tice, however, that the interaction contribution vanishes if u ( r ) is constant in space, as expected from the trans- lational invariance of the interaction. and dynamical exchange-correlation (xc) effects. This non-interacting system is known as the “Kohn-Sham ref- erence system” and its ground-state density coincides with the exact ground-state density of the interacting system, i.e. n$_{0}$ ( r ). The potential that produces this ex- act ground-state density in the Kohn-Sham reference sys- tem is known as the static Kohn-Sham potential and is usually written as The formulas presented in the preceding subsections simplify dramatically in one-dimensional systems, where the displacement field has only one component, u ( x ), the strain tensor reduces to the derivative of the displace- ment field u ′ ( x ), and the equilibrium kinetic stress tensor reduces to a scalar Then the combination on the last line of Eq. (53) vanishes and we are left with the simpler expression where the primes denote derivatives with respect to x . The complete equation of motion for one dimensional systems is thus where K ( x, x ′ ) is given by the one-dimensional version of Eq. (57). We will make use of this form of the equation of motion in the model applications presented below. where ψ$_{ℓ}$ ( r ) are Kohn-Sham orbitals for the ground-state and the sum over ℓ runs over the occupied orbitals. Assuming that the Kohn-Sham ground-state has been obtained by one of the available approximations for V$_{s,}$$_{0}$ , the remaining problem is to find a suitable approximate where V$_{H,}$$_{1}$ is the first-order term in the expansion of the time-dependent Hartree potential, and E$_{xc,}$$_{1}$ is the first order term in the expansion of E$_{xc}$ in powers of u ( r ). The non-interacting kinetic force density − δT$_{s}$$_{2}$ [ u ] /δ u ( r ) is given by Eq. (53) in which, however, the equilibrium kinetic stress tensor T$_{µν,}$$_{0}$ is replaced by the correspond- ing quantity for the Kohn-Sham reference system, i.e The idea is now to apply our continuum mechanics formulation directly to the Kohn-Sham reference sys- tem. There is a small technical problem in doing this, namely the time-dependent xc vector potential A$_{xc}$ ( r , t ) that acts on the Kohn-Sham system has in general a transverse component, which cannot be represented as the gradient of a scalar potential. Indeed, a complete representation of A requires that we introduce both an electric field E$_{xc}$ and a magnetic field B$_{xc}$ . The inclusion of the xc magnetic field does not create any difficulties in principle (see Ref. 20), and leads to the appearance of a Lorentz-force term in the equation of motion for the current. But this Lorentz force term can be safely disre- garded in the linear response approximation, because it has the form j × B which is of second order in the devi- ation from equilibrium. Thus we can take into account dynamical xc effects simply by adding the force − e E$_{xc}$ to the driving force − n$_{0}$ ∇ V$_{1}$ on the right hand side of Eq. (15). All this considered, our equation of motion takes the form where V$_{H,}$$_{0}$ and V$_{xc,}$$_{0}$ are, respectively, the Hartree poten- tial and the xc potential of the ground-state. These static potentials should not to be confused with the additional dynamical Hartree and xc potentials, which appear when the system is not in equilibrium. E. Equation of motion for one-dimensional systems III. CURRENT-DENSITY FUNCTIONAL APPROACH T$_{0}$ ( x ) = 1 m [ ∂$_{x}$∂$_{x}$$_{′}$ρ ( x, x ′ ) ∣ x = x ′ − n ′′ 0 4 ] . (59) − δT$_{2}$ [ u ] δu$_{µ}$ = (3 T$_{0}$u ′ ) ′ − 1 4 m ( n$_{0}$u ′′ ) ′′ (60) mn$_{0}$∂ 2 $_{t}$u = − n$_{0}$uV ′′ 0 + (3 T$_{0}$u ′ ) ′ − 1 4 m ( n$_{0}$u ′′ ) ′′ + ∫ $_{dx}$′$_{K}$$_{(}$$_{x, x}$ ′$_{)[}$$_{u}$$_{(}$$_{x}$$_{)}$ − u ( $_{x}$′$_{)]}$ − n$_{0}$V ′ 1 , (61) V$_{s,}$$_{0}$ ( r ) = V$_{0}$ ( r ) + V$_{H,}$$_{0}$ ( r ) + V$_{xc,}$$_{0}$ ( r ) , (62) m∂ 2 $_{t}$u ( r , t ) + ( u · ∇ ) ∇ V$_{s,}$$_{0}$ = − 1 n$_{0}$ δT$_{s}$$_{2}$ [ u ] δ u ( r ) − ∇ [ V$_{1}$ ( r , t ) + V$_{H,}$$_{1}$ ( r , t )] − e E$_{xc,}$$_{1}$ ( r , t ) , (63) T s µν, 0 = 1 2 m ( ∂$_{µ}$∂ ′ ν + ∂$_{ν}$∂ ′ µ ) ∑ ℓ ψ ∗ $_{ℓ}$( r ) ψ$_{ℓ}$ ( r ′ ) ∣ r = $^{r}$′ − 1 4 m ∇ $^{2}$n$_{0}$ δ$_{µν}$ (64) 8 Our discussion thus far has not relied on time- dependent current density functional theory, except on a very abstract level, i.e. as a basis for the statement that the stress tensor must be a functional of the current den- sity. The formulas presented in the last two subsections relied on the knowledge of the exact density matrices ρ and ρ$_{2}$ of the many-body ground-state – two quanti- ties that are amenable to treatment by powerful numer- ical techniques (e.g. the quantum Monte Carlo method) which have little in common with DFT. Before proceed- ing, we wish to clarify how the time-dependent CDFT can help us in more concrete ways when the exact ρ and ρ$_{2}$ are not known, which is by far the most common case. One of the main ideas of TDCDFT is that the current and density evolutions of the interacting many-body sys- tem can be simulated in a non-interacting many-body system subject to an effective time-dependent vector po- tential which includes the Hartree electrostatic potential