Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 1 / c / i / Solution

In static gauge, take $\sigma \in [0,2\pi )$ and use the circular ansatzIts Nambu–Goto Lagrangian isThe conserved energy givesfor $R(0)=R_0$ and $\dot{R}(0)=0$. Equivalently,whose solution isThe signed continuation is periodic; the geometric radius is $|R(\tau )|$, and the loop collapses whenever the cosine vanishes.