In the following we demonstrate that the equations of motion derived above are equivalent to the non- dissipative supersolid hydrodynamics developed by An- dreev and Lifshitz, 1 Saslow, 4 and Liu. 5 First, the taking the gradient of Eq. (11) produces the Josephson equation use Eqs. (10) and (13) to eliminate α and β$_{i}$ . We then obtain 3 where we have used the thermodynamic relation for ∂U$_{ss}$/∂ρ given in Eq. (3). Second, we derive the momen- tum conservation equation; the following identity simpli- fies the derivation: where f ≡ U$_{SS}$ − T s satisfies the thermodynamic relation When cast in this form, we see that the coupling be- tween the superfluid and the normal fluid [the fourth term in Eq. (21)] is − ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ ) v$_{n}$$_{i}$v$_{s}$$_{j}$ ; this is a “current- current” interaction, where the coupling constant is the normal fluid density. This coupling coefficient is univer- sal –it is determined by conservation laws and Galilean invariance. As mentioned earlier, several other authors have re- cently proposed Lagrangian descriptions for supersolids. Son 35 used symmetry-based arguments to derive an ef- fective Lagrangian for a supersolid. To connect to Son’s results, we first invert Lin’s constraint, Eq. (8), to obtain where D/Dt ≡ ∂$_{t}$ + v$_{i}$∂$_{i}$ . Take D$_{n}$/Dt of Eq. (14), and eliminate the Lagrange multipliers by using Eqs. (7), (8), (13)-(15). The result is Third, combine Eq. (18) with the thermodynamic rela- tion, Eq. (3), the continuity equation, Eq. (5), and the Josephson equation, Eq. (16). After some algebra, we obtain the momentum conservation law where j$_{i}$ is the mass current given in Eq. (6), and Π$_{ij}$ is the (non-dissipative) stress tensor where p$_{i}$ ≡ ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ )( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) and ϵ satisfies a ther- modynamic relation given by Eq. (B3). Note that the Josephson equation, Eq. (16), and the momentum con- servation equation, Eq. (19), are Eqs. (9) and (12) of Andreev and Lifshitz 1 [Andreev and Lifshitz neglected nonlinear strain terms, effectively replacing R$_{ik}$ by δ$_{ik}$ in the last term of Eq. (20) above]. Moreover, the mo- mentum conservation equation is equivalent to Eq. (4.16) of Saslow 4 when v$_{s}$ is taken as a Galilean velocity, and Eq. (3.40) of Liu 5 in the case where the superthermal current vanishes. The Lagrangian density used to derive the hydrody- namics of supersolids, Eq. (9), can be recast into a more familiar and compact form by using the equations of mo- tion, as illustrated for an ideal fluid in Appendix A. To see this, we integrate the terms involving the Lagrange multipliers by parts (neglecting boundary terms), and In this section we discuss the propagation of collec- tive modes in supersolids by examining the response to small fluctuations away from equilibrium. The number of collective hydrodynamic modes of a system can be inferred by enumerating the system’s conservation laws and broken symmetries. 38 Since we are more interested in the effect of density or defect fluctuations than ther- mal fluctuations, for simplicity we ignore thermal fluctu- where R − 1 ji ≡ ∂x$_{i}$/∂R$_{j}$ and R$_{ij}$ R − 1 jk = δ$_{ik}$ . Substituting this into Eq. (21), we obtain a Lagrangian similar in form to Eq. (23) of Son’s paper (however, our energy density f depends upon ρ , ρ$_{s}$ , and T in addition to R$_{ij}$ ). A differ- ent approach was used by Josserand et al. , 36 who applied a homogenization procedure to a nonlocal version of the Gross-Pitaevskii equation to systematically derive a long- wavelength Lagrangian for a supersolid. On the whole, our Eq. (21) agrees with their Eq. (4), once we identify their ρ ( n ) with our normal fluid density ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ . Fi- nally, Ye 37 proposed a phenomenological supersolid La- grangian; however, the Lagrangian in his Eq. (1) is not manifestly Galilean invariant, so that his coupling con- stants a$_{αβ}$ are arbitrary. His approach also misses certain nonlinear terms that are important when treating topo- logical defects. III. QUADRATIC LAGRANGIAN DENSITY AND THE LINEARIZED HYDRODYNAMICS OF SUPERSOLIDS Π$_{ij}$ = ρv$_{s}$$_{i}$v$_{s}$$_{j}$ + v$_{s}$$_{i}$p$_{j}$ + v$_{n}$$_{j}$p$_{i}$ − R$_{ik}$λ$_{jk}$ − [ ϵ − T s − µρ − ( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) p$_{j}$ ] δ$_{ij}$, (20) v$_{ni}$ = − R − 1 ji ∂$_{t}$R$_{j}$, (23) ∂$_{t}$j$_{i}$ + ∂$_{j}$ Π$_{ij}$ = 0 , (19) D$_{n}$ Dt [ ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ )( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) ] = − s∂$_{i}$ ( ∂U$_{SS}$ ∂s ) − ∂$_{i}$R$_{j}$∂$_{k}$ ( ∂U$_{SS}$ ∂R$_{kj}$ ) − ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ )( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) ∂$_{k}$v$_{n}$$_{k}$ − ( ρδ$_{jk}$ − ρ$_{s}$$_{jk}$ )( v$_{n}$$_{k}$ − v$_{s}$$_{k}$ ) ∂$_{i}$v$_{n}$$_{j}$. (18) df = − sdT + [ µ + 1 2 ( v$_{n}$$_{i}$ − v$_{s}$$_{i}$ ) 2 ] dρ − λ$_{ik}$dR$_{ik}$ − 1 2 ( v$_{n}$$_{i}$ − v$_{s}$$_{i}$ )( v$_{n}$$_{j}$ − v$_{s}$$_{j}$ ) dρ$_{s}$$_{ij}$. (22) L SS = − ρ∂$_{t}$φ − 1 2 ρ$_{s}$$_{ij}$∂$_{i}$φ∂$_{j}$φ + 1 2 ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ ) v$_{n}$$_{i}$v$_{n}$$_{j}$ − ( ρδ$_{ij}$ − ρ$_{s}$$_{ij}$ ) v$_{n}$$_{j}$∂$_{i}$φ − f ( ρ, ρ$_{s}$$_{ij}$, T, R$_{ij}$ ) , (21) ∂$_{t}$v$_{s}$$_{i}$ = − ∂$_{i}$µ − 1 2 ∂$_{i}$v$_{s}$ $^{2}$, (16) D ( a∂$_{i}$b ) Dt = ∂$_{i}$b Da Dt + a∂$_{i}$ ( Db Dt ) − a∂$_{j}$b∂$_{i}$v$_{j}$, (17)