Assume you want to find the extremals for the functional
$$
y \rightarrow \int_a^by(x)\left[\int_a^xy(\xi)\, d\xi\right]\, dx
$$
where $[a,b]\subset \mathbb{R}$ and $y\in \mathcal{C}^1\left([a,b],\mathbb{R}\right)$.
How can you write down the Euler Lagrange equations associated with this problem?
Best Answer
OP's functional is non-negative $$ F[y]~:=~\int_{[a,b]} \!\mathrm{d}x~y(x)\int_{[a,x]}\!\mathrm{d}\xi ~y(\xi) ~=~\iint_{[a,b]^2} \!\mathrm{d}x~\mathrm{d}\xi ~\theta(x\!-\!\xi) ~y(x)y(\xi)$$ $$~=~\frac{1}{2}\iint_{[a,b]^2} \!\mathrm{d}x~\mathrm{d}\xi ~y(x)y(\xi) ~\stackrel{(2)}{=}~\frac{G[y]^2}{2} ~\geq~ 0,\tag{1}$$ where $$G[y]~:=~\int_{[a,b]} \!\mathrm{d}x~y(x).\tag{2}$$
The functional/variational derivative is $$\frac{\delta F[y]}{\delta y(x)}~\stackrel{(1)}{=}~G[y]\frac{\delta G[y]}{\delta y(x)} ~\stackrel{(2)}{=}~G[y]1_{[a,b]}(x),\tag{3} $$ which is a more general notion than the Euler-Lagrange derivative.
OP's sought-for equation is the vanishing of the functional derivative (3). Evidently, a stationary configuration $y:[a,b]\to \mathbb{R}$ has $$G[y]~\stackrel{(3)}{=}~0,\tag{4}$$ which is clearly a minimum (and in particular an extremum) for OP's functional (1).