Poisson brackets in condensed matter physics

A general method of deriving nonlinear equations of hydrodynamics for both normal liquid and superfluid ⁴He and ³He, equations of the elasticity theory, equations for spin waves in magnets and spin glasses, liquid crystals, and so on is described. The method is based on the use of the Poisson “hydrodynamic” brackets. Hydrodynamic brackets are on the one hand, a classical limit of quantum commutators, on the other hand, Poisson brackets of certain symmetry groups inherent in the given problem: groups of general coordinate transformations for hydrodynamics and elasticity theory, groups of local spin rotations for spin waves, etc. Along with well-known examples nonlinear equations of the elasticity theory for bodies with impurities, dislocations and disclinations, and equations of motion for spin glasses and multisublattice magnets are studied.