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I'm trying to remember a problem in classical mechanics involving a special surface that allows a ball to roll to the top and lose all it's momentum in finite time.

This leads to some interesting problems with time reversibility, as it implies the ball will spontaneously roll down the surface.

I'm not looking for an explanation, so much as a name and link to study it some more.


marked as duplicate by John Rennie, user36790, knzhou, Qmechanic Jun 20 '16 at 10:30

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  • $\begingroup$ why would there be a problem with time reversibility? the ball will roll down due to gravity, you cannot tell if the movie is forward or in reverse. $\endgroup$ – Wolphram jonny Jun 19 '16 at 15:25
  • $\begingroup$ @Wolphramjonny Presumably the issue is what happens when a system reaches a stationary state; all information about how it got there is effectively lost, and therefore so is time-reversibility... $\endgroup$ – lemon Jun 19 '16 at 15:30
  • $\begingroup$ Norton's dome ? $\endgroup$ – John Rennie Jun 19 '16 at 15:38
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    $\begingroup$ I think you are looking for the Norton Dome -- pitt.edu/~jdnorton/Goodies/Dome $\endgroup$ – user55515 Jun 19 '16 at 15:38
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    $\begingroup$ Possible duplicate of Non-deterministic particle system $\endgroup$ – John Rennie Jun 19 '16 at 15:54

I believe the analysis of the Norton Dome is flawed (as many philosophers thought experiments). The ball does not stay at rest and start to move spontaneously in the absence of any force. If there were no forces it will stay there forever. The reason it starts to move is some small perturbations. They could be either external (random variations in pressure around, or just nonisotropic temperature fluctuations; there are plenty of choices). So if you had full information of the system and its surroundings you should be able to predict (in theory, not in practice) which way the ball would move and when. The system is deterministic.

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    $\begingroup$ Possibly true, but this isn't an answer to the question. $\endgroup$ – John Rennie Jun 19 '16 at 15:55
  • $\begingroup$ then I did not understand the question $\endgroup$ – Wolphram jonny Jun 19 '16 at 15:57
  • $\begingroup$ @Wolphramjonny: agreed. There's no breaking with determinism there. A lot of hot air though, enough to disturb a 'metastable' ball! ;-) $\endgroup$ – Gert Jun 19 '16 at 16:21
  • $\begingroup$ Risking going off-topic: if a valid solution to Newton's equations can be found, does that imply that the solution can be realized physically? Maybe I should make that a question. I think closed time-like curves are solutions to Einstein's equations, but does that mean that they must exist? $\endgroup$ – garyp Jun 19 '16 at 16:45
  • $\begingroup$ Agreed, these kinds of systems are still important to teach that in practice even classical systems are almost always irreversible in a practical sense because their starting points have such a small volume in phase space that the system never truly goes back there on any physically relevant time scale. $\endgroup$ – CuriousOne Jun 19 '16 at 21:26

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