Clebsch parameterization from the symplectic point of view

Abstract

This Letter proposes an alternative systematic way to obtain a canonical Lagrangian formulation for rotational systems. This will be done in the symplectic framework and with the introduction of extra variables which enlarge the phase space. In fact, this formalism provides a remarkable and new result to compute the canonical Lagrangian formulation for rotational systems, i.e., the obstruction to the construction of a canonical formalism is solved in a straight and systematic way. In order to illustrate this systematic formalism, which is not dimension dependent, the 3-dimensional rotational fluid dynamics, whose vorticity is nonvanishing, is chosen and the usual result obtained by using Clebsch parameterization is reproduced. Further, we also argue that this new formalism might provide a set of dynamically equivalent Lagrangian descriptions for rotational systems, which can be related by canonical transformations.