  We formulate equations of time-dependent density functional theory (TDDFT) in
the co-moving Lagrangian reference frame. The main advantage of the Lagrangian
description of many-body dynamics is that in the co-moving frame the current
density vanishes, while the density of particles becomes independent of time.
Therefore a co-moving observer will see the picture which is very similar to
that seen in the equilibrium system from the laboratory frame. It is shown that
the most natural set of basic variables in TDDFT includes the Lagrangian
coordinate, $\bm\xi$, a symmetric deformation tensor $g_{\mu\nu}$, and a
skew-symmetric vorticity tensor, $F_{\mu\nu}$. These three quantities,
respectively, describe the translation, deformation, and the rotation of an
infinitesimal fluid element. Reformulation of TDDFT in terms of new basic
variables resolves the problem of nonlocality and thus allows to regularly
derive a local nonadiabatic approximation for exchange correlation (xc)
potential. Stationarity of the density in the co-moving frame makes the
derivation to a large extent similar to the derivation of the standard static
local density approximation. We present a few explicit examples of nonlinear
nonadiabatic xc functionals in a form convenient for practical applications.
