ELI5: why do they go to the trouble of constructing a dome when a simple cone would have the same properties? It seems to me that the motion of the object is not infinitely differentiable in either case, and the dome shape only serves to obscure this fact.
The easiest way to see it is to consider the time-reversed version. You chuck the ball up the shape so it perfectly stops balanced at the top. It turns out this is possible for the weird dome in question, but not for pretty much any other shape - including a cone. Going back to time running normally it turns out that this means that a ball balanced perfectly on a cone only has one option consistent with Newton's laws, it'll stay balanced there forever, while there are multiple trajectories consistent with Newton's laws for the dome.
Why is it not possible for a cone? Slide it up the (frictionless) cone with kinetic energy equal to mgh and it should stop on the point.
It never reaches the top in finite time on a cone, it just keeps going up and up getting slower and slower.
That makes sense; thanks.
It's the reverse of a point mass sliding up a hemispherical dome. It takes infinite time to reach the top.
Because the cone is intuitively ridiculous? If I say I’m balancing an object (presumably of finite size) perfectly centered on a cone, then the obvious question is “you did what”? It just seems more absurd than balancing an object on a continuous surface.