energies ω$_{n}$$_{0}$ . The sum rule reads: where the “oscillator strengths” and where u$_{λ}$ ( r ) is the solution of the elastic eigen- value problem normalized with respect to the scalar product (67), ω$_{λ}$ is its eigenvalue, and the sum runs over all the exact eigenfunctions. Further, the oscillator strengths satisfy the sum rule for all λ . The proof of these results is presented in Ap- pendix E. From this vantage point we see that the elastic ap- proximation is the extension of the well-known collective approximation 40 of the homogeneous electron gas to in- homogeneous systems. Each eigenvalue ω$_{λ}$ of the elastic equation of motion is a weighted average of exact exci- tation energies, with a weight controlled by the overlap of the exact current matrix element with the eigenmode u$_{λ}$ ( r ). Since u$_{λ}$ ( r ) form an orthogonal basis in the space of displacements, we can say that ω$_{λ}$ represents the av- erage energy of exact excitations in the “direction” λ . Thus, the full excitation spectrum is replaced by a set of spectral lines, one for each orthogonal direction in dis- placement space, and each one carrying the entire and exact spectral weight for that particular direction. For orientation we now examine the application of our theory to a few simple models. In a homogeneous electron gas the ground-state den- sity n$_{0}$ = n is independent of position. The equilibrium kinetic stress tensor has a constant value where t ( n ) is the kinetic energy per particle. The two particle density matrix ρ$_{2}$ ( r , r ′ ) is a function of | r − r $^{′}$| . In such a homogeneous system the displacement eigen- functions are simply plane waves where ω 2 p = 4 πne $^{2}$/m is the square of the plasmon fre- quency and ˆ q is the unit vector in the direction of q . Notice that this potential is assumed to vary only in the direction of ˆ q , because it is only in this direction that the displacement generates boundary charges: the system re- mains perfectly homogeneous in the direction perpendic- ular to q . 45 The corresponding force in the equation of motion (15) is Finally, in order to take into account the neutralizing background of positive charge (required for the stability of the electron gas), we add the external potential Thus, we get where v ( q ) = 4 πe $^{2}$/q2 is the Fourier transform of the Coulomb potential in three dimensions, and ρ$_{2}$ ( q ) – the Fourier transform of ρ$_{2}$ ( r − r ′ ) – is related to the static structure factor S ( q ) in the following manner: 35 where The force density from potential energy, Eq. (58), is given by characterized by a wave vector q . Of these there are two kinds: longitudinal, in which ˜ u is parallel to q , and transverse, in which ˜ u is perpendicular to q . The expres- sion (53) for the kinetic force density reduces to 11 V. MODEL APPLICATIONS A. Homogeneous electron gas T$_{µν,}$$_{0}$ = 2 3 nt ( n ) δ$_{µν}$ , (84) u ( r , ω ) = ˜ u ( q , ω ) e i ( q · r − ωt ) (85) u · ∇ ∂$_{µ}$V$_{0}$ = mω 2 $_{p}$(ˆ q · u )ˆ q$_{µ}$ . (92) V$_{0}$ ( r ) = m 2 ω 2 $_{p}$( r · ˆ q ) 2 , (91) − δW$_{2}$ [ u ] δ u = − n ∫ d q ′ [ S ( q − q ′ ) − S ( q ′ )] v ( q ′ ) q ′ [ q ′ · ˜ u ( q )] . (90) ρ$_{2}$ ( q ) = n [ S ( q ) − 1] . (89) K$_{µν}$ ( q ) = − ∫ d q ′ (2 π ) $_{3}$ρ$_{2}$ ( q − q ′ ) q ′ $_{µ}$q ′ $_{ν}$v ( q ′ ) , (88) − δW$_{2}$ [ u ] δ u$_{µ}$ = [ K$_{µν}$ ( 0 ) − K$_{µν}$ ( q )]˜ u$_{ν}$ , (87) − δT$_{2}$ [ u ] δ u = − 2 3 nt ( n )[2 q ( q · ˜ u ) + q $^{2}$˜ u ] + nq 2 4 m q ( q · ˜ u ) . (86) ω 2 λ = ∑ n f λ $_{n}$ω 2 λ 0 (81) f λ n = 2 m ∣ ∫ d rj$_{0}$$_{n}$ ( r ) · u$_{λ}$ ( r ) ∣ ∣$^{2}$ ω$_{n}$$_{0}$ , (82) ∑ n f λ n = 1 (83)