Variational approach to supersolid hydrodynamics

On the basis of the consistent unification of the principles of non-equilibrium thermodynamics and classical mechanics, we construct a Lagrangian, describing the Hamiltonian dynamics of supersolids. This Lagrangian enables to obtain the closed algebra of the Poisson brackets for the local thermodynamic variables and to derive the Hamilton equations of motion. These equations, in the leading order in spatial gradients of the local thermodynamic variables, result in the non-dissipative supersolid hydrodynamics. The derived hydrodynamic equations do not assume any dynamic symmetry associated with Galilean or Lorentz invariance. The constraint on the thermodynamic potential related to Galilean invariance leads to Andreev-Lifshitz hydrodynamics. We also require relativistic invariance of a constructed theory and obtain a relativistically-invariant supersolid hydrodynamics. (author)