OK, but why? (I think that is part of the thought experiment).
E.g., if h=\frac{2}{3}r^{3/2}, conservation of energy gives \ddot{r}=g\sqrt{r}. Let’s keep r positive and try to roll the ball from r(0)=1 up to r=0 at some finite time. I get something like r(t) = (1-\sqrt{\frac{g}{12}}t)^4 as a particular solution. But then it has zero velocity at the top.
The paper I linked to above has the equations. The differential equation for the radial acceleration is just:
\frac{d^2r}{dt^2} = r^{1/2}
One solution is just:
r(t) = 0
But there’s an entire other class of solutions:
r(t) = \frac{1}{144}(t-T)^4 \ for \ t \ge T
“An object at rest” and so on are adequate heuristics most of the time, but really the machinery of Newtonian mechanics is in differential equations. And it’s clear that there are multiple valid solutions in this case.
I agree those are mathematical solutions to the equation for some initial conditions; however, the initial conditions are are both r(t) = 0 and frac{dr(t)}{dt} = 0. So integrating once frac{dr(t)}{dt} = frac{3}{2} r^{3/2} + frac{dr(0)}{dt} and the second term is zero. It no longer integrates to the cited value of r(t).
ETA: Sorry I guess I don’t know how to write the math. I copied what was in your post but it doesn’t work.
I think it does, or at least the problem is not with his integration [rather with lack of Lipschitz continuity of the force]: if r(0)=0 and v(0)=0 where v=dr/dt, then (at least where r>0 to avoid any nasty singularity 
) dv/dt=\sqrt{r} yields v\,dv = \sqrt{r}\,dr so v^2 = \frac{4}{3}r^{3/2} + a constant, which we conveniently set to 0 because we want the velocity to vanish at r=0. Then we easily get solutions 144 r = {(t+C)}^4 after separating the variables a second time. Also, you can directly check that @Dr.Strangelove 's solution satisfies the equation d^2r/dt^2=\sqrt{r}, just like your solution r(t) = 0, and in both cases v(0)=0.
It is clear that the surface is not smooth at 0 , though, which may largely be the “point” of this scenario.
I looked at the paper, and have no doubt that for the given potential, the equations of motion do permit more than one solution passing through r = 0, r’=0. The potential was picked to not satisfy the conditions of the existence-uniqueness theorem for ODEs.
I’d argue that the potential itself is unphysical, except as an approximation. The real dome is made up of real molecules so the forces are necessarily a sum of intermolecular forces which all obey the conditions of the E-U theorem, and since they’re all continuity/differentiability conditions, they also do so in aggregate. The lack of uniqueness comes from making an approximation to the actual potential that ignores the fine detail.
Right, the cows are spheres.
One I always loved was the Newtonian 5-body system, where one star gets gravitationally slung to infinity, in a finite time.
Yeah. I read an article in Scientific American about that scenario, decades ago, and found it fascinating.
And here’s the first page of the article, from 1991
I think that “marble on a dome” potential only “works” (i.e., takes a finite time for the marble to come to rest exactly on the peak) if the “marble” has zero size. Otherwise, you have a region on the top where it acts identically to a spherical marble on a conical point, which means that the actual shape of the potential function there is just spherical, and the time to come to rest is again infinite.
On one hand, that’s clearly true of the universe we live in. But on the other hand, it seems somehow unsatisfying to me. Instead of being ruled out by the basic laws, situations like this are only ruled out by the specific details of our universe. One has to go in manually and verify that the E-U theorem applies to all of the potentials that exist. Thought experiments usually don’t require that kind of unpleasant detail work.
Related in some sense, there is, in (relativistic) physics a “cosmic censorship hypothesis”, the idea being that one would like to rule out a situation where there are singularities visible to observers. It is known at least that one can construct black holes with a Cauchy horizon inside where it is unpredictable what happens to a particle falling past it (cf https://arxiv.org/pdf/1710.01722)
Are links to sci-hub allowed here? I can’t remember. In any case, if one did have a means of accessing various academic archives, the DOI for this article is doi:10.1038/scientificamerican1291-144.
Thanks. I was able to get the whole document from a computer with legitimate access to academic archives earlier today.
Can someone explain to me how Prof. Norton derived the magnitude of the gravitational force tangential to the surface of the dome? He writes,
The dome of Figure 1a sits in a downward directed gravitational field, with acceleration due to gravity g. The dome has a radial coordinate r inscribed on its surface and is rotationally symmetric about the origin r=0, which is also the highest point of the dome. The shape of the dome is given by specifying h, how far the dome surface lies below this highest point, as a function of the radial coordinate in the surface, r. For simplicity of the mathematics, we shall set h = (2/3g) r^{\frac{3}{2}}. […]
At any point, the magnitude of the gravitational force tangential to the surface is F = \frac{d(gh)}{dr} = r^{\frac{1}{2}} and is directed radially outward. There is no tangential force at r=0.
Where did he get F = \frac{d(gh)}{dr}? Why is delta r, a radial coordinate, in the denominator?
~Max
Force is the derivative of potential energy with respect to distance (from the definition of work/energy as the integral of force over distance). The relevant distance is radial in this case.
P.S. gh is the standard expression for gravitational potential energy in a situation where gravity is constant .