Line-shaped objects and their balances related to gauge symmetries in continuum theories

摘要

Within the Lagrange formalism Noether's theorem is a well-known tool for connecting symmetries of a physical system with homogeneous balance equations. In the context of this paper we call them balance equations of the volume type. They are associated with symmetry groups of the Lie type. However, in physics there are a lot of different balance equations which we call balance equations of the area type. Physically, they are associated with the dynamics of line-shaped objects. In this paper a general theory is presented which supplements Noether's theorem in so far as the area-type balances are associated with regauging symmetry groups of the non-Lie type. The theory is demonstrated for three prominent examples: for the Helmholtz laws of the vortex dynamics in an ideal fluid, for the dislocation dynamics in the dynamical eigenstress problem of elastic crystals, and for the homogeneous Maxwell equations. The theory will also be a valuable tool for solving inverse variational problems, i.e. to construct Lagrangians for physical systems. [References: 16]