It’s frustrating as the text does not go anywhere near demonstrating any “simple violation of determinism in Newtonian mechanics”. The core issue of the discussion linked in the story is that there are mathematical solutions to Newton’s second law that are inconsistent with other bits of classical physics (in this case that a particle at rest in a given Galilean frame of reference cannot just start moving without something else applying some force to it). That is entirely true, interesting, and there is nothing wrong with this.
What this does not tell us is how it somehow violates any kind of determinism.
> Then there are many solutions of Newton’s law F = ma. In one the ball remains at rest on top of the dome. But in others, it starts to roll down the dome in some arbitrary direction! Moreover it can start rolling at any time.
That is just not going to happen in (classical) reality, though. Because once you properly set the initial state of the ball (force=velocity=0, or any other values), then the solution becomes unique and that’s it, there is one possible trajectory. The ball starting to move without anything acting on it would violate other principles of classical mechanics. It’s not going to happen regardless of whether that trajectory is consistent with Newton’s second law.
What this does not tell us is how it somehow violates any kind of determinism.
> Then there are many solutions of Newton’s law F = ma. In one the ball remains at rest on top of the dome. But in others, it starts to roll down the dome in some arbitrary direction! Moreover it can start rolling at any time.
That is just not going to happen in (classical) reality, though. Because once you properly set the initial state of the ball (force=velocity=0, or any other values), then the solution becomes unique and that’s it, there is one possible trajectory. The ball starting to move without anything acting on it would violate other principles of classical mechanics. It’s not going to happen regardless of whether that trajectory is consistent with Newton’s second law.