For example: https://physics.stackexchange.com/questions/39632/nortons-do...
To work the curvature of the dome is infinite is at the apex, which then breaks many things. There’s a lot of disagreements around this paradox and much older related examples because Newtonian physics is somewhat ill defined: https://philsci-archive.pitt.edu/8833/1/dome_v3.pdf
It has no reason to roll unless the placement was uneven, and if it was uneven, it would not break determinism.
see https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... which was linked to from that new question:
> If we start at an arc length of 1/144 for example, it will run up the dome and arrive at the apex in 1 second. As we’ve seen, it has zero velocity and zero acceleration at this point, but moves off after anyway because it still has a positive value for snap.
The form of the Norton's dome does not matter. The so-called paradox is just a random example of the fact that there exist multiple functions of a variable that have in the origin the same values for the function and for the first 2 derivatives, e.g. various pairs of polynomials of the 4th order.
Therefore if you accept any function of time as describing a possible motion, you can always find motions that at some moment in time have the same position, velocity and acceleration.
This is not an example of indeterminacy in classic mechanics, because one of the axioms of the classic mechanics is that all the forces that exist in nature are such that the state of a mechanical system is completely determined by the positions, velocities and accelerations of its components (in other words, a mechanical system must be described by a system of differential equations of the second degree that has a unique solution).
There is no difficulty of imagining other kinds of forces, for which this assumption is not true, but a theory where such forces exist is no longer the Newtonian mechanics, in the same way as any geometry where Euclid's axiom of parallels is not true is no longer an Euclidean geometry.
If Newtonian mechanics were a correct model for the World, a ball would remain forever on the top of the dome, without ever falling. In reality, even assuming the validity of Newtonian mechanics, the main reason why any attempt to test this experimentally would fail is the thermal motion, due to which a ball can never be at rest, so it would always start immediately to fall in a random direction.
The violation of the axioms is why the so-called different solutions are not solutions within Newtonian mechanics.
On the other hand the argument that the initial state could be obtained by launching the ball towards the top, and then time reversal would demonstrate a valid solution, it is also wrong, because the so-called solution cannot be obtained by time reversal.
If the ball is launched with only enough energy to reach the top, so it will come to rest, then it requires an infinite time to reach the top. Reversing the time means that the ball will remain on the top for an infinite time, without falling, as expected.
TL;DR: Magic breaks Newton's laws
As it does for you for different reasons, this also matches my lay intuition of physics: sometimes things just spontaneously occur, and a system in dynamic equilibrium simply will not hold still forever.
Because that's legal according to the laws of motion. The intuitive answer is that it's the time reversed situation to a ball being carefully rolled UP the dome so that it stops and comes to rest on the apex. The shape function of the dome was carefully constructed so that this process takes finite time. So if it's legal in one direction it must be legal in the other.
Obviously this is a statement about math and not physics (since the underlying physical theory here is, after all, wrong!) What we thought were a bunch of well-constructed rules for classical dynamics turn out to have some holes.
That's nonsense. The arrow of entropy always goes forward. Sure, the ball comes to the top of the dome to rest but it also carries direction, momentum and a lot of other properties that you have to put in as well in your hypothetical entropy-arrow-now-goes-back scenario.
This is high-school grade physics, come on. It's surprising some people still take John Norton seriously, not because of the dome, but because of his many other "controversial" takes on physics that fail miserably on their foundations.
Norton's dome is a surprising mathematical situation in very conventional classical mechanics. It doesn't matter what else Norton has done, this observation is trivial to verify for every undergrad maths/physics student.
This has absolutely nothing to do with entropy or the arrow of time.
The mathematical situation is of no practical relevance because it's "density zero": Generic deviations will destroy this peculiar behaviour.
Good one, chap! How about you argue with substance instead ...
Explain, what makes the ball suddenly start rolling down the dome? Do not hand-wave, just give a direct answer to this question, based on your purported understanding of the problem.
Considering they were replying to a post that was, effectively, arguing "nuh uh!", their response seems reasonable.
> Explain, what makes the ball suddenly start rolling down the dome?
That's _literally_ the entire point. Nothing does. There is nothing that causes the ball to start rolling. But the Newtonian laws of physics indicate it will.
Pedantically: they indicate it can. The situation where the ball spontaneously starts rolling[1] at any specific moment in time, without any application of force or interaction with any other part of the system, are perfectly legal and well-defined by the laws of motion. They just can't be predicted determinically.
[1] FWIW it's not even a ball in this case, as the rotational mechanics of a sphere with non-zero moment of inertia would destroy the very carefully constructed function required for the potential energy field.
The next state is not uniquely determined by the prior state, so asking what makes the ball roll shows that you don't understand the claim (non-determinism ) at hand.
If you were able to perfectly balance the ball on a perfectly constructed dome, blah blah, would the ball stay static indefinitely or would it start rolling down some arbitrary path?
This is completely contrary to our intuition about Newtonian mechanics. The question "given this situation, what would happen?" typically has a unique answer is typical. If it does, we have determinism. The observation of Norton's dome is that mathematically this question does not have a unique answer in all situations.
Not a different question, not an essay, not hand-waving, just focus on that very concrete question.
Are you asking about a fictional universe governed by the Newton equations and nothing else? Then I can not answer your question because the question builds on a faulty assumption: That this universe is deterministic and that what is determines what will be. Mathematics shows that to not be the case.
The only possible answer to your question in the second case is: It can not be known or predicted what the ball will do.
And yet, the video in question seems to make it _very_ clear that this has been debated over and over, across various papers and people, and _nobody_ has been able to provide proof as to why it's wrong.
The arrow of what now?[1] This is classical dynamics we're doing.
I repeat, this is a math result, not an argument about physical systems.
[1] Edit as this was clearly missed: THIS IS SARCASM. Thermodynamics and statistical mechanics are excellent theories and worth studying as they tell us deep and profound things about the natural world. This particular novelty is a result from classical dynamics where they don't apply. The "arrow of time" in Newtonian mechanics is absolutely reversible, and there is no Newtonian idea of "entropy".
Read. Then post.
>I repeat, this is a math result, not an argument about physical systems.
Did you even care to read the title of the post?
the discussion is about hypothetical results from classical mechanics, which, along with the rest of physics, is a mathematical model that may be incongruous with observations.
Sorry, I didn't take my crazy pills today.
To me it is very clear that the question involves physics from the start.
Regarding your post about entropy. The reason it does not apply is because entropy is a concept from statistical mechanics which is about the statistics of ensembles of many (even non-classical) particles. It's a concept invented after Newton dynamics, but does not apply to describing the equations of motion of a single particle (try to define the entropy of the single particle system). Time reversal is a core tenent of Newton dynamics.
>What situations in classical physics are non-deterministic?
Is "statistical mechanics" contained within "classical physics"?
Yes or no? No need for a nonsensical philosophical essay.
In the case of classical physics, we come to a singularity in which there are several solutions for how the system resolves. This doesn't make classical physics nondeterministic, this simply means if you come to such a solution, then classical physics have no answer for what happens next.