Local symmetries and conservation laws
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Abstract
Starting with Lie's classical theory, we carefully explain the basic notions of the higher symmetries theory for arbitrary systems of partial differential equations as well as the necessary calculation procedures. Roughly speaking, we explain what analogs of ‘higher KdV equations’ are for an arbitrary system of partial differential equations and also how one can find and use them. The cohomological nature of conservation laws is shown and some basic results are exposed which allow one to calculate, in principle, all conservation laws for a given system of partial differential equations. In particular, it is shown that ‘symmetry’ and ‘conservation law’ are, in some sense, the ‘dual’ conceptions which coincides in the ‘self-dual’ case, namely, for Euler-Lagrange equations. Training examples are also given.
AMS (MOS) Subject classifications (1980)
35A30 58G05 58G35 58H05Key words
System of partial differential equations differential operator infinitely prolonged equation contact structure ofkth order (infinitesimal) symmetry transformation conservation law De Rham cohomologyPreview
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