System symmetries and inverse variational problems in continuum theory

Abstract

The aim of the conventional Inverse Problem in Lagrange formalism is to find a Lagrangian, the associated Euler-Lagrange equations of which are equivalent to a given set of partial differential equations of a physical system. In contrast, I am dealing with a different type of an inverse problem. I look for a Lagrangian which is associated with a given set of balance equations. My approach is based on general relations between symmetry groups (geometrical and gauge symmetries) and its associated balance equations. I follow two different mathematical lines: The first one is Noether's theorem: Universal Lie symmetry groups like translations (spatial and temporal), rotations and Galllei transformation are connected with the fundamental conservation laws for energy, linear momentum, angular momentum and center of mass motion. All of these balances are of the "volume-type". The second line takes account of a relationship between non-Lie symmetry groups (e.g. regauging of potentials) and balances of the "area- type". These are physically associated with line-shaped objects like vortex lines and dislocations. Following both lines in an inverse manner I derive the relevant symmetry properties of a yet unknown Lagrangian for a given set of balance equations of volume- and area-types. Consequently, a rough scheme for the analytical structure of the Lagrangian can be given. As an example, a Lagrangian for the elastic deformation of a body with eigenstresses due to fixed dislocations is constructed.