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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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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.

Eph
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