(Posted to MSE [here](https://math.stackexchange.com/questions/4801880/what-variational-problem-does-the-parabolic-suspension-bridge-solve), no answers)

The catenary curve $y(x)$ minimizes the gravitational potential energy
$$\int \rho g y ds=\int \rho g y \sqrt{1+y'^2}dx,$$
subject to a fixed length, $L=\int \sqrt{1+y'^2}dx.$

It is known that in a suspension bridge the cable has a parabolic shape. This can be derived from forces. But what potential does that minimize?