The crooked road to effective stress

Abstract

Two Lagrangian functions are said to be variationally equivalent if they differ by a null Lagrangian (a Lagrangian whose associated Euler-Lagrange equations are identically satisfied). Kibble [1] noted in his seminal paper of 1961 that variationally equivalent Lagrangians lead to inequivalent gauge field theories, after which this important observation was actively ignored. There is an understandable reason for this situation; variationally equivalent Lagrangian functions are distinguished only by their distinct natural Neumann data, while elementary particle physics rarely if ever considers problems with imposed Neumann data. On the other hand, problems with imposed Neumann data demand inclusion of appropriate null Lagrangians in order that the imposed data be made variationally natural. It is thus clear that gauge theories with imposed Neumann data must make due allowances for the gauge-theoretic inequivalence of variationally equivalent Lagrangians. A specific case in point is the gauge theory of materials with defects that are subjected to imposed tractions on their boundaries.