= One-dimensional twisted nematic energy
{title2=$\mathcal L=\tfrac12 f(\theta)\theta'^2+\tfrac12g(\theta)\phi'^2-c\sin^2\theta$}
For a laterally uniform <nematic director> $(\cos\theta\cos\phi,\cos\theta\sin\phi,\sin\theta)$, its <divergence> is $\cos\theta\,\theta'$ and its twist invariant is $-\cos^2\theta\,\phi'$. Thus $f=K_1\cos^2\theta+K_3\sin^2\theta$ and $g=K_2\cos^4\theta+K_3\sin^2\theta\cos^2\theta$. A normal <electric field> contributes $-c\sin^2\theta$. The <Euler-Lagrange equations for two fields> are $(g\phi')'=0$ and $f\theta''+f_\theta\theta'^2/2-g_\theta\phi'^2/2+2c\sin\theta\cos\theta=0$.
Back to article page
