So, for example, for large enough r, the gravitational force \sqrt(r) will exceed the free fall accelleration g?
More importantly, does this additional branch of solutions that satisfies the initial conditions, survive under the small deformations of this dome shape? The perfect Dome shape certainly does not exist.
But by your definition, even something as simple as a cylinder or sphere doesn't exist in nature, because basic mathematical relationships like radius to circumference won't survive "small deformations".
I don't know what point you're trying to make. Norton's Dome "exists in nature" as much as a sphere or a cube does. If it doesn't exist in nature, then no geometric form does.
In the sense that one cannot create an ideal dome and make an experiment, whether a point particle placed exactly at the top later randomly starts to fall.
One has to study if this class of solutions survives deformations of the ideal dome, to make such an experiment (neglecting quantum effects).
As I already explained, this is only non-deterministic in classical mechanics. And the world is quantum. It is an entirely theoretical distinction to begin with. It does not require experimentation.
Nevertheless, you can construct such a dome the same as you can construct a sphere. It's just a regular old geometric shape.
Well, that's what we're saying. The distinction lies in ideal/perfect, vs imperfect. For example, does a perfect cube shape exist? If you closely examine any cube in existence, it has small deformities if you look close enough, as the very edges and corners which make up a cube are made of atoms, which are non-cubical in shape (not to even mention quanta). A perfect cube relies on an cubical shape at infinite scale, but as you mentioned above:
>when we say something doesn't exist in nature, we mean it's fundamentally incompatible
>or is infinite along some dimension
Norton's dome requires an infinitesimal point of sorts for the math to work out. Does that exist in reality? Idk, but it certainly seems dubious.
No, that's not. "Exist in nature" doesn't mean "perfect". Totally different concepts.
> Norton's dome requires an infinitesimal point of sorts for the math to work out. Does that exist in reality? Idk, but it certainly seems dubious.
A cone requires a point at the pointy end. Does that exist in reality? I've certainly seen a lot of objects we call "cones". And they were pointy.
The point is, if you say Norton's dome doesn't exist then you mean cubes don't exist. And we all agree cubes do exist. A perfect Norton's dome doesn't exist, just like a perfect cube doesn't exist. But a regular Norton's dome certainly does exist. Just like a regular cube. Again, it's not an exotic shape. But it doesn't need to be perfect to exist -- otherwise nothing would exist at all!