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Recently I came across the problem of Norton's dome.
I thought of two questions, for which I found no answer.

  1. Does there exist a newtonian initial value problem, where the total force on each body is non-zero everywhere, with more than one solution and/or broken symmetry?
  2. Given a newtonian system, is the property of an initial condition to have a unique solution generic?
    (i.e. is the subset of initial conditions with a unique solution open and dense?)

Thanks,
Shay

Shay Ben Moshe
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    Related to non-uniqueness & Norton's dome: http://physics.stackexchange.com/q/39632/, http://physics.stackexchange.com/q/141111 – Kyle Kanos Feb 06 '16 at 16:24
  • @KyleKanos, thanks for you comment. I read this thread a few days ago. It was useful, in understanding Norton's dome, but unfortunately it does not answer my questions. – Shay Ben Moshe Feb 06 '16 at 16:25
  • Where did I say it would answer your questions? – Kyle Kanos Feb 06 '16 at 16:26
  • Never said you did :), I also thanked you for it, and said that is a useful thread. Take it easy. – Shay Ben Moshe Feb 06 '16 at 16:29
  • You made the claim that it does not answer my questions, as if I had posted it to be an answer to the queries. All I said was that those two links were related to this one due to their contents. – Kyle Kanos Feb 06 '16 at 16:30
  • I'm sorry that it came out this way, I agree that these are related and relevant for some that may read this thread. Thanks again. – Shay Ben Moshe Feb 06 '16 at 16:33
  • While this doesn't answer your question, either, the uniqueness of the initial value problem is of little help (and I admit that I don't know for which classes of potentials it will hold) because what we are interested in are the long term maps, which, unfortunately, are chaotic. – CuriousOne Feb 06 '16 at 18:52
  • Sorry, but what do you mean by long term maps? Also, I think that uniqueness is important, since at least I thought newtonian physics was completely deterministic (in that sense), and I find it very peculiar to have non-unique solutions in classical physics. – Shay Ben Moshe Feb 06 '16 at 19:04
  • @Timaeus. There's no real reason. If you have an example for 1 I would like to see it. Regarding the second, the topology is the product topology on $\mathbb{R}^n$. What I mean by this question is: do such phenomena required fine-tuning of initial conditions? – Shay Ben Moshe Mar 21 '16 at 21:16
  • @ShayBenMoshe Q1 yes. Q2 no. But you asked two questions in one post, so I'm not sure which you wanted people to answer. – Timaeus Mar 22 '16 at 00:22

1 Answers1

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In order to have non-uniqueness, you must have a discontinuity in the derivative of the potential. As mentioned in the dome paper, a local maximum, or saddle point will not be sufficient, as any perturbation would take an infinite time to manifest. Only initial conditions that result in exactly reaching the discontinuity produce not unique outcomes. Thus in order for the set of initial conditions that result in unique outcomes to not be dense, the subset of the potential where the derivative is continuous would need to not be dense. While there are curves whose derivative is not densely continuous, I'm struggling trying to get a newtonian potential with the same property. I would guess that it's not possible.

Rick
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