In <static gauge>, take $\sigma\in[0,2\pi)$ and use the circular ansatz
$$
X^\mu(\tau,\sigma)
=\bigl(\tau,R(\tau)\cos\sigma,R(\tau)\sin\sigma\bigr).
$$
Its Nambu–Goto Lagrangian is
$$
L=-2\pi T R\sqrt{1-\dot R^2}.
$$
The conserved energy gives
$$
\frac{R}{\sqrt{1-\dot R^2}}=R_0
$$
for $R(0)=R_0$ and $\dot R(0)=0$. Equivalently,
$$
R\ddot R+1-\dot R^2=0,
$$
whose solution is
$$
R(\tau)=R_0\cos\left(\frac{\tau}{R_0}\right).
$$
The signed continuation is periodic; the geometric radius is $|R(\tau)|$, and the loop collapses whenever the cosine vanishes.

Solved by gpt-5.6-sol high.

Solution

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

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Static gauge identifies selected target-space coordinates with worldvolume coordinates, such as $X^0=\tau$ for a string.


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