Off the top of my head (haven’t verified carefully):
What they’re really saying is that they have an initial value problem in classical mechanics that does not have a unique solution.
Fortunately, the theory of such things is very well established.
From the article, the dome results in motion like this:
> d2r/dt2 = sqrt(r) [reformatted as text]
This function is interesting at r=0 — it’s not differentiable, and it’s not even locally Lipschitz. So one would not generally expect solutions to the initial value problem to be unique, and math and classical mechanism still work.
(The differential equation in question is second order and is in two spatial dimensions. The standard transformation to an ordinary vector differential equation applies, and you end up with four variables (x, v_x, y, v_y or however you like to name them) plus time, and the time derivatives of v_x and v_y as functions of everything else are not Lipschitz in any open set containing the origin.)
And, intuitively, what’s going on is that the acceleration of the mass is exquisitely sensitive to position around the top of the dome (it varies infinitely quickly with a small displacement), which is precisely what breaks the uniqueness of the initial value problem.
What they’re really saying is that they have an initial value problem in classical mechanics that does not have a unique solution.
Fortunately, the theory of such things is very well established.
From the article, the dome results in motion like this:
> d2r/dt2 = sqrt(r) [reformatted as text]
This function is interesting at r=0 — it’s not differentiable, and it’s not even locally Lipschitz. So one would not generally expect solutions to the initial value problem to be unique, and math and classical mechanism still work.
You can read more about the theory here:
https://en.m.wikipedia.org/wiki/Initial_value_problem
(The differential equation in question is second order and is in two spatial dimensions. The standard transformation to an ordinary vector differential equation applies, and you end up with four variables (x, v_x, y, v_y or however you like to name them) plus time, and the time derivatives of v_x and v_y as functions of everything else are not Lipschitz in any open set containing the origin.)
And, intuitively, what’s going on is that the acceleration of the mass is exquisitely sensitive to position around the top of the dome (it varies infinitely quickly with a small displacement), which is precisely what breaks the uniqueness of the initial value problem.