[nicf]

I'm not sure if this will make you feel any better, but there's an interesting mathematical corner case at work with Norton's Dome that's responsible for the breakdown in the intuition you're expressing in (1).

You could formalize this intuition as the statement that, if I'm trying to describe a function f(t) and I know (a) the value of f and its derivative at t=0 along with (b) a second-order differential equation that f has to satisfy, this should be enough to nail down the entire function.

A big theorem, which many people just call something like "existence and uniqueness of solutions of ordinary differential equations", says that in most ordinary situations this is indeed true, and basically for the reason you probably intuitively think: you can imagine using the differential equation to make tiny "updates" to the value of f to move a little bit forward in time, and take the limit as the size of your time increment goes to zero. (You can read more about it in this somewhat technical Wikipedia article: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_t....)

But there is a condition on the theorem which limits its scope: the right side of the differential equation has to be something called "Lipschitz continuous". The vast majority of differential equations that appear in Newtonian physics satisfy this condition, but the equation you get in the Norton's Dome example doesn't, and this is what's responsible for the lack of uniqueness in the solution. It turns out that there are many different trajectories for the particle that satisfy both the initial condition and the differential equation.

What relevance does this have to the actual universe? Personally, I think very little; it's a fact about a model of physics, not a fact about the actual universe. There are all sorts of reasons why you can't literally build Norton's Dome: matter is not actually continuous because it's made of atoms, and classical physics isn't an exact model of the universe anyway. But it's interesting to see that a feature of Newtonian physics that we usually take for granted isn't actually always true.

[meroes]

Wow thank you. You’re right there’s likely some hidden assumptions that I had taken for granted that a unique solution is relying on when solving DE’s. I will have to read up to make this more clear mathematically, but at least mathematically I think you’ve answered my concern about 1). Now whether things break down when we model a potentially discrete world with continuous math, that’s for another day like you said. And what it means for something “at rest” to start moving if all its position derivatives are 0. But those might be more philosophical.

[nicf]

You're welcome! There's actually one more point that I thought of after sending that reply, which since you mentioned it again I should maybe flag.

Totally apart from physics, it may seem intuitively plausible that if you have a function f and (a) all f's derivatives exist everywhere, and (b) f(0), f'(0), f''(0), etc. are all zero, then f must be the zero function. This is actually also not true! For a counterexample, you can look at this article on bump functions: https://en.wikipedia.org/wiki/Bump_function.