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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.
Reviewer: W.M.Oliva (Lisboa)

MSC:

35A30 Geometric theory, characteristics, transformations in context of PDEs
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
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