Poisson-bracket formulation of the dynamics of polar liquid crystals

We develop the dynamical theory of polar liquid crystals with local $C_{\infty v}$-symmetry using the general Poisson-bracket formalism. We obtain dynamical equations for the slow macroscopic fields that govern the dynamics in both the polarized and the isotropic phases. Starting from a microscopic definition of an alignment vector proportional to the polarization, we obtain Poisson bracket relations for the director field. The hydrodynamic equations differ from those of nematic liquid crystals ($D_{\infty h}$) in that they contain terms violating the ${\bf{n}} \rightarrow -{\bf{n}}$ symmetry. We find that the $\mathcal{Z} _2$-odd terms induce a general splay instability of a uniform polarized state in a range of parameters.