Classical helical minimum of the J1-J2 XY chain ($\cos (qc)=-J_1/(4J_2)$)

For classical fixed-length planar spins with energy per site $-S^2[J_1\cos (qc)+J_2\cos (2qc)]$, the substitution $x=\cos (qc)$ makes the problem a quadratic on $[-1,1]$. A noncollinear minimum exists for $J_2<0$ and $|J_2|>|J_1|/4$. Its two signs of $q$ have opposite chirality and equal energy. This is a classical energy minimum, not a finite-temperature long-range-order theorem for a one-dimensional quantum chain.