Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 2 / e / Solution

Every term in the free energy contracts the vector components with the Euclidean inner product: it depends only on $|\mathbf{p}{|}^2$, $|\mathbf{p}{|}^4$, $({\nabla }_i{p}_j)({\nabla }_i{p}_j)$, and $|{\nabla }^2\mathbf{p}{|}^2$. A constant $\mathcal{R}\in SO(3)$ preserves all these contractions and commutes with spatial differentiation. Thereforeso every constant rotation of candidate (ii) has the same free energy. The orientation of the rotation plane of the polar helical smectic is thus continuously degenerate.