What are the tools for searching for functional extrema if the functional is nonlinear?
I am especially interested in how the area/circumference ratio optimization problem (of which the circle is the solution) is attacked in the language of variational calculus. In polar coordinates,
$$ R[ r(\phi) ] = \frac{\frac{1}{2} \displaystyle\int_0^{2\pi}r^2(\phi) d \phi}{\displaystyle\int_0^{2\pi}d\phi\sqrt{r^2(\phi)+\dot{r}^2(\phi)}} $$
with constraint
$$r(0) = r(2\pi) = 1$$
How to find $r(\phi)$ such that $R$ is maximized? It does not seem the standard Lagrange-Euler equations work.
TL;DR: OP's variational problem is unbounded from above.
Sketched proof: Consider e.g. the test function $$r(\phi)~=~K(\pi-|\phi|)+1, \qquad K~\in~\mathbb{R}_+, \qquad \phi~\in~[-\pi,\pi].$$ It satisfies the boundary conditions $$ r(\phi\!=\!\pm\pi)~=~1.$$ Derivative: $$ \dot{r}(\phi)~=~-K{\rm sgn}(\phi). $$ Area: $$ A~=~\int_0^{\pi}\! \mathrm{d}\phi~(K\phi+1)^2~=~\frac{\pi^3K^2}{3}+\pi^2K+\pi.$$ Circumference: $$ C~=~2\int_0^{\pi}\! \mathrm{d}\phi\sqrt{(K\phi+1)^2+K^2}~\leq~2\pi\sqrt{(K\pi+1)^2+K^2}.$$ A short calculation shows that OP's fraction $A/C\to\infty$ for $K\to\infty$.$\Box$