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"Norton's Dome is a thought experiment that exhibits a non-deterministic system within the bounds of Newtonian mechanics. "

A ball rolled to the top can reach it in finite time with zero energy at the end, which means if you time reverse it the ball appears to star moving for no reason. By don't use just use an inverted cone with a point at the top? Whouldn't this be much simpler?

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  • $\begingroup$ An inverted cone would not have a first derivative at he top. $\endgroup$
    – trula
    Commented Jul 23, 2023 at 12:24
  • $\begingroup$ @trula isn’t the key point that there is no 2nd derivative at the equilibrium point (here the top) on the surface constraining the motion? How would an inverted cone not satisfy this? $\endgroup$
    – ZeroTheHero
    Commented Jul 23, 2023 at 12:51

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For all shapes other than Norton's dome, it takes a ball infinite time to come to rest at the top. Therefore the time reversed path starts infintly long ago, not at an arbitrary time. This removes the indeterminacy from the solution.

Or so it seems. There is still nothing needed to make the motion start. Many paths rolling up tthe cone from different sides lead to the same state in the infinite future. Time reversal can lead to any of them.

This still does not show the universe is indetermanistic. This uses classical Newtonian mechanics and perfectly rigid objects on infinitely precise trajectories. These are approximations and idealizations. Nondetermanism is only found in corner cases where the approximations do not match the universe well. Of course, quantum mechanics does show the universe is nondetermanistic.

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  • $\begingroup$ Infinite time? It certianly doesn't take infinite time to reach the top of a cone, even for a trajectory that has zero velocity at the end. The acceleration is constant. $\endgroup$
    – blademan9999
    Commented Jul 23, 2023 at 15:01
  • $\begingroup$ I didn't calculate the time. I was just repeating the Wikipedia article you linked. But it makes sense for a perfect sphere. It would be tangent to the surface until it reached the vertex, and then rock on the vertex until upright. When it was $0.1^o$ from upright, it would have a small angular velocity. When $0.1^o$ from upright, even smaller. I am taking the articles word for it that the time to come to a complete stop would be infinite. $\endgroup$
    – mmesser314
    Commented Jul 23, 2023 at 21:13
  • $\begingroup$ The Wikipedia article mentions that it takes infinite time for hemispheres. $\endgroup$
    – blademan9999
    Commented Jul 24, 2023 at 5:03

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