[SlySherZ]

The article is hard to follow for me, but if I understood it correctly, this is not true:

> 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

If you set velocity = velocity = 0, then the ball staying at the top is a valid solution, AND the ball rolling down the hill (in any direction) is also a valid solution.

If this sounds confusing (it did for me), look at the example at the end, it's possible to do the reverse - send the ball rolling up the hill with perfect velocity, such that it stops at the very top after time T. And if that is possible, the opposite is also possible because NM is time reversible.

[kergonath]

> The article is hard to follow for me, but if I understood it correctly, this is not true

You are right, I was missing some conditions. The higher order derivatives need to be zero as well.

> If you set velocity = velocity = 0, then the ball staying at the top is a valid solution, AND the ball rolling down the hill (in any direction) is also a valid solution.

It is a valid solution to the f=ma equation. It is not a valid trajectory in Newtonian physics because it violates other principles. It is a “gotcha” only if you think that Newton’s second law is the entirety of classical mechanics.

> If this sounds confusing (it did for me), look at the example at the end, it's possible to do the reverse - send the ball rolling up the hill with perfect velocity, such that it stops at the very top after time T.

This paragraph is confusing. And does not demonstrate much of anything, instead asserting facts that we are supposed to believe.

In the time-reversal “experiment”, where the particle comes from the rim towards the apex, it ends up at the apex with a non-zero fourth derivative, because of the pathological shape of the dome. It cannot stay on the apex for any length of time, even with a velocity of 0. It is completely different from a particle starting at rest on the apex.

> And if that is possible, the opposite is also possible because NM is time reversible.

It is not.

[sgregnt]

> It is a valid solution to the f=ma equation. It is not a valid trajectory in Newtonian physics because it violates other principles. It is a “gotcha” only if you think that Newton’s second law is the entirety of classical mechanics

Could you please elaborate which Newtonian principles it does violate?

[lisper]

> If you set velocity = velocity = 0, then the ball staying at the top is a valid solution, AND the ball rolling down the hill (in any direction) is also a valid solution.

Yes, that is exactly right. Not only in any direction, but beginning at any time.

The easiest way to see this is described at the end: imagine the ball is initially in motion and the initial conditions are precisely those that bring it precisely to rest at the apex of the dome at some time T. (Making this possible is the reason the dome has to be a specific shape. Not all shapes allow this.) The time-reversal of this motion is the ball beginning to move in some arbitrary direction at some arbitrary time.

[kergonath]

> The easiest way to see this is described at the end: imagine the ball is initially in motion and the initial conditions are precisely those that bring it precisely to rest at the apex of the dome at some time T.

This is a red herring. It sounds plausible, but there is no trajectory that does this. This is the weakest paragraph in the original post, and I am not sure whether this is intentional (because the demonstration sounds truthy if you don’t go too deep in the details) or whether it was not entirely thought out. There is some discussion about the time-reversal thing here: https://blog.gruffdavies.com/2017/12/24/newtonian-physics-is... . There isn’t much to discuss however, because ultimately it is just a distraction.

[wruza]

If you throw a ball into a bowl, it will also find the (anti-)apex. And the time-reversal of that is the ball arbitrarily choosing a direction to jump off the center of the bowl. So what? Why is it important to mention in case of a non-stable equilibrium?

[eesmith]

> The time-reversal of this motion ... at some arbitrary time.

The "ball rolling to the top of the sphere" requires infinite time. "Some arbitrary time" is an expression of a finite time.

You cannot simple mix ideas of finite and infinite and have the result make sense, as anyone who has stayed at the Hilbert Hotel knows. https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand...