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2005•Symmetry Integrability and Geometry Methods and ApplicationsOpen access

Connections Between Symmetries and Conservation Laws

George W. Bluman

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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.

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What this paper is about

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.

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Available 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.

Key concepts: Noether's theorem, Conservation law, Homogeneous space, Mathematics, Symmetry (geometry), Invertible matrix, Law, Transformation (genetics)

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