On the hydrodynamic canonical formalism of the Gross-Pitaevskii field

We derive a canonical formalism for the hydrodynamic representation of the Gross-Pitaevskii field (nonlinear Schr"odinger field), where the density and the phase of the condensate form a canonical pair of conjugate field variables. To do so, we treat the meanfield as a singular Lagrangian system and apply both the Dirac-Bergmann and Faddeev-Jackiw methods. The Faddeev-Jackiw method is found to be a more direct approach to the problem.