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.
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.
> The article is hard to follow for me, but if I understood it correctly, this is not true
You are right, I was missing some conditions. The higher order derivatives need to be zero as well.
> 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.
It is a valid solution to the f=ma equation. It is not a valid trajectory in Newtonian physics because it violates other principles. It is a “gotcha” only if you think that Newton’s second law is the entirety of classical mechanics.
> 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.
This paragraph is confusing. And does not demonstrate much of anything, instead asserting facts that we are supposed to believe.
In the time-reversal “experiment”, where the particle comes from the rim towards the apex, it ends up at the apex with a non-zero fourth derivative, because of the pathological shape of the dome. It cannot stay on the apex for any length of time, even with a velocity of 0. It is completely different from a particle starting at rest on the apex.
> And if that is possible, the opposite is also possible because NM is time reversible.
> It is a valid solution to the f=ma equation. It is not a valid trajectory in Newtonian physics because it violates other principles. It is a “gotcha” only if you think that Newton’s second law is the entirety of classical mechanics
Could you please elaborate which Newtonian principles it does violate?
The simplest one is that a particle on its own keeps a linear trajectory with a constant speed. A change in that (like going from rest to any motion) requires interacting with another particle: things do not start moving for no reason. This is a generalisation of one of the formulations of Newton’s first law, which states that things that don’t move don’t start moving without being pushed (rough translation).
This is related to another formulation of Newton’s first law: if there is a force that pushes the ball at some time T, it implies that there is another body that felt the opposite force.
Another one is a bit more involved, but basically a mechanical system cannot change its symmetry by itself. In this case, the initial state with a ball at rest has a radial symmetry with a centre on the apex of the dome. This is not true anymore if the ball moves in one direction. This is related to the conservation of momentum.
There are a couple of points that can be solved easily, but are clearly defects in the original formulation of the problem. for example, the height according to the equations is not a length, which is not a problem itself (we can just multiply by an arbitrary factor with the right dimensions) but an indication of sloppy thinking and hand waving. Similarly, the force is not bounded in the original formulation. Again, this can be fixed by restricting the valid range for r, but is rather messy.
Theres a couple of mistakes here. Firstly the particle is not on its own, it is being acted on by the dome and by gravity.
The thing about symmetry breaking also doesn't make much sense. I guess you're trying to appeal to Noether's theorem, but Noether's theorem in classical mechanics is a consequence of f = ma. You derive the Lagrangian formulation of mechanics from f=ma and Noether's theorem from that. However the weird solution when then ball suddenly randomly falls down the dome after staying put for an arbitrary time is completely consistent with f=ma, so that can't help you here.
In any case the radial symmetry you're looking for (the system is invariant under rotations around the peak of the dome) implies conservation of angular momentum about this point, and not about any other point (since the setup is manifestly not symmetric under rotations about any other point). However (one can easily check) that for both the static solution and the randomly starts moving solution, the angler momentum about the axis through the peak of the dome is always zero.
The particle does not undergo any net force while at rest. If you understand "acted upon" to be a causal statement, then no, the particle should not be able to leave a resting position, because when it is at rest there is no net force acting upon it.
The argument is basically assuming that the particle moves, showing that it moves in a way that respects the second law, then restating the first law to be a special case of the second to avoid the causal language it contains and to make it completely redundant.
There is no time when the particle is accelerating, or even moving, while not experiencing a net force in this setup. The argument you'd have to make would be to change Newtonian mechanics so the first law is no longer a special case of the second law, but actually says something nontrivial about all of the time-derivatives of position rather than just the first two. This (in my opinion) would no longer be Newtonian mechanics, but some extention.
Even then I'm not sure that would save you in general, since it should be possible to cook up examples where the motion is some non-analytic thing like a portion of a "bump function".
Edit: by the way, a more modern formulation of the first law is essentially that there are no privileged inertial frames. All inertial frames are equally valid. This is straight-up false in this setup since obviously the rest frame of the dome is privileged (since the dome is given infinite inertial mass).
> There is no time when the particle is accelerating, or even moving, while not experiencing a net force in this setup.
I agree! But at the same time, there is no moment at which a net force compels the particle to leave a state of rest. The criteria you are putting forwards with that sentence isn't that of the first law, it's just a consequence of the second law. The first law makes a statement that force is required to cause a body to leave a state of rest, but that's not what we have here.
We just postulate that it must stop being at rest immediately after T, and then we find a force as a result of it not being at rest anymore. This inverts the causality that NFL requires, and because it doesn't have give a cause for the particle to leave the state of rest, which the 1st law requires, it's not a valid physical solution in Newtonian mechanics.
Here is Wikipedia's translation of the first law, which is as good as any:
> Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.
In our trajectory, the body is not compelled to change its state by a force impressed on it. It's in a stable state, then we arbitrarily decide that it will change it's state immediately after T, and then we show that this leads to a trajectory which satisfies the second law. But since it changes state without being compelled by a force, it's no longer a trajectory which satisfies the first law.
> The argument you'd have to make would be to change Newtonian mechanics so the first law is no longer a special case of the second law
I'm not changing Newtonian mechanics - Newton's first law is not a special case of the second law. It makes a statement about cause and effect. The second law doesn't - it's just a differential equation, which does not have such a content. The article redefines the first law to strip it of its causal content and instead make it a trivial statement, but that's not what the first law plainly states. In Newtonian physics, force is necessary to cause a change in velocity, it's not simply that force and acceleration co-occur.
> but actually says something nontrivial about all of the time-derivatives of position rather than just the first two
No, this isn't necessary. If you want, you can instead understand it as something about the nth derivative of position, and that way you can recover perfect reversibility, but you don't have to. You can just keep Newtonian mechanics as is, and recognize that they describe a causal system, and not just a system of differential equations (which make no statements about causation in and of themselves), but then you lose perfect reversibility for some pathological trajectories like this one.
> Even then I'm not sure that would save you in general, since it should be possible to cook up examples where the motion is some non-analytic thing like a portion of a "bump function".
I'm not arguing for the infinite derivative modification either, but I don't see how that's the case - if you have a non-analytic function then some derivatives don't exist, and if they don't exist then they wouldn't be able to satisfy the infinite derivative formulation. Seems to me that such a law would directly eliminate all non-analytic trajectories.
This is not correct. Momentum is conserved by the spurious solution and there's still an equal but opposite force on another body (the body producing the gravitational force).
I think this example just illustrates a case where the Newtonian model of reality simply does not describe reality itself
Even if it does, that would amount to a contradiction in Newtonian mechanics. You don't get to simply ignore that the ball starting to roll after arbitrary (non deterministic) time T is a solution to these equations.
(Note that the article goes on at length separating Newtonian mechanics from the "real world" or whatever)
> 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.
> 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.
This is a red herring. It sounds plausible, but there is no trajectory that does this. This is the weakest paragraph in the original post, and I am not sure whether this is intentional (because the demonstration sounds truthy if you don’t go too deep in the details) or whether it was not entirely thought out. There is some discussion about the time-reversal thing here: https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... . There isn’t much to discuss however, because ultimately it is just a distraction.
There's a lot of minor points in that post, but it seems like both authors largely agree on the meaning, but are using different language. From Dr. Davies' post:
>To remain Newtonian and preserve determinism, we can exclude the singular point by constraining the higher orders to zero whenever the net force is zero. We lose time symmetry for this special case if we do this. If we wish to keep that, then we have to accept that Newtonian mechanics is incomplete and consider higher order differentials.
And from Dr. Norton's article:
>The solutions (3) are fully in accord with Newtonian mechanics in that they satisfy Newton's requirement that the net applied force equals mass x acceleration at all times.
>An important feature of Newtonian mechanics is that it is time reversible, or at least that the dynamics of gravitational systems invoked here are time reversible.
Dr. Davies is saying that there's three options: a) relaxing time-reversal symmetry (at singularities) from Newtonian mechanics, by interpreting Newton's First Law to apply to higher derivatives; b) considering Newtonian mechanics to be incomplete, and make (unspecified) choices about what trajectories of higher-order derivatives are acceptable; or c) accept non-determinism.
Dr. Norton is defining "Newtonian mechanics" as necessarily having time-reversal symmetry, which prevents the first solution. He is also defining it as specifying acceleration only (which I think is quite reasonable), preventing the second solution. Therefore he's concluded the third solution: This mathematical stating of Newtonian mechanics is non-deterministic.
You are entirely right, whether something that depends on higher-order derivatives can be called Newtonian is debatable. Personally I don’t really care either way, as this is just a label. Newton did not mention higher-order derivatives but on the other hand they are a trivial extension to the mathematical framework. It is difficult to call a body at rest if any of the derivatives of the position is not zero, because then it will start moving instantaneously so it is hard to read the first law otherwise. And the second law does not care about anything other than acceleration. And there certainly isn’t anything that prevents us from using clever shape to roll balls on, as long as the shape make physical sense.
What this does not change, however, is that the dome does not demonstrate non-determinism. The apparent demonstration hinges on logical errors that remain errors regardless of the framework used, be it classical or quantum mechanics, or relativity.
If you throw a ball into a bowl, it will also find the (anti-)apex. And the time-reversal of that is the ball arbitrarily choosing a direction to jump off the center of the bowl. So what? Why is it important to mention in case of a non-stable equilibrium?
> The "ball rolling to the top of the sphere" requires infinite time.
No it doesn't, because it's not a sphere. The dome is specifically designed so that it takes finite time. There's zero involvement of infinity, or mixing infinity, here.
I don't know what kind of answer you're looking for. The equation was explicitly chosen/derived to have this property. I assume the mathematical proof of that isn't something that fits in a few sentences in an HN comment.
Back when I was a little smithling who knew more math than physics, I complained about an assignment whose solution didn't make mathematical sense. My teacher commented that I needed to think like a physicist, that is, understand that certain mathematical issues didn't exist in the real world, so could be ignored.
That doesn't mean that if it's invertable it's in physics.
Are the inverse dynamics of this system still in Newtonian physics? For example, is is the inverse path actually on the described surface or does it detach? How does a moving mass have an instantaneous jerk with no change in velocity?
The article explicitly discusses the fact that this is possible specifically because of the shape of the dome, and does not work on a hemisphere, precisely for reason you bring up.
The next step would be to verify that the paths always stay on the surface. The mathematics shown says the point always follows the surface, but I don't see a demonstration that that's true.
I no longer have the skills to easily do this calculation.
EDIT: Oh man, I used to be a lot better at this. I remember the mgh = 1/2 m v^2 and the slope calculation, but can't figure out how tell when the falling point mass detaches from the slope. If it detaches at h=0 then there's no physically viable reversed path on the surface.
> 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.
As the linked article points out, that is not true. Even for force = velocity = 0, there are solutions for which the ball starts rolling. Additionally, even adding the first law does not help, since the body has F = v = a = 0 at any time before T, and it has a > 0 only for times where it also has F > 0. So the body spontaneously starting to move is perfectly consistent with classical mechanics.
A better attack on the example is that the shape of the dome may not be possible to construct from actual matter. The math only works with a shape that has a certain type of infinite smoothness with a single bump, which is probably not possible even in principle to construct from real matter.
> 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.
Not true, and the trick is very simple, you just contrive a force field that admits x''(0) = 0 but some non-zero higher derivative. Indeed, if you look at the proposed solution r = 1/144 (t - T)^4 for t > T, you'll see that:
r(T) = 0
r'(T) = 0
r''(T) = 0
r'''(T) = 0
r''''(T) is ill-defined (but Newtonian mechanics AFAIK says nothing to forbid this)
We can see that r''''(T) is either 0 or 1/6, depending on if we go with the top or bottom equation. That does look like some sort of hidden state change, there.
Interestingly, the spherical dome he mentioned (which doesn't yield to this non-determinism) forces all derivatives of r(t) to be continuous ...
> but Newtonian mechanics AFAIK says nothing to forbid this
In the Wikipedia article on Newton's laws of motion, the first law is stated as "A body remains at rest, or in motion at a constant speed in a straight line, unless acted upon by a force.". Here it would seem that we leave the state of rest not due to a force, but due to some other cause, which the first law would forbid. So I think that the particular interpretation of Newtonian mechanics used by the author is a bit of a strawman.
We must ask ourselves what "at rest" means since there are multiple trajectories that give x(0) = 0, x'(0) = 0 and it seems that some posters believe this cannot fully constrain the particle to be either at rest or in motion at t=0.
There are multiple ways to resolve this dissonance. We could demand that the higher derivatives are also zero. We could derive some elaborate rule excluding non-zero x''(t) in the neighborhood of t=0. Or something else altogether. The issue IMO with these resolutions is that they're quite complicated and Newton almost certainly did not have any of them in mind.
It's much simpler to just content ourselves that NFL is a special case of F=ma (where F=0). I'm not sure why we should contort ourselves to preserve determinism since we know that gets thrown out the window with QM anyway.
Newton didn't need to have an understanding of these solutions to foresee the problem. If you don't specifically state that force is a cause for leaving the state of rest, it's relatively straightforward to foresee issues where someone would assume the trajectory first and then imply a scheme for force that leads to unprompted movement, which is something that he definitely wanted to avoid. I also don't think he would have kept the first law if he understood it to be completely redundant and useless as it's entirely contained by the second law. I think it was very deliberately there to introduce force as causative to acceleration and not the opposite.
Who says that just because it's nondeterministic, it's acausal? Gravity clearly causes it to leave its state of rest. At literally every moment that it has a non-zero acceleration, it has a non-zero net force due to gravity (and the normal force). There is no instant at all where we can say Newton's laws were violated. At t=T the particle has zero net force and is at rest. For every e greater than 0, at t=T+e the particle has net force and is accelerating.
NFL isn't quite redundant since many people understand it to define an inertial frame. So if you observe F = 0 but x'' != 0 then you know you're not in an inertial frame.
> Who says that just because it's nondeterministic, it's acausal?
No one - I'm saying that it's the opposite. It is acausal, and that is why its nondeterministic.
> t literally every moment that it has a non-zero acceleration, it has a non-zero net force due to gravity (and the normal force). There is no instant at all where we can say Newton's laws were violated. At t=T the particle has zero net force and is at rest. For every e greater than 0, at t=T+e the particle has net force and is accelerating.
Yes, and you only find this because you assume a trajectory first, and then find force as a result of the trajectory. The force isn't causing the movement, you first assume on your trajectory that the ball stops being at rest immediately after T. Newton's first law is literally that an object stays at rest in its inertial reference frame unless a force causes it to leave its state of rest, yet you have to assume that immediately after T it leaves rest without any valid physical reason.
> NFL isn't quite redundant since many people understand it to define an inertial frame. So if you observe F = 0 but x'' != 0 then you know you're not in an inertial frame.
The existence of inertial reference frames is a consequence of the first law saying that force is necessary to exit the state of rest. It's not equivalent to the definition of inertial reference frames.
Here is Wikipedia's translation of the first law:
> Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.
This obviously does imply the definition of inertial reference frames, but it doesn't just do that, it also clearly requires force to cause a body to leave uniform motion. In the trajectories suggested by the article, the body leaves the state of rest, but that isn't because a force compels it, it's just by assertion that it must do so after T, and the force is then a result of it no longer being in the same position. I don't see how that is in accordance with the first law. It's clearly leaving rest immediately after the moment T by pure assertion, which makes it an unphysical trajectory. Then the article gets around this by providing a new NFL which has no causative language and is simply a statement about the second derivative of position, and that isn't what it is.
> Yes, and you only find this because you assume a trajectory first, and then find force as a result of the trajectory.
Sure, the choice of solution is arbitrary. No one (not even the author) argues otherwise. In fact it's the whole point.
> The force isn't causing the movement, you first assume on your trajectory that the ball stops being at rest immediately after T.
If the gravity were not there, this could not be a valid solution. The net force from gravity causes the acceleration. It is only a necessary condition and not sufficient, but the fact remains that gravity acts on the particle whenever it sets in motion.
> Here is Wikipedia's translation of the first law: > Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.
So there are naturally multiple translations of Principia, but I just cracked open the first free one I could find on Google[1] and it doesn't mention anything about compelling, merely endeavoring. Even if we generously accept that it is "compel," we must then reckon with the fact that in the solution where it sets in motion, the net force has in fact compelled it to do so. I wouldn't hang my hat on the exact wording implying anything about determinism.
> Newton's first law is literally that an object stays at rest in its inertial reference frame unless a force causes it to leave its state of rest, yet you have to assume that immediately after T it leaves rest without any valid physical reason.
It leaves rest for the perfectly valid physical reason that there's a net force. You are trying to pick apart the timeline to find a moment where the particle was impelled to move with no net force, but no such time exists. This argument absolutely conflates nondeterminism (the particle could leave at any time) with acausality (nothing caused the particle to move).
[1] https://redlightrobber.com/red/links_pdf/Isaac-Newton-Princi... The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line. This force is ever proportional to the body whose force it is ; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertia, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition ; and the exercise of this force may be considered both as resistance and impulse ; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished ; nor are those bodies always truly at rest, which commonly are taken to be so.
> anything acting on it would violate other principles of classical mechanics
What other principles of classical mechanics does it violate though? I think that's the point of the exercise. To reflect on part of our mathematical modeling would prevent this.
>The ball starting to move without anything acting on it would violate other principles of classical mechanics.
Isn't gravity acting on the ball? Ideally a ball balanced a the tip of a bowl has 0 net force, but only because it has the force of gravity pulling it into the bowl and an equal force from the bowl pushing back. This would require assuming a reality where there is no smallest particle, no atoms making up the bowl or ball as they are both perfect mathematical objects comprised of infinite points no matter what resolution you look at them at.
The issue is that such a system, if it can be created by rolling the ball up the bowl with perfect precision from any direction, indicates that it is unstable and may reverse at any point, but without any obvious reason causing it to do so. So either the system has some non-deterministic factor which allows for the perfect stability to break at an arbitrary time in an arbitrary direction or it isn't time reversible as the ability to roll the ball up the bowl cannot be reversed.
Looking at it another way, can you tell the difference between a ball I perfectly rolled to the top of the bowl 10 years ago and one I did 10 seconds ago?
I do wonder if we've added so many assumptions, with mathematically perfect objects and infinite precision forces that we have created some sort of paradox. We have hit levels of perfectly spherical cows that cause even the perfectly spherical cows to complain about unrealistic standards.
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.
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.)
> This seems like a mathematical trick -- pretending like perfectly smooth functions exist in reality -- which is then extrapolated from in bizarre and unilluminating ways.
Given that it is common to make certain claims about the model of NM (e.g. determinism and reversibility) a mathematical trick that demonstrates this is not true is a valid refutation of claims about the model.
You can use NM to model other physically impossible systems (such as objects with arbitrarily high rigidity or arbitrarily low friction), but they are not internal contradictions within the NM model, so are less valid for criticism of NM.
This example is not practically that different from chaotic systems e.g. the double pendulum, which end up in states that can't be time-reversed back to the initial state without infinite precision. In the case of a double-pendulum pointing straight up at t=0, the same argument holds: it will eventually move in an indeterminate direction. In both cases, reversing time to recover the initial state is impossible without infinite precision.
I don't think anybody is losing sleep over the non-determinacy of a double-pendulum. It's just understood as evidence that your model is only as deterministic as your ability to say that something happens at t=0 or x=0 instead of |t| < epsilon or |x| < epsilon.
> This example is not practically that different from chaotic systems...
This is wildly different from chaotic systems in that it cannot be time reversed even with infinite precision.
For cryptography, compare to a one-time-pad. There are plenty of practical attacks on one-time-pads, but they are theoretically unbreakable. If someone were to show an actual weakness in OTP in theory it would be a big deal, even if (in practice) OTPs are seldom used due to practical considerations.
Well I agree that it's not "theoretically" different from chaotic systems, that's why I said "practically". Practically, infinite precision doesn't exist.
Just saying. Doesn't really affect anything, though.
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
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.
Somewhat interested, but tbh, the reason I thought of it is the same reason that they thought of it -- that it's obvious to anyone who studies physics from a philosophical angle that it really has to be that way. They just put it in a lot of fancy words so that it has academic rigor.
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.
(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.
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.
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.
The ball could stay there forever, but it could also start moving down along the shape at any point in time
Only if the fourth derivative spontaneously changes from zero to nonzero. It doesn't seem any more surprising than the conditions f(0)=f'(0)=f''(0)=0 not uniquely determining f(x) for all x.
The condition imposed by the construction of the problem and the laws of motion is that f''(t) = sqrt(t), and that f''(t) = 0 => F(t) = 0. The function given as an example in the article, f(t) = {(1/144) (t-T)^4, t >= T | 0, t < T}, obeys both laws, just as much as f(t) = 0 does.
I'm not sure what the fourth derivative has to do with this argument.
Well, yeah, but my point is that it shouldn't be surprising. If you think about it, if it is even possible to bring a particle to rest for a finite time in the Platonic ideal then that plus time reversal necessarily entails non-determinism. So non-determinism should be no more surprising than the fact that it is actually possible to bring a particle to rest for a finite time.
I think the only reason this example surprises people is that everyone just assumes that bringing a particle to rest is possible/easy without really thinking through what this would actually require in the Platonic ideal. It's actually very challenging to stop things from moving without friction.
> then that plus time reversal necessarily entails non-determinism.
No. Generally speaking in the "Platonic ideal", we assume that if we reversed time, the particle would move from rest in a deterministic way, that there's only one solution for its movement. The surprising part here is that there are multiple valid solutions. Which takes a very-specially designed "dome" to demonstrate -- which is very surprising indeed.
> It's actually very challenging to stop things from moving without friction.
No it's not. All it takes is a billiard ball in motion hitting another billiard ball at rest. The first billiard ball will now have stopped moving. Which is kind of the canonical example of how we expect things to be deterministically reversible.
Yes, you're right, and this occurred to me after I posted but while I was in the shower so I couldn't correct it :-) I should have said "if it is possible to bring all of the constituent particles in a system to rest for a non-zero time..."
You're just reframing the article as though it's obvious and then claiming that it is obvious. The problem is - the article already does just that.
Further, you have changed your stance from one of "of course it moves from some noise in the environment" to one more closely resembling the article's main points.
You basically aren't making sense. Your initial abuse of "platonic ideals" is the sort of thing that gives philosophical arguments a bad reputation.
This is just one example of non-uniqueness condition. There are first-order differential equations that don't have unique solution for a given initial condition.
Maybe I'm just stupid but I found it very surprising that such a shape exists (i.e. stopping a ball at the top in finite time) even in a platonic ideal.
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.
Consider how many pop-sci singularitans think their god in a box could predict the future through mere calculation. Many people do have a naive clockwork-universe model of causality.
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.
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.
Because the cone is intuitively ridiculous? If I say I’m balancing an object (presumably of finite size) perfectly centered on a cone, then the obvious question is “you did what”? It just seems more absurd than balancing an object on a continuous surface.
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.
> 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.
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.)
> You’re welcome not to think so! I’m just pointing out the relevant context.
It isn’t, though. The argument is not “lol Newtonian physics are dumb” and you’ll see nothing of the sort in the Norton argument. The argument is instead “Newtonian physics can be non-deterministic”, which is something we can demonstrate regardless of the validity range of Newtonian physics.
> This isn’t an argument in a vacuum, there’s an academic discipline that has thoroughly engaged with it.
Sure. How is that linked to your point?
> But the dome is rather boring. P1, this math describes a deterministic system; P2, here is a non-deterministic result; QED, P1 is false.
Indeed.
> 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.
That’s a strange position to take from a philosophical point of view. Why things are wrong matters more than whether they are because, again, everything is wrong to some extent. So, of course in some abstract epistemological sense anything that can be formulated within Newtonian mechanics is wrong. But then nothing is right, so who cares?
The initial question is much more interesting from a philosophical point of view: does the old Newtonian mechanics, which is still the closest to our daily experience, contains seeds of non-determinism? But then the logic is flawed: the issue is not that Newtonian mechanics are wrong, it is that, in your formalism, P2 does not follow from P1 and is actually wrong.
Mmm, I promise I haven’t taken any position. Is that a position? In which case, mea culpa, I really don’t care if the dome holds. Even if I cared, I wouldn’t care, because space invaders already gets us all that plus sound effects.
Not for nothing, even historians and philosophers of physics broadly don’t care if the dome holds. Some do, of course, but unless you're thinking of a specific paper it’ll probably be less frustrating for everyone to leave it there— with no hard feelings!
> We already know that no theory is complete and perfect, so we can say the same thing about any theory.
If you mean this as an appeal to Gödel’s incompleteness results, the things those show can’t happen aren’t the same kinds of things that a “fully complete theory of physics” would have to satisfy.
That’s not to say that I expect that we will ever (in this world) have a complete description of the physics of this world,
But I’m quite confident that Gödel’s incompleteness theorems do not pose a fundamental barrier to the laws of physics of a world being perfectly known by entities in that world.
I’ll take my lumps for saying this: If you’ve downvoted me I’d love to know why!
Did I sound preachy? I really don’t mean to! I’m not an educator, and it’s hard to communicate a discipline’s “common knowledge” without coming off a bit patronizing.
It’s just, every time the dome comes up people want to talk about the physics of it— but the author isn’t a physicist, the journal isn’t for physicists, it’s not making any claims in physics…
It’s a (famous) philosophy paper, specifically philosophy of science, specifically philosophy of physics. If you aren’t a philosopher, physics is annoyingly irrelevant here.
There’s so much more to say— space invaders! It’s way weirder than this! The math still checks out! Philosophy is cool actually! Sorry to take up your time!
> this is an interesting result in philosophy of physics, not in physics.
Yes. Although it has consequences. It's a demonstration that some physical variables have to be quantized or probabilistic to avoid divide by zero errors in reality.
This becomes clear when you do idealized Newtonian physics with impulses.
An impulse is an infinite force applied over zero time with finite energy transfer. That's not something that can exist in the physical universe.
It's just asking for divide by zero problems. It's also why impulse-constraint physics engines for games have some rather strange semantics.
Indeed. Bittersweet consequences, I’ve always thought, since it implies we wasted centuries recording empirical data that didn’t add up, while pure logical coherence was pointing straight at quantization and a universal speed limit.
Modern philosophers are (obviously?) most interested in how this historical fact should affect our handling of, say, our current best model for QG. That’s a fascinating thing for lay people to ponder, which is one reason I wish the context weren’t so opaque to HN.
(I should caveat that I’m less familiar with the “history of philosophy of science” side of this— there may be a live debate as to whether this would obtain with the logic Newton himself had access to, for example.)
Thanks for that! What a fascinating set of problems.
It is interesting that all of the ones listed in that section involve either infinities or infinitesimally small points -- except for Norton's dome. Which really makes it a great example for that reason.
Yeah, exactly! You can see why the dome is the famous one, even though it’s the most recent. Previously there were models to finitize whatever term one was worried about specifically, but they all had an infinity somewhere. Even since Cantor, that makes philosophers antsy.
The dome is analogous to a loophole-free Bell test— most people already accept the conclusion, so it’s not a surprising result, but having it in one simple argument closes the door to a lot of nitpicking.
"...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.
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.
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?)
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.
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.
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.
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?
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.
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'Alember...
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!
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.
> 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.
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.
Although discrete models may not exist, discreetness is expected because without there's likely unsolvable informational problems.
This is why Beckenstein was so brilliant, he was among the first to understand physical systems must fit into information theory. If they don't, physics has far worse and profound issues.
Any description of the universe is informational. We should hope that the universe is informational so that humanity can have a complete description of the cosmos. If the universe is not informational, since description by nature is informational, there would be no way to describe the universe completely.
You missed my point. It’s not a thing in a continuous universe. And again, there is no proof whatsoever that ours is not, and we don’t need to invoke any spacetime quantisation to explain any of the current established theories. I am not saying that the universe is not discrete (I just don’t know), but if it’s discrete character we’re this trivial to observe, it would have been settled for a long time.
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.