[johnp314]

Thanks for the reference to the video. I watched it a few weeks ago and was befuddled by it. How can the ball just randomly start rolling in a random direction? It seemed to me that an obvious explanation would be that there is air flow in the environment and with the ball balanced in an unstable position that some air movement would easily nudge the ball off balance. I understand the diff eq of motion with the singularity but it seems to me that a ball balanced at the apex of any radially symmetric convex surface would eventually commence rolling, due to fluctuations in the air flow.

[enlightens]

Norton’s Dome plays fast and loose with the math. It could be a halfway decent way of modeling a ball that randomly started moving, but that’s not actually how anything works.

For example: https://physics.stackexchange.com/questions/39632/nortons-do...

[jdhwosnhw]

Norton’s dome is a valid paradox, in the sense that the math really does admit two valid equations of motion. The link you provided doesn’t dispute that fact (other than commenters pointing out that you need a proportionality constant to make the units work out). My favorite intuitive explanation for the presence of the paradox is well summarized by the Wikipedia article on the dome: “To see that all these equations of motion are physically possible solutions, it's helpful to use the time reversibility of Newtonian mechanics. It is possible to roll a ball up the dome in such a way that it reaches the apex in finite time and with zero energy, and stops there. By time-reversal, it is a valid solution for the ball to rest at the top for a while and then roll down in any one direction.”

[flatline]

If you watched the video, you would have the answer. Well, an answer. Which is that there does not need to be a cause! Even in an idealized dome with no air, friction, or external forces.

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.

[josh-sematic]

The point is that this is not discussing a physically realizable situation, but an idealized one. The engineering and manufacturing precision that would be required to actually achieve this setup are infinite and unattainable. In the idealized setup there is no air, no surface imperfections, no deviations from central positioning, etc.. And yet despite this idealized perfection, a case can still be made that under this construction the ball might spontaneously move in an undetermined direction. The discussion of whether that case “holds water” and what that means if so is an abstract philosophy discussion rather than one with any obvious practical implications.

[ajross]

> 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.

[moralestapia]

>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.

[Asooka]

It's an entirely nonsense argument. Akin to arguing that algebra is nondeterministic with "zero divided by zero is a random number, because any number times zero is zero".

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.