Starting from a microscopic definition of an alignment vector proportional to the polarization, we discuss the hydrodynamics of polar liquid crystals with local $C_{\infty v}$ symmetry. The free energy for polar liquid crystals differs from that of nematic liquid crystals $\left(D_{\infty h}\right)$ in that it contains terms violating the $\mathbf{n}\to -\mathbf{n}$ symmetry. First we show that these ${\mathcal{Z}}_2$-odd terms induce a general splay instability of a uniform polarized state in a range of parameters. Next we use the general Poisson-bracket formalism to derive the hydrodynamic equations of the system in the polarized state. The structure of the linear hydrodynamic modes confirms the existence of the splay instability.

• Received 25 October 2005

DOI:https://doi.org/10.1103/PhysRevE.73.031708