# Converting order parameter to director (nematics)

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.

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.