−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.01 0 100 200 300 400 500 600 700 (D  / c$_{NS}$ 2 )  Normalized Density−Density Correlation Function normal solid $_{s}$ /   = 0.01 $_{s}$ /   = 0.02 FIG. 2: (Color online). Splitting of the Rayleigh peak (black dashed line)due to the defect diffusion mode in the normal solid phase into the Brillouin doublet of the second sound modes in the supersolid phase. The red dash-dot line is for ρ s /ρ = 1%, and the green solid line ρ s /ρ = 2%. earized version of this action to calculate the second of our important results–the density-density correlation and response functions for a model supersolid (with isotropic elastic properties), see Eq. (87). In complete analogy with a superfluid, we showed that the onset of superso- lidity causes the zero-frequency defect diffusion mode to split into propagating second sound modes, and from our calculation we can determine the spectral weight in these modes as well as the weight in the “normal” longitudinal sound waves in a solid. The authors would like to thank K. Dasbiswas, J. Dufty, P. Goldbart, D. Goswami, Y. Lee, M. Meisel, and J. Toner for helpful discussions and comments. A.T.D. would like to thank the Aspen Center for Physics, where part of this work was completed. This work was sup- ported by NSF Grant No. DMR-0705690. This appendix demonstrates the simplicity of the varia- tional principle for deriving the hydrodynamic equations of motion and the Lagrangian density for continuum sys- tems. Consider the simplest case of an ideal fluid (IF) which is irrotational, inviscid and incompressible. The Lagrangian density for the ideal fluid is This equation of motion can be derived from the La- grangian density For comparison, consider a system of weakly interact- ing bosons with a condensate wave function ψ ( r , t ) that satisfies Gross-Pitaevskii equation 42 where P is the pressure. The Lagrangian density may be cast into an equivalent form by substituting v = ∇ φ into Eq. (A3) and integrating by parts, with the result which implies that there is no vorticity, as expected for an ideal fluid. We can derive the Euler equation from Eq. (A4) by taking its gradient, using Eq. (A7), and then the Gibbs-Duhem relation to obtain From Eq. (A5) we obtain the velocity field Then the equations of motion are obtained from varia- tions of the action S [ ρ, v , φ ] with respect to ρ , v , and φ : T his constraint is incorporated into the Lagrangian den- sity by introducing a Lagrange multiplier φ : w here ρ is the mass density, v the velocity field, and U$_{IF}$ the internal energy density which satisfies dU$_{IF}$ = µdρ . From the Lagrangian density we construct the action S = ∫ dt ∫ d $^{3}$x L $_{IF}$. The variational principle states that the equations of motion are derived from variations of the action with respect to all the dynamical variables. The naive application of this principle to the ideal fluid leads to the trivial equation of motion v = 0. The difficulty is that the dynamical variables ρ and v are not indepen- dent, but constrained by the conservation of mass, 10 ∂$_{t}$ρ + ∂$_{i}$ ( ρv$_{i}$ ) = 0 . (A2) L IF = 1 2 ρv 2 − U$_{IF}$ ( ρ ) + φ [ ∂$_{t}$ρ + ∂$_{i}$ ( ρv$_{i}$ ) ] . (A3) δS δρ = 1 2 v 2 − ∂U$_{IF}$ ∂ρ − Dφ Dt = 0 , (A4) δS δv$_{i}$ = ρv$_{i}$ − ρ∂$_{i}$φ = 0 , (A5) δS δφ = ∂$_{t}$ρ + ∂$_{i}$ ( ρv$_{i}$ ) = 0 . (A6) v$_{i}$ = ∂$_{i}$φ, (A7) ρ Dv$_{i}$ Dt = − ∂$_{i}$P, (A8) L IF = − ρ∂$_{t}$φ − 1 2 ρ ( ∂$_{i}$φ ) 2 − U$_{IF}$ ( ρ ) . (A9) i $_{ℏ}$∂ψ ∂t = − ℏ 2 2 m ∇ $^{2}$ψ + g | ψ | $^{2}$ψ. (A10) L GP = i ℏ 2 [ ψ $^{∗}$∂$_{t}$ ψ − ψ∂$_{t}$ψ $^{∗}$] − ℏ 2 2 m ( ∇ ψ $^{∗}$) · ( ∇ ψ ) − g 2 ( ψ $^{∗}$ψ ) $^{2}$. (A11) L IF = 1 2 ρv 2 − U$_{IF}$ ( ρ ) , (A1) Acknowledgments Appendix A: Variational principle for an ideal fluid and the Gross-Pitaevskii action