  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.
