Second Noether theorem for quasi-Noether systems

TL;DR

This paper extends the Second Noether Theorem to quasi-Noether systems with infinite symmetries, showing many conservation laws are trivial and analyzing examples like Euler and Navier-Stokes equations.

Contribution

It provides a theoretical extension of the Second Noether Theorem to quasi-Noether systems with infinite symmetries involving arbitrary functions.

Findings

Infinite symmetries with arbitrary functions lead to trivial conservation laws.

The associated differential systems are under-determined.

Examples include Euler and Navier-Stokes equations with trivial conservation laws.

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.