(Posted to MSE here, 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, $L=\int \sqrt{1+y'^2}dx.$

On the other hand, 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?