described at the end: imagine the ball is initially in motion and the initial conditions are precisely those that bring it precisely to rest at the apex of the dome at some time T. (Making this possible is the reason the dome has to be a specific shape. Not all shapes allow this.) The time-reversal of this motion is the ball beginning to move in some arbitrary direction at some arbitrary time.

lisper 3y ago

I don't understand why anyone would think this is even the slightest bit weird. The situation described is dynamically unstable. The object remaining at rest forever is only possible in the Platonic ideal: zero friction, infinitely rigid materials, no thermal motion, no external perturbations. As soon as the state diverges from the Platonic ideal in any way, positive feedback will amplify that divergence. So the prediction for the Platonic ideal is exactly what one would expect: the object might stay still, or it might start to move at any time without any apparent cause. Of course, this situation can never actually be observed because even if you could somehow construct it (and good luck with that), you couldn't actually look at it because merely shining a light on the object would give it a nudge.

divs1210 3y ago
> merely shining a light on the object would give it a nudge so you're saying you would change the outcome by measuring the system?
- bee_rider 3y ago
  Sure, but more in the “I knocked the plank off the desk by accident while measuring it, by hitting it really hard with the ruler” sense, and less in the quantum physics is weird sense.
kp1197 3y ago
The article is asking us to consider an idealized situation
- lampiaio 3y ago
  In an idealized situation though, why would the ball not stay at rest?
  - tsimionescu 3y ago
    The whole point is that Newtonian mechanics doesn't uniquely predict the motion of the ideal ball on that ideal shape. The ball could stay there forever, but it could also start moving down along the shape at any point in time - both are valid possibilities in the idealized model. This is the unintuitive part.

tylerneylon 3y ago

The article is pointing out that, if we think of Newtonian mechanics as a set of math equations, then there exist simple systems in which the future of the system has multiple equally-valid solutions to those equations. Mathematically, it’s possible to have a differential equation with an initial condition, and still have multiple different solutions. That’s what the author creates as well. I don’t think this article is about continuous vs discrete physics, as some commenters suggest - nor about reality at all (not directly). Rather, this is pointing out a surprising property of the model of Newtonian mechanics.

koryk 3y ago

I am not a physicist, but the "no probabilities" problem could just be solved with a constant offset right? Or just say that whichever direction the marble rolls is the zero degree mark. I don't see how "no cause" problem is fully true either. The system has potential energy due to the marble being places at the top of the dome, so that was the cause.

romwell 3y ago
To see "no probabilities" easier: there's no probability distribution for when that ball starts rolling.

kp1197 3y ago

I don't buy the argument that merely because solutions exist, they can spontaneously be "chosen" by the system, even if such choosing doesn't break rules. We are talking about an idealized setup using disproven physics, so it's weird to talk about correctness. But I think the author is invoking a variety of Murphys law: Anything that can happen, might happen spontaneously. If that's true in idealized Newtonian physics world, maybe we should "fix" that? Or maybe the question of what to do about weird multiple solution sotuations is simply not in the scope of Newtons theory.

sebzim4500 3y ago
This is just what determinism means: that for a given intial condition there is only a single solution obeying the rules. The article demonstrates that newtonian mechanics is not deterministic, which is surprising at least to me.
- mvaliente2001 3y ago
  I think parent comment (by @sebzim4500) is the clearest most concise summary of the argument stated in the article.

ajkjk 3y ago

This seems like a mathematical trick -- pretending like perfectly smooth functions exist in reality -- which is then extrapolated from in bizarre and unilluminating ways. The only reasonable reading of Newton's laws (and descriptions of e.g. physically-constructed curves like this dome) is that they are true up to some small epsilon length scale. No matter how small the epsilon, as long as it is not literally zero, this doesn't work. (for instance if things aren't perfectly smooth then there is some small force proportional to the discrepancy ~O(e^2) or O(sin e) or whatever, which moves the ball off the dome) If I was teaching physics from sceatch, I would state on day one: physics is the practice of building models whose low-order approximations give correct predictions about reality. There is no such thing as a perfect model to infinite decimal places. (That one 9 decimal place calculation from QFT doesn't invalidate this: given a model, the calculations may be perfectly accurate! But the model is still fuzzy because of fuzzy inputs. In that case, it has been possible to make it very very not fuzzy.)

spekcular 3y ago
You may be interested to know that this exact objection has been made in the philosophical literature. See "Causal Fundamentalism in Physics" by Zinkernagel (2010). Available here: https://philsci-archive.pitt.edu/4690/1/CausalFundam.pdf https://philsci-archive.pitt.edu/4690/1/CausalFundam.pdf At the end, the author notes (as you do) that if you consider a finite difference equation with small time steps, there are no pathological solutions. He also mentions that Newton takes this difference equation approach when solving problems in his Principia. See also "The Norton Dome and the Nineteenth Century Foundations of Determinism" by van Strien: >> Abstract. The recent discovery of an indeterministic system in classical mechanics, the Norton dome, has shown that answering the question whether classical mechanics is deterministic can be a complicated matter. In this paper I show that indeterministic systems similar to the Norton dome were already known in the nineteenth century: I discuss four nineteenth century authors who wrote about such systems, namely Poisson, Duhamel, Boussinesq and Bertrand. However, I argue that their discussion of such systems was very different from the contemporary discussion about the Norton dome, because physicists in the nineteenth century conceived of determinism in essentially different ways: whereas in the contemporary literature on determinism in classical physics, determinism is usually taken to be a property of the equations of physics, in the nineteenth century determinism was primarily taken to be a presupposition of theories in physics, and as such it was not necessarily affected by the possible existence of systems such as the Norton dome.

selimthegrim 3y ago

I’m pretty sure this is a failure to understand virtual work and virtual displacement.

gus_massa 3y ago
I agree, ut I think your comment needs more details. I'll try: The idea is that the Newton's Laws an be rewritten and get equivalent equations that look very different. One of the rewrites is "Lagrangian Mechanics". Instead of a differential equation, you have a calculus of variations. In this new formulation it's more clear that the solution is not unique, and you get the particle that rest on top of the dome, the one that falls down, and even some solutions where the particle waits a little and then "decides" to fall down without an eternal trigger. For this formulation, two of the man concepts are virtual work and virtual displacement. More details in: https://en.wikipedia.org/wiki/Lagrangian_mechanics#D'Alembert's_principle https://en.wikipedia.org/wiki/Lagrangian_mechanics#D'Alember...

amluto 3y ago

Off the top of my head (haven’t verified carefully): What they’re really saying is that they have an initial value problem in classical mechanics that does not have a unique solution. Fortunately, the theory of such things is very well established. From the article, the dome results in motion like this: > d2r/dt2 = sqrt(r) [reformatted as text] This function is interesting at r=0 — it’s not differentiable, and it’s not even locally Lipschitz. So one would not generally expect solutions to the initial value problem to be unique, and math and classical mechanism still work. You can read more about the theory here: https://en.m.wikipedia.org/wiki/Initial_value_problem https://en.m.wikipedia.org/wiki/Initial_value_problem (The differential equation in question is second order and is in two spatial dimensions. The standard transformation to an ordinary vector differential equation applies, and you end up with four variables (x, v_x, y, v_y or however you like to name them) plus time, and the time derivatives of v_x and v_y as functions of everything else are not Lipschitz in any open set containing the origin.) And, intuitively, what’s going on is that the acceleration of the mass is exquisitely sensitive to position around the top of the dome (it varies infinitely quickly with a small displacement), which is precisely what breaks the uniqueness of the initial value problem.

jasonhansel 3y ago

Doesn't this just prove that Newton's First Law is not, in fact, equivalent to the claim that "in the absence of a net external force, a body is unaccelerated"? (There's also the question of whether r(t) in (3) is differentiable at t=T, right?)

mathieutd 3y ago
There is an external force here: gravity.
- sudosysgen 3y ago
  Yes, hence the two not being the same - there is no net force.

Cushman 3y ago

Hoping I’m not too late to head off the usual confusion: this is an interesting result in philosophy of physics, not in physics. We already know that classical mechanics is non-physical. This result (and others) show that it is not even internally consistent— that is, you shouldn’t need any empirical evidence to know that there’s something else going on. That’s interesting to philosophers and historians, but since you and I already know empirically that it’s non-physical, it shouldn’t come as much of a surprise. Anyway, if you enjoyed learning about the dome, you may also look up the lesser-known “space invaders”, in which arbitrary objects can appear at infinity, with infinite velocity, and then be brought to rest at any time T. But again, don’t look for reasons that doesn’t actually happen— it means the theory is wrong.

kergonath 3y ago
> We already know that classical mechanics is non-physical. This result (and others) show that it is not even internally consistent— that is, you shouldn’t need any empirical evidence to know that there’s something else going on. I don’t think this is a right approach. We already know that no theory is complete and perfect, so we can say the same thing about any theory. Even worse than that, we can make the philosophical point that any theory, being conceived within our limited brains, physically cannot be anything other than models and approximations. The logical conclusion of this argument is then that we should throw our hands in the air and stop discussing anything. It’s also wrong in this case specifically because there is absolutely no reason why this thought experiment cannot be proven or disproven within Newtonian mechanics.
- Cushman 3y ago
  > I don’t think this is a right approach. You’re welcome not to think so! I’m just pointing out the relevant context. This isn’t an argument in a vacuum, there’s an academic discipline that has thoroughly engaged with it. Part of that is a vast body of literature that in turns agrees and disagrees with your observations— to take a side, I’d start clicking links from “theory-ladenness”. But the dome is rather boring. P1, this math describes a deterministic system; P2, here is a non-deterministic result; QED, P1 is false. Since we know that P1 doesn’t describe the universe, it should be hard to have a strong opinion unless you’re invested in a philosophical position about pre-modern scientific practice. (I’m not invested, so I don’t have an opinion other than that it isn’t trivially refutable as stated.)

nabla9 3y ago

More generally: There are first-order differential equations that don't have unique solution for a given initial condition. This is just one example of non-uniqueness condition in ordinary differential equations. Picard–Lindelöf theorem tells when there are unique solutios

deleted 3y ago

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bjornsing 3y ago

Interesting… If you take the initial conditions of the ball sitting still at the top of the dome and integrate Newton’s laws stepwise forward in time, then surely the ball will just sit there forever. But for some reason this is not certain in continuous time?

JohnDeHope 3y ago

"...with acceleration due to gravity g... A point-like unit mass slides frictionlessly over the surface under the action of gravity." If you make up funny physics problems that don't look like the real world, then the math works out funny. What about heat, light, the motion of the larger entire system? There are a lot of forces they are just hand waving away. I think it's disingenuous to hand wave away much of the complexity of the real world and still expect your algebra problem to not be overly simplistic.

aidenn0 3y ago

ELI5: why do they go to the trouble of constructing a dome when a simple cone would have the same properties? It seems to me that the motion of the object is not infinitely differentiable in either case, and the dome shape only serves to obscure this fact.

eigenket 3y ago
The easiest way to see it is to consider the time-reversed version. You chuck the ball up the shape so it perfectly stops balanced at the top. It turns out this is possible for the weird dome in question, but not for pretty much any other shape - including a cone. Going back to time running normally it turns out that this means that a ball balanced perfectly on a cone only has one option consistent with Newton's laws, it'll stay balanced there forever, while there are multiple trajectories consistent with Newton's laws for the dome.
- aidenn0 3y ago
  Why is it not possible for a cone? Slide it up the (frictionless) cone with kinetic energy equal to mgh and it should stop on the point.
  - scatters 3y ago
    It's the reverse of a point mass sliding up a hemispherical dome. It takes infinite time to reach the top.
  - eigenket 3y ago
    It never reaches the top in finite time on a cone, it just keeps going up and up getting slower and slower.
    - aidenn0 3y ago
      That makes sense; thanks.
amluto 3y ago

dang 3y ago

Related: The Dome: A Simple Violation of Determinism in Newtonian Mechanics - https://news.ycombinator.com/item?id=26507118 https://news.ycombinator.com/item?id=26507118 - March 2021 (2 comments)

MightyBuzzard 3y ago

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sudosysgen 3y ago

I'm not satisfied with the argument that this satisfies Newton's first law. The fact that it has zero acceleration before T and at the moment T is necessary but it isn't sufficient. I take issue with the interpretation of Newton's first law. What's happening here is some sleight of hand that allows the author to interpret Newton's first law in such a way that it becomes completely empty - it simply becomes a restatement of the second law. Indeed, we look at it in its instantaneous form, which means that what we actually confirm is that dv/dt=k(t)*F(t), where F(t) is force at any time T and k(T) is a nonzero function - this is the functional equivalent of the statement that "In the absence of a net external force, a body is unaccelerated.". This is of course just a more general statement for F=ma where k(T) = 1/m. So that leads me to conclude that either this instantaneous version of the law isn't sufficient, or that it's a misinterpretation of the law, or that Newton is an idiot and didn't realize that his first law is redundant. Personally I just think that this is a misreading of the first law - I read it as saying that force is causative to changes in velocity, and not the reverse, and hence that the logic in this argument suddenly stops making sense, and only perpetual rest is a correct solution, since we cannot resolve the trajectory from the forces anymore, we have to presuppose the trajectory before finding implied forces, thus breaking the causative relationship implied by the first law. I don't think that when Newton wrote that rest in the inertial frame is something that must be broken by action, he meant something that is straightforwardly implied by the second law, as of course he knew the elementary calculus necessary to see that a weaker statement is redundant.

majikaja 3y ago

I kind of doubt Newton was that concerned of cases like this when he defined his laws of mechanics and if he was given such an example he could have easily made an alteration to the laws if he wanted. This discussion is only happening because he did not get around to doing that. It's not important.