Second Noether theorem for quasi-Noether systems

Abstract

Quasi-Noether differential systems are more general than variational systems and are quite common in mathematical physics. They include practically all differential systems of interest, at least those that have conservation laws. In this paper, we discuss quasi-Noether systems that possess infinite-dimensional (infinite) symmetries involving arbitrary functions of independent variables. For quasi-Noether systems admitting infinite symmetries with arbitrary functions of all independent variables, we state and prove an extension of the Second Noether Theorem. In addition, we prove that infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial and that the associated differential system is under-determined. We discuss infinite symmetries and infinite conservation laws of two important examples of non-variational quasi-Noether systems: the incompressible Euler equations and the Navier-Stokes equations in vorticity formulation, and we show that the infinite sets of conservation laws involving arbitrary functions of all independent variables are trivial. We also analyze infinite symmetries involving arbitrary functions of not all independent variables, prove that the fluxes of conservation laws in these cases are total divergences on solutions, and demonstrate examples of this situation.

Sub-Symmetries and Conservation Laws

“…In [25] and [26], an approach based on the Noether operator identity (2.8) was suggested to relate symmetries to conservation laws for a large class of differential systems that may not have well-defined Lagrangian functions; in [35] these systems were called quasi-Noether.…”

“…for any differential system of class (2.14). Thus, the condition (2.14) can be considered as defining quasi-Noether systems, see also [35]. A system (5.1) is quasi-Noether if there exist functions (differential operators) β a such that the condition (2.14) is satisfied.…”

“…In the recent paper [35], we studied differential systems that may not have well-defined Lagrangian functions (quasi-Noether systems), but possess infinite symmetries involving arbitrary functions of all independent variables. It was shown that the approach based on the Noether operator identity allows for the derivation of an extension of the Second Noether theorem for non-Lagrangian differential systems, [35].…”

Quasi-Noether Systems and Quasi-Lagrangians

“…The condition (3.3) can be considered as defining quasi-Noether systems, see also [16]. A system (3.1) is quasi-Noether if there exist functions (differential operators) β a such that the condition (3.3) is satisfied.…”

On the triviality of potential vorticity conservation in geophysical fluid dynamics

“…For instance, Müller (1995) demonstrated that the evolution of the potential vorticity charge density ρ q is governed by a conservation equation which is a mathematical identity, although he still used Noether's first theorem to associate potential vorticity conservation with particle-relabelling. Rosenhaus and Shankar (2016) also analyzed the triviality of potential vorticity conservation in the context of incompressible flows.…”