Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 13C (Variational Principles)

Put $W=1+{\rho }^{\prime 2}+{\rho }^2{\varphi }^{\prime 2}$. The Euler–Lagrange equations areWhen $n=n(\rho )$, the helical ansatz $\rho =R$, $\varphi ={\varphi }_0+\omega z$ automatically satisfies the second equation. The first becomesso $a=1$.

The endpoints have the same cylindrical angle exactly when $\omega L=2\pi k$ for some integer $k$. Thus a positive $L$ with $L=2\pi k/\omega (R)$ admits the two special solutions of opposite handedness, using $\omega$ and $-\omega$.