Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 354 / 2 / b

Translation invariance in $y$ lets the surface be written $z=z(x)$. Per unit $y$ length, its area and entropy functionals areThe boundary conditions are $z(\pm R)=0$, understood with an ultraviolet cutoff, and reflection symmetry gives $z^{\prime }(0)=0$ at $z(0)=z_{\max }$.

The Lagrangian $\mathcal{L}=z^{-2}\sqrt{1+z^{\prime 2}}$ has no explicit $x$ dependence, so its Beltrami identity is conserved:At the turning point $z=z_{\max }$ and $z^{\prime }=0$, the constant is $z_{\max }^{-2}$. Rearrangement gives