The connection between variational principles in Eulerian and Lagrangian descriptions

The question of whether there exists a connection between variational principles in Eulerian and Lagrangian descriptions is investigated. By having recourse to a proper view of the Eulerian description it is shown that a variational principle in one description holds whenever a corresponding variational principle in the other description is given. This theoretical conclusion is operative in that a precise rule for writing the new Lagrangian is exhibited. As an application, a new Lagrangian for fluid dynamics in the Eulerian description is determined.