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?<p>Edit: To answer my own question, the shape of the dome has been specifically chosen to avoid this problem, as described here:<p><a href="https://www.reddit.com/r/Physics/comments/2cueh3/nortons_dome_a_simple_violation_of_determinism_in/cjjmbfk?utm_medium=android_app&utm_source=share&context=3" rel="nofollow">https://www.reddit.com/r/Physics/comments/2cueh3/nortons_dom...</a>
I think there's an easier thought experiment to generate non-determinism in classical mechanics without some continuity assumption:<p>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.<p>If there is a discontinuity at the nth decimal place, then X_T+1 is not determined by X_T.<p>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.<p>OR: *classical* mechanics is non-deterministic.
This blog post provides an interesting analysis: <a href="https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is-deterministic-sorry-norton/" rel="nofollow">https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is...</a><p>The concluding paragraph:<p><i>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.</i>
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.<p>It shows that Newtonian mechanics is only an approximation of the real world.
This is a relevant thread:
<a href="https://news.ycombinator.com/item?id=28191408" rel="nofollow">https://news.ycombinator.com/item?id=28191408</a>
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?<p>BTW entropy was mentioned in another thread, but this thought-experiment is frictionless, so if entropy still comes up that would really be interesting.