Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 1 (2005), 011, 16 pages      math-ph/0511035      http://dx.doi.org/10.3842/SIGMA.2005.011

Connections Between Symmetries and Conservation Laws

George Bluman
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T1Z2

Received July 29, 2005, in final form October 22, 2005; Published online October 26, 2005

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 words: conservation laws; linearization; nonlocal symmetries; Noether's theorem.

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