> 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.
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 "ball rolling to the top of the sphere" requires infinite time. "Some arbitrary time" is an expression of a finite time.
You cannot simple mix ideas of finite and infinite and have the result make sense, as anyone who has stayed at the Hilbert Hotel knows. https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand...
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.
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.
Can you explain why?
The skit at https://www.youtube.com/watch?v=xPzR_D9qKeo gives some examples. The one at https://youtu.be/xPzR_D9qKeo?t=165 is pretty close to this example "if it's in physics, it's invertable."
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?
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.
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.
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 not.
Could you please elaborate which Newtonian principles it does violate?
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.
I think this example just illustrates a case where the Newtonian model of reality simply does not describe reality itself
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 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.
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).
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.
(Note that the article goes on at length separating Newtonian mechanics from the "real world" or whatever)
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.
No, you've missed the point entirely. There are circumstances under which the solution is not unique, and the article describes such a circumstance.
The original tumblr post linked to this wikipedia article: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_t... The initial conditions here are _not_ sufficient to pin to a unique solution.
> 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.
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)
r(t) = 0 ; t <= T
r(t) = (1/144)(t-T)^4 ; t >= T
Interestingly, the spherical dome he mentioned (which doesn't yield to this non-determinism) forces all derivatives of r(t) to be continuous ...
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.
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.
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.
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.
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.
And the math does check out. There really are many solutions to this system, i.e. it is nondeterministic under the theory.
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.
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.