I am struggling to understand the directional derivative of a functional in direction of a 'path' $\omega$:
Let $\omega\in \mathcal{C}([0,T],\mathbb R^{d}) \;$ and $F: \mathcal{C}([0,T], \mathbb R^{d})\times[0,T]\to \mathbb R$ be some functional
How should I interpret$\;\nabla_{\omega} F(\omega,t)$? What changes from differentiating with respect to a vector and differentiating with respect to an actual function? Is there any graphical intuition?
A surprising amount changes when moving from the finite-dimensional derivation to infinite-dimensional derivation. Lets first define the variation of the functional $F$ in the direction $h$, $$(\delta_{w} F)(w, t, h): C([0, T], \mathbb{R}^d) \times [0, T] \times C([0, T], \mathbb{R}^d) \rightarrow \mathbb{R}$$ $$ (\delta_{w} F)(w, t, h) = \underset{x\rightarrow 0^+}{\lim} \frac{1}{x}\left[F(w + xh, t) - F(w, t)\right] $$ you can interpret this as the miniscule increase/decrease in the function when adding a miniscule (positive) perturbation in the direction $h$ to the first argument $w$.
Let's take an example: Assume we have a long beach for which the waves in the ocean are uniform in the direction parallel to the beach. We can simplify and consider the problem as one-dimensional in the direction into the ocean.
We now want to find the gnarliest place to surf. We define the height of the water at distance $t \in [0, T]$ from the beach as $w(t)$. $T$ is some upper limit set by the coast guard. Now assume we have some measurability of gnarlyness $$f(z) = z^2$$ such that larger waves get increasingly more gnarly the larger they are. We wish to define the gnarlyness for all possible states of the ocean at all possible distances from the beach and thus define $$ F(w, t) = w(t)^2 $$ for which we can also define the variation $$ (\delta_{w} F)(w, t, h) = \underset{x\rightarrow 0^+}{\lim} \frac{1}{x}\left[F(w + xh, t) - F(w, t)\right]. $$ This can be interpreted as the increase / decrease in gnarlyness at location $t$ when the ocean is in state $w$ and is perturbed in the direction of some other state $h$. Let's assume the ocean is in state $w(t) = 1 + \sin(t)$. A very strange tide start to come in perturbs the ocean in the direction $h(t) = -\sin(t)$. We compute the variation $$ (\delta_{w} F)(w, t, h) = \underset{x\rightarrow 0^+}{\lim} \frac{1}{x}\left[(1 + \sin(t) - x \sin(t))^2 - (1 + \sin(t))^2\right] \\ = \underset{x\rightarrow 0^+}{\lim} \frac{1}{x}\left[(1 + \sin(t) - x\sin(t) + \sin(t) + \sin(t)^2 - x\sin(t)^2 - x\sin(t) -x\sin(t)^2 + x^2\sin(t)^2 - 1 - 2\sin(t) - \sin(t)^2\right] \\ = \underset{x\rightarrow 0^+}{\lim} \frac{1}{x}\left[ - x\sin(t) - x\sin(t)^2 - x\sin(t) -x\sin(t)^2 + x^2\sin(t)^2 \right] \\ = -2\sin(t)(1 + \sin(t)) $$ And observe that we will be seeing a decrease in gnarlyness for $t\in [2 k \pi, (2k +1)\pi]$ and an increase in gnarlyness for $t\in [(2k +1) \pi, (2k +2)\pi]$.
The power of thinking like this is that we can define the variation before assuming some state $w$. Instead if doing the somewhat cumbersome calculation above we could have used some observed that $$ (\delta_{w} F)(w, t, h) = h(t) \frac{\partial w^2}{\partial w}(t) = 2 h(t) w(t) =-2 \sin(t) (1 + \sin(t)) $$ Note that this only happens under some conditions. If you are interested in what these conditions are then I suggest you find yourself a cool course on infinite-dimensional optimization and / or the calculus of variations if you aren't already doing so.