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

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