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 ($D_{ \infty h}$) in that it contains terms violating the ${ \bf{ n}}\to -{ \bf{ n}}$ 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.
Categories: cond-mat.soft cond-mat.stat-mech