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Local symmetries and conservation laws. (English) Zbl 0534.58005

A survey article intended for specialists in mathematical physics as an introduction to the geometric theory of nonlinear partial differential equations. Recently, a vast amount of experimental material has been accumulated in the literature on symmetries and conservation laws, and this is an attempt to bring it together on the foundation of the author’s theory, which comprises geometry of jet manifolds and infinitely prolonged differential equations equipped with an analogue of contact structure, and the calculus of the so-called \({\mathcal C}-differential\) operators (generalizing the notion of full derivatives). In Section 2, Lie’s classical theory of symmetries is described in modern terms. In Section 3, it is extended to a theory of higher symmetries. In Section 4, the notion of a conservation law is shown to have homological nature and the machinery of algebraic topology (complexes, homology, spectral sequences) is shown to be relevant. Throughout the text, ample motivations of the concepts introduced are given. Differential geometric and algebraic-topological considerations are widely used; however, the paper is written to be comprehensible to a physicist who can get useful algorithms for computation of symmetries and conservation laws, converting the former into the latter and vice versa (via generalized direct and inverse Noether’s theorems), finding invariant and partially invariant (with respect to a Lie algebra of higher symmetries) solutions of differential equations, and generating new solutions from the known ones. As an example, the Burgers’ equation is most extensively treated; some interesting remarks pertain also to the heat, Laplace and Cauchy- Riemann equations.
Reviewer: S.V.Duzhin

MSC:

58A20 Jets in global analysis
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q99 Partial differential equations of mathematical physics and other areas of application
58A12 de Rham theory in global analysis
58J70 Invariance and symmetry properties for PDEs on manifolds
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