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?