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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. 

## Appendix A: Variational principle for an ideal fluid and the Gross-Pitaevskii action 

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 

## Acknowledgments 

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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, 

T his constraint is incorporated into the Lagrangian den- sity by introducing a Lagrange multiplier φ : 

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Then the equations of motion are obtained from varia- tions of the action S [ ρ, v , φ ] with respect to ρ , v , and φ : 

From Eq. (A5) we obtain the velocity field 

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 

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 

For comparison, consider a system of weakly interact- ing bosons with a condensate wave function ψ ( r , t ) that satisfies Gross-Pitaevskii equation 42 

This equation of motion can be derived from the La- grangian density 

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