I'm looking at Arnold's Mathematical Methods of Classical Mechanics at the beginning of chapter 3 p 55 which defines when a functional is differentiable. Slightly paraphrasing and skipping a couple of details:
A functional $\Phi$ is said to be differentiable if $\Phi(\gamma+h)-\Phi(\gamma)=F+R$
where $F(\gamma,h)$ is linear in $h$ and $R\sim O(h^2)$ in the sense that for $|h|<\epsilon$ and $\left|\frac{dh}{dt}\right|<\epsilon$ then $|R|<C\epsilon^2$.
I've been telling myself to think of $\gamma$ as a curve, $h$ as a slight variation of the curve, and $F$ as the differential or "principal variation" of the functional, and $R$ the "error."
This is my first time seeing this definition of "differentiable" and also of $O(h^2)$, and I have a pair of questions about the latter.
In the 50 pages preceding, I can't seem to find out what $|\cdot |$ means here. Is $|h|$ total variation of the curve $h$, or something like that?
How can I make sense of the constraints $|h|<\epsilon$ and $\left|\frac{dh}{dt}\right|<\epsilon$ controlling $R$? The best I've come up with is "if $h$ does not go up and down too much and the speed doesn't go up and down too much, then $R$ will be under control." It might be helpful to have a prototypical example of a situation where $h$ slow but too wavy, and a situation where $h$ is not wavy but the speed varies wildly. Good heuristics are welcome too.