TLDW
A paper, by philosopher of physics John Norton, describes a scenario in which a ball is placed at the apex of a perfectly smooth and frictionless dome. The paper argues that, according to Newton's laws, the ball will spontaneously start rolling down the dome at some random time, without any external force acting on it. Norton's claim suggests that Newtonian mechanics may not a deterministic theory or that there are multiple possible interpretations.
https://sites.pitt.edu/~jdnorton/papers/DomePSA2006.pdf
This was a very interesting read !
I get the impression that the author considers Newtonian theory as completely disconnected from reality just because we know it to not hold in every context, and hence practical considerations should not be used to call the apex motion unphysical.
However, the needed infinite precision would make the experiment impossible to set up even if Newtonian mechanics held at microscopic level, just because of measurement erros, as well as wind, earth shaking, ...
The discussion about whether Newton's first law actually states that the forces are first causes for the motion is very interesting. However, I do not find the comparison with the harmonic oscillator appropriate : vanishing instantaneous acceleration at one point in time (like in the harmonic oscillator) is not the same thing as an inertial motion.