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Comment by ajross | Hacker News Reader

> How can the ball just randomly start rolling in a random direction?

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.

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>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.

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.

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.

>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.

> 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.

> 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.

Nitpick: Indicate it can. Not that it will.

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.

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?

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.

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.

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.

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!

> The arrow of entropy always goes forward.

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".

https://en.wikipedia.org/wiki/Entropy_as_an_arrow_of_time

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?

did you? the title and content of this post is about math results. you should really consider the possibility that you're very wrong here.

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.

>What situations in classical physics are non-deterministic?

Sorry, I didn't take my crazy pills today.

To me it is very clear that the question involves physics from the start.

I suggest you read the HN guidelines, you are quite abrasive and aggressive in your posts.

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.

Read the title of this post, and the title of the question in the exchange, I will post it yet again, *sigh*

>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.

Maybe you should read my most, I didn't say that statistical mechanics is nonclassical. I said statistical mechanics does not apply to the discussion of a single particle rolling up or down a slope. Tell me which of the states has more entropy the one with the particle at the top od the Norton dome or the one at the bottom?

> This is high-school grade physics, come on.

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.

If you even knew the least bit about math and its history, you would know that "no proof given by anyone in 500 years" doesn't mean a thing.

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