  We present a constructive derivation of a time-dependent deformation
functional theory -- a collective variable approach to the nonequalibrium
quantum many-body problem. It is shown that the motion of infinitesimal fluid
elements (i.e. a set of Lagrangian trajectories) in an interacting quantum
system is governed by a closed hydrodynamics equation with the stress force
being a universal functional of the Green's deformation tensor $g_{ij}$. Since
the Lagrangian trajectories uniquely determine the current density, this
approach can be also viewed as a representation of the time-dependent current
density functional theory. To derive the above theory we separate a
"convective" and a "relative" motions of particles by reformulating the
many-body problem in a comoving Lagrangian frame. Then we prove that a properly
defined many-body wave function (and thus any observable) in the comoving frame
is a universal functional of the deformation tensor. Both the hydrodynamic and
the Kohn-Sham formulations of the theory are presented. In the Kohn-Sham
formulation we derive a few exact representations of the exchange-correlation
potentials, and discuss their implication for the construction of new
nonadiabatic approximations. We also discuss a relation of the present approach
to a recent continuum mechanics of the incompressible quantum Hall liquids.
