[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.

[hexane360]

There's a lot of minor points in that post, but it seems like both authors largely agree on the meaning, but are using different language. From Dr. Davies' post:

>To remain Newtonian and preserve determinism, we can exclude the singular point by constraining the higher orders to zero whenever the net force is zero. We lose time symmetry for this special case if we do this. If we wish to keep that, then we have to accept that Newtonian mechanics is incomplete and consider higher order differentials.

And from Dr. Norton's article:

>The solutions (3) are fully in accord with Newtonian mechanics in that they satisfy Newton's requirement that the net applied force equals mass x acceleration at all times.

>An important feature of Newtonian mechanics is that it is time reversible, or at least that the dynamics of gravitational systems invoked here are time reversible.

Dr. Davies is saying that there's three options: a) relaxing time-reversal symmetry (at singularities) from Newtonian mechanics, by interpreting Newton's First Law to apply to higher derivatives; b) considering Newtonian mechanics to be incomplete, and make (unspecified) choices about what trajectories of higher-order derivatives are acceptable; or c) accept non-determinism.

Dr. Norton is defining "Newtonian mechanics" as necessarily having time-reversal symmetry, which prevents the first solution. He is also defining it as specifying acceleration only (which I think is quite reasonable), preventing the second solution. Therefore he's concluded the third solution: This mathematical stating of Newtonian mechanics is non-deterministic.

[kergonath]

You are entirely right, whether something that depends on higher-order derivatives can be called Newtonian is debatable. Personally I don’t really care either way, as this is just a label. Newton did not mention higher-order derivatives but on the other hand they are a trivial extension to the mathematical framework. It is difficult to call a body at rest if any of the derivatives of the position is not zero, because then it will start moving instantaneously so it is hard to read the first law otherwise. And the second law does not care about anything other than acceleration. And there certainly isn’t anything that prevents us from using clever shape to roll balls on, as long as the shape make physical sense.

What this does not change, however, is that the dome does not demonstrate non-determinism. The apparent demonstration hinges on logical errors that remain errors regardless of the framework used, be it classical or quantum mechanics, or relativity.

[archgoon]

It is not a trivial extension. If you need all infinite number of derivatives to predict motion than your theory is non predictive.

[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?

[scatters]

A ball rolls through the bottom of the bowl and out the other side. It doesn't come to rest.

[cma]

In a time reversal situation where the ball is at maximum magnitude momentum, it will just go the opposite way, right?

[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...

[crazygringo]

> The "ball rolling to the top of the sphere" requires infinite time.

No it doesn't, because it's not a sphere. The dome is specifically designed so that it takes finite time. There's zero involvement of infinity, or mixing infinity, here.

[kergonath]

> No it doesn't, because it's not a sphere. The dome is specifically designed so that it takes finite time.

Can you explain why?

[crazygringo]

I don't know what kind of answer you're looking for. The equation was explicitly chosen/derived to have this property. I assume the mathematical proof of that isn't something that fits in a few sentences in an HN comment.

[eesmith]

Back when I was a little smithling who knew more math than physics, I complained about an assignment whose solution didn't make mathematical sense. My teacher commented that I needed to think like a physicist, that is, understand that certain mathematical issues didn't exist in the real world, so could be ignored.

The skit at https://www.youtube.com/watch?v=xPzR_D9qKeo gives some examples. The one at https://youtu.be/xPzR_D9qKeo?t=165 is pretty close to this example "if it's in physics, it's invertable."

That doesn't mean that if it's invertable it's in physics.

Are the inverse dynamics of this system still in Newtonian physics? For example, is is the inverse path actually on the described surface or does it detach? How does a moving mass have an instantaneous jerk with no change in velocity?

[xyzzyz]

The article explicitly discusses the fact that this is possible specifically because of the shape of the dome, and does not work on a hemisphere, precisely for reason you bring up.

[eesmith]

You are right - I misread it.

The next step would be to verify that the paths always stay on the surface. The mathematics shown says the point always follows the surface, but I don't see a demonstration that that's true.

I no longer have the skills to easily do this calculation.

EDIT: Oh man, I used to be a lot better at this. I remember the mgh = 1/2 m v^2 and the slope calculation, but can't figure out how tell when the falling point mass detaches from the slope. If it detaches at h=0 then there's no physically viable reversed path on the surface.

[kergonath]

The article explicitly discusses this without demonstrating anything. On it’s face this argument has the weight of these demonstrations.