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Lie-Bäcklund and Noether symmetries with applications. (English) Zbl 0912.35011
Authors’ abstract: New identities relating the Euler-Lagrange, Lie-Bäcklund and Noether operators are obtained. Some important results are shown to be consequences of these fundamental identities. Furthermore, we generalize an interesting example presented by Noether in her celebrated paper and prove that any Noether symmetry is equivalent to a strict Noether symmetry, i.e. a Noether symmetry with zero divergence. We then use the symmetry based results deduced from the new identities to construct Lagrangians for partial differential equations. In particular, we show how the knowledge of a symmetry and its corresponding conservation law of a given partial differential equation can be utilized to construct a Lagrangian for the equation. Several examples are given.
##### MSC:
 35A30 Geometric theory for PDE, characteristics, transformations 58J72 Correspondences and other transformation methods (PDE on manifolds) 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.