A description of collective degree of freedom as a dynamical variable

We propose a Hamiltonian method on an ignorable redundant variable, associated with a certain invariance. In part I, we consider elementary mechanical examples with certain invariance properties and introduce a simple procedure, which may be called “the extrication of the redundant variable”. In this method, the redundant variable can be treated as a dynamical variable. In part II, we generalize our method to physical systems with many degrees of freedom. It is demonstrated that the auxiliary variables introduced by Bohm and Pines are nothing but the extricated redundant variable. This fact suggests a general method of extracting collective modes in many-body systems. In part III, suggested by the argument in part II, we attempt at extracting hydrodynamical variables from the Schrödinger field with a self-interaction.