On the correspondence between variational principles in Eulerian and Lagrangian descriptions

TL;DR

This paper explores the relationship between variational principles in Eulerian and Lagrangian descriptions of continuum mechanics, revealing that the Lagrangian system relates to nonlocal variables and that variational principles may not always transfer between descriptions.

Contribution

It establishes a connection between Eulerian and Lagrangian variational principles using differential coverings and discusses the implications for symplectic structures.

Findings

Lagrangian description involves nonlocal variables.

No direct variational principle transfer from Lagrangian to Eulerian systems.

Differential coverings link the two descriptions.

Abstract

A relation between 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 corresponding Lagrangian description is related to introducing nonlocal variables. The connection between these descriptions is obtained in terms of differential coverings. The relation between 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 Eulerian variables.