On the Correspondence between the Variational Principles in the Eulerian and Lagrangian Descriptions

The relationship between the variational principles for equations of continuum mechanics in Eulerian and Lagrangian descriptions is considered. It is shown that, for a system of differential equations in Eulerian variables, the corresponding Lagrangian description is related to introducing nonlocal variables. The connection between the descriptions is obtained in terms of differential coverings. The relation between the variational principles of a system of equations and its symplectic structures is discussed. It is shown that, if a system of equations in Lagrangian variables can be derived from a variational principle, then there is no corresponding variational principle in the Eulerian variables.