(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?
Quite generally, the shape $y(x)$ of the cable and the shape $w(x)$ of the suspended deck are governed by the Melan equation, a fourth order ODE. A variational formulation is given in Variational formulation of the Melan equation (2016). The catenary shape is obtained if the deck has neglible mass compared to the cable, while the parabolic shape follows if the cable has negligible mass compared to the deck. In the latter case the deck remains flat, $w(x)=\text{constant}$ and the variational principle reduces to $y''(x)=\text{constant}$. This miminizes $U=\int[\mu y- (y')^2]\,dx$ (with $\mu$ a constant proportional to the mass density of the suspended deck).
