= Gravitational potential producing a circular hanging chain
{title2=$\Phi(z)=\Phi_0+\kappa/(h-z),\quad\kappa>0$}
For uniform <linear mass density> $\mu$, a chain minimizing $\mu\int\Phi(z)\,ds$ at fixed length has <Beltrami identity> $(\mu\Phi-\lambda)/\sqrt{1+z'^2}=C$. In the domain $z<h$, the displayed potential has downward <gravitational acceleration> $-\Phi'(z)=-\kappa/(h-z)^2$. A circular arc $z=h-\sqrt{a^2-x^2}$ satisfies the identity because $\sqrt{1+z'^2}=a/(h-z)$.
This arc is a global minimum among admissible competitors of the same length with the same endpoints in $z<h$. Set $w=h-z>0$. The variable part of the energy is $\mu\kappa\int ds/w$, a <hyperbolic length> in the <Poincaré half-plane model>. The <calibration for semicircular hyperbolic geodesics> $S=\operatorname{arsinh}(x/w)$ has $|\nabla S|=1/w$. The <Cauchy-Schwarz inequality> gives $\int ds/w\geq S(B)-S(A)$, with equality along the circle traversed with increasing $x$. The constant-potential part is unchanged by the fixed-length constraint.
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