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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 $U=\int[y- (y')^2]\,dx$.