[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!