Generalization of Noether's theorem in modern form to non-variational partial differential equations

Abstract

A general method using multipliers for finding the conserved integrals for any system of partial differential equations (PDEs) is reviewed and further developed in several ways. Multipliers are expressions whose (summed) product with a PDE (system) yields a local divergence identity which has the physical meaning of a continuity equation involving a conserved density and a spatial flux for solutions of the PDE (system). On spatial domains, the integral form of a continuity equation yields a conserved integral. When a PDE (system) is variational, multipliers correspond to symmetries of the variational principle, and the local divergence identity relating a multiplier to a conserved integral is the same as the variational identity used in Noether's theorem for connecting conserved integrals to invariance of a variational principle. From this viewpoint, the multiplier method is shown to constitute a modern form of Noether's theorem in which the variational principle is not used. When a PDE (system) is non-variational, multipliers are shown to be an adjoint counterpart to symmetries, and the local divergence identity relating a multiplier to a conserved integral is shown to be an adjoint generalization of the variational identity underling Noether's theorem. Two main results are established for PDE systems having a solved-form for leading derivatives, which encompasses all typical PDEs of physical interest. First, all non-trivial conserved integrals are shown to arise from non-trivial multipliers in a one-to-one manner, taking into account certain equivalence freedoms. Second, a scaling formula based on dimensional analysis is derived to obtain the conserved density and spatial flux in any conserved integral, just using the corresponding multiplier and the given PDE system. The derivations use a few basic tools from variational calculus, for which a self-contained formulation is provided.