Converting order parameter to director (nematics)

Question

If I have a differential equation for the order parameter of a uniaxial nematic, like

$\frac{\partial S_{\alpha \beta}}{\partial t} = f(S_{\alpha \beta})$,

for

$S_{\alpha \beta} = S(n_{\alpha}n_{\beta} - \frac{1}{3}\delta_{\alpha \beta})$ ,

is it possible to convert this to a differential equation for $n_{\alpha}$? Meaning, can I perform some transformation to get an equation of the form

$\frac{\partial n_{\alpha}}{\partial t} = g(n_{\alpha})$.

If not, is there a way to insert angles $\phi$ and $\theta$ into a 3-D $S_{\alpha \beta}$, where $\phi$ is in one direction, and $\alpha$ the perpendicular direction (imagine essentially the angles of spherical coordinates)? These angles are the angles that define the director $n_{\alpha}$, as a unit vector of constant magnitude 1.

Answer 1

For future physicists wondering the same; you can in fact do so. You need only to dot in a power of the director on your equation! The tensor $\mathsf{S}$ dotted with $n$ results in a power of $n$, so all works out in the end.