  We present nonlinear dynamic equations for nematic and smectic $A$ liquid
crystals in the presence of an alternating electric field and explain their
derivation in detail. The local electric field acting in any liquid-crystalline
system is expressed as a sum of external electric field and the fields
originating from feedback of liquid crystal order parameter, and a field,
created by charged impurities. The system tends to decrease the total electric
field, because it lowers the energy density. This basically nonlinear problem
is not a pure academic interest. In the realm of liquid crystals and their
applications, utilized nowadays modern experimental techniques have progressed
to the point where even small deviations from the linear behavior can be
observed and measured with a high accuracy. Hydrodynamics is the macroscopic
description of condensed matter systems in the low frequency, long wavelength
limit. Nonlinear hydrodynamic equations are well established to describe simple
fluids. Similar approaches (with degrees of freedom related to the broken
orientational or translational symmetry included) have been used also for
liquid crystals. However to study behavior of strongly perturbed well above the
thresholds of various electro-hydrodynamic instabilities of liquid crystals the
nonlinear equations should include soft electromagnetic degrees of freedom as
well. The self-consistent derivation of the complete set of the nonlinear
electro-hydrodynamic equations for liquid crystals became an actual task. The
aim of our work is to present these equations, which is a mandatory step to
handle any nonlinear phenomenon in liquid crystals.
