Connections Between Symmetries and Conservation Laws

TL;DR

This paper explores the relationship between symmetries and conservation laws in differential equations, introducing a direct method to find conservation laws and analyzing their implications for system linearization.

Contribution

It presents the Direct Construction Method for deriving conservation laws directly from differential equations, expanding beyond Noether's theorem limitations.

Findings

The method provides multipliers and integral formulas for conserved densities.

Symmetries acting on conservation laws can generate new laws.

Conservation laws help determine linearizability of systems.

Abstract

This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for any given system of differential equations. This method yields the multipliers for conservation laws as well as an integral formula for corresponding conserved densities. The action of a symmetry (discrete or continuous) on a conservation law yields conservation laws. Conservation laws yield non-locally related systems that, in turn, can yield nonlocal symmetries and in addition be useful for the application of other mathematical methods. From its admitted symmetries or multipliers for conservation laws, one can determine whether or not a given system of differential equations can be linearized by an invertible transformation.