Identities from infinite-dimensional symmetries of Herglotz variational functional

Abstract

This paper formulates and proves a theorem which provides the identities corresponding to infinite-dimensional symmetry groups of the functional defined by the generalized variational principle of Herglotz in the case of several independent variables. It contains the classical second Noether theorem as a special case. The equations satisfied by the extrema of this functional, when the Lagrangian density depends on first and second order partial derivatives of the argument functions, are found. We apply the theorem to find two new identities satisfied by the four-potential of the electromagnetic field propagating in a conductive medium. We also obtain an identity corresponding to a gauge symmetry of the Klein-Gordon equation with dissipation/generation. This identity becomes a continuity law when the wave function is a solution of this equation.