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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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Volume 18, Issue 2


Lie algebras of conservation laws of variational partial differential equations

Emanuele Fiorani
  • Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino (Macerata), Italy
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/ Sandra Germani / Andrea Spiro
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  • Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino (Macerata), Italy
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Published Online: 2018-03-26 | DOI: https://doi.org/10.1515/advgeom-2018-0004


We establish a version of Noether’s first Theorem according to which the (equivalence classes of) conserved quantities of given Euler–Lagrange equations in several independent variables are in one-to-one correspondence with the (equivalence classes of) vector fields satisfying an appropriate pair of geometric conditions, namely: (a) they preserve the class of vector fields tangent to holonomic submanifolds of a jet space; (b) they leave invariant the action from which the Euler–Lagrange equations are derived, modulo terms identically vanishing along holonomic submanifolds. Such a bijective correspondence Φ͠ between equivalence classes comes from an explicit (non-bijective) linear map Φ from vector fields into conserved differential operators, and not from a map into divergences of conserved operators as it occurs in other proofs of Noether’s Theorem. Where possible, claims are given a coordinate-free formulation and all proofs rely just on basic differential geometric properties of finite-dimensional manifolds.

Keywords: Generalized infinitesimal symmetry; Noether’s first theorem; Poincaré–Cartan form

MSC 2010: 70S05; 70S10; 70G65


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About the article

Received: 2016-04-21

Published Online: 2018-03-26

Published in Print: 2018-04-25

Funding: This research was partially supported by the Project MIUR “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis” and by GNSAGA of INdAM.

Citation Information: Advances in Geometry, Volume 18, Issue 2, Pages 207–228, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0004.

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