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By reading Norton dome "proof" about non-determinism of Newton laws, i've started to think about how much Norton system is physical at all. Let's push it to the limits. Let's say we have vertical cone and a particle sitting exactly on cone apex :

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Usually normal force is defined as normal vector perpendicular to surface where body is placed. However, here body-particle is placed not on surface at all, but just at cone apex, meaning that it's just a singular point. So, surface normal will not help here. Best bet is to calculate vertex normal which can be defined as averaging all surface normals at some cone cross-section :

$$ \hat{\textbf{n}}_{vertex} = \frac{\sum_{i}{\vec{N}_{surface_{_i}}}}{||\sum_{i}{\vec{N}_{surface_{_i}}}||} $$

So we have got a unit vertex normal vector. This is perfectly valid in mathematical sense and used a lot in computer graphics too. But, the question is - Is it physically meaningful to use vertex normal here as a unit normal force vector (substitute for SURFACE normal force) ?

Agnius Vasiliauskas
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  • If there is no normal (perpendicular) direction then it does not make sense to call it normal, just call it the force at the tip. If everything is symmetric, then it must point up.of course, ideal objects do not exist though. –  Oct 29 '19 at 15:41
  • But if there is no normal, then we can't judge about net force acting on particle, thus - this system as well as Norton dome is doomed. How do you know that it MUST point up ? Maybe simply in Norton dome and similar cases normal force is undefined, and as such Newton laws does not apply (i.e. incomplete) here ? Or maybe on such edge cases normal force acts randomly ? How we can be sure ? – Agnius Vasiliauskas Oct 29 '19 at 17:21
  • I mean let us not call it normal force, just force of the tip. And if there is symmetry it must point up, it is not undefined. If the ball is not perfectly homogeneous or is not positioned perfectly so the tip is at the center, then the direction will change –  Oct 29 '19 at 17:24
  • I'm not so sure about "tip force" definition. Maybe it is undefined (or random), that's why particle slips of the apex for "no apparent reason" in Norton dome. Normal force random fluctuation between valid interval would explain such phenomenon in Newton law's way. – Agnius Vasiliauskas Oct 29 '19 at 17:31
  • I dont think the math here supports a time reversible result like in norton's dome, but I am not going to discuss norton's dome here –  Oct 29 '19 at 17:45
  • Time reversibility doesn't explains why particle should start moving of the apex without experiencing no net force. That's against Newton second law. And Norton dome has same apex singularity problems as in here. Just here they are explicitly stated and more obvious. – Agnius Vasiliauskas Oct 29 '19 at 18:47
  • yes, because you must be able to roll it up and get it to rest –  Oct 29 '19 at 21:38

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