Eulerian and Lagrangian Phonons

Abstract

Hydrodynamic motion can be described either in the formulation of Euler or of Lagrange. Quantization leads to phonons, which in Euler’s case carry a momentum $\hbar {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{k}}$ (k = wave vector) and in Lagrange’s case have no momentum. In spite of this difference, physically observable effects must be the same since the two representations are equivalent. This conclusion is confirmed in an explicit comparison of the two cases for the microscopic derivation of the two-fluid equations in liquid He II.