Application of Noether's theorem to the transport equation

Abstract

Abstract The Euler-Lagrange equations are derived for a variational integral whose integrand is a function of integral as well as differential operations on the field functions. An analog of Noether's theorem is then derived for the same integral. Using the derived Euler-Lagrange equations, a Lagrangian density is constructed which yields the single speed, one dimensional transport equation and its adjoint. The Lagrangian density is found to be invariant under a gauge group and Noether's theorem is then used to derive a conserved current as a consequence of this invariance.