[Certhas]

Seriously, you don't seem to know much about things you speak confidently about.

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.

[moralestapia]

>you don't seem to know much about things you speak confidently about

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.

[RHSeeger]

> Good one, chap! How about you argue with substance instead ...

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.

[ajross]

> 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.

[Certhas]

Nitpick: Indicate it can. Not that it will.

[Certhas]

Nothing. The Newton equations predict that, given the Norton potential, there are two possible trajectories that solve them.

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.

[aipatselarom]

I'm trying to wrap my head around your logic. So, I'll go step by step to make sure I get it.

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?

[Certhas]

The point is that the equations don't tell you what would happen. Both options would be valid according to the equations.

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.

[aipatselarom]

Could you just answer the question in turn?

Not a different question, not an essay, not hand-waving, just focus on that very concrete question.

[Certhas]

Are you asking about the real world? Then the answer is that you can not perfectly craft the dome and what happens depends on the imperfections.

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.

[ajross]

It is a concrete question! Here's a potential energy function describing a "hill". Here's an object on the hill at this location. How will it move? The question is well formed and complete. And it has more than one answer!