
SIGMA 1 (2005), 011, 16 pages mathph/0511035
https://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 nonlocally 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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