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Norton's Dome

by miobrien on 2/13/23, 6:00 PM with 36 comments

  • by keskival on 2/15/23, 12:47 PM

    The argument for time reversal assumes that the ball can reach the top of the dome in a finite time, while its speed becomes zero at that limit. Sure, you can calculate the kinetic energy needed to exactly reach the top like described, but would such a ball reach the top in a finite time, as it slows down in the process?

    Edit: To answer my own question, the shape of the dome has been specifically chosen to avoid this problem, as described here:

    https://www.reddit.com/r/Physics/comments/2cueh3/nortons_dom...

  • by mjburgess on 2/15/23, 9:59 AM

    I think there's an easier thought experiment to generate non-determinism in classical mechanics without some continuity assumption:

    Take a chaotic system (eg., the moon of one of our solar system planets) and let it evolve for some time, T. Track the position with coordinate X. Let T be large enough that the nth decimal place of X_T is significant to determining X_T+1.

    If there is a discontinuity at the nth decimal place, then X_T+1 is not determined by X_T.

    For quite observable T, n quickly becomes "sub-quantum". So, if classical mechanics is deterministic, and describes nature, nature must be continuous at arbitary depth.

    OR: *classical* mechanics is non-deterministic.

  • by layer8 on 2/15/23, 5:16 AM

    This blog post provides an interesting analysis: https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...

    The concluding paragraph:

    Position, velocity and acceleration will be zero at t = 0 for every equation of polynomial form of order 3 and above, but non zero everywhere else. Particles following these trajectories move to and from an unstable equilibrium where Newton’s laws fail to be fully descriptive at the singular point t = 0 where the implied force is zero.

  • by gowld on 2/14/23, 11:33 PM

    This example assumes a physically impossible infinitely-sharp corner and an Zeno-paradoxical infinitely small jump from the initial position to some other position after time T.

    It shows that Newtonian mechanics is only an approximation of the real world.

  • by onos on 2/14/23, 10:36 PM

    Very interesting. I wonder if something analogous to this could be behind wave function collapse in quantum mechanics.

  • by rylittle on 2/15/23, 12:53 AM

    Why can't it just be a triangle? Is there a reason it needs to be curved?

  • by coef2 on 2/16/23, 6:53 AM

  • by moring on 2/15/23, 6:29 AM

    I think the mathematical approach to this paradox would be to line up the reasoning for time-reversal side-by-side with the predictions for Norton's Dome, and find a flaw in either of them. Are we even sure that the reasoning behind time-reversal in Newton's laws is solid?

    BTW entropy was mentioned in another thread, but this thought-experiment is frictionless, so if entropy still comes up that would really be interesting.