Principle of stationary action in the theory of superfluid systems with spontaneously broken translational symmetry
The paper is devoted to the construction of a Lagrangian describing the low-frequency dynamics of a system with broken phase and translational symmetries using some concepts of the microscopic theory. In order to obtain the equations of an ideal hydrodynamics when dissipative processes can be neglected the approach is based on the principle of local statistical equilibrium. This assumes that all thermodynamic quantities are slowly varying functions of the coordinates and time. The principle is used to construct the Lagrangian and find the hydrodynamic equations. The local statistical equilibrium of the system with broken phase and translational symmetries of the equilibrium state is described by the densities of the additive integrals of motion and by additional local thermodynamic variables related to the broken symmetries. The choice of the dynamical variables plays an important role in constructing of the Lagrangian and the corresponding principle of stationary action. In addition to the dynamical variables determining the hydrodynamic fields, the authors introduce cyclic variables corresponding to certain nonphysical fields on which the Hamiltonian (the thermodynamic energy) of the system is independent. The hydrodynamic variables themselves are defined by basic nonequilibrium thermodynamics relation. The kinematic part of the Lagrangian states the Poisson brackets of the system while the Hamiltonian describes its dynamics. In order to obtain the hydrodynamic equations of motion, there are used the Euler-Lagrange equations to being a consequence of the principle of stationary action. After obtainment of the motion equations the Gibbs thermodynamic potential is introduced which is proper to study the relativistic invariance of condensed matter states. The relativistic-invariant hydrodynamic equations are formulated by using the motion equations in which the density of the additive integrals of motion and their fluxes are expressed in terms of the Gibbs thermodynamic potential density. Due to there is no requirement on that the thermodynamic potential density be relativistically invariant, the equations obtained formally hold for a system with any dynamical symmetry. The equations obtained present a closed system of relativistic-invariant hydrodynamic equations for a superfluid with a crystalline structure and are generalized to the case of a gravitational field. Finally, the authors consider the relativistic equations of elasticity and the relativistic hydrodynamic equations for a superfluid and for a normal fluid.
scientific article; zbMATH DE number 5626881