A description of collective degree of freedom as a dynamical variable

The method of extricated redundant variable is now applied to the Schrödinger field. The longitudinal collective mode is separated as an extricated redundant variable in such a manner that the Galilei invariance is satisfied. It is shown that the longitudinal mode satisfies a Galilei invariant non-linear equation which reduces to the usual phonon equation in linear approximation. The total system consisting of the longitudinal and the individual modes is expressed in terms of hydrogynamical variables to obtain the two-fluid model. The canonical formalism is introduced.