Starting from a microscopic definition of an alignment vector proportional to
the polarization, we discuss the hydrodynamics of polar liquid crystals with
local Cβvβ-symmetry. The free energy for polar liquid crystals
differs from that of nematic liquid crystals (Dβhβ) in that it
contains terms violating the nββn symmetry. First we show
that these Z2β-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.Comment: 9 pages, corrected typos, added references, revised content, to
appear in PR