Jump to ContentJump to Main Navigation
Show Summary Details
More options …

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. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

4 Issues per year


IMPACT FACTOR 2016: 0.552

CiteScore 2016: 0.61

SCImago Journal Rank (SJR) 2016: 0.564
Source Normalized Impact per Paper (SNIP) 2016: 1.021

Mathematical Citation Quotient (MCQ) 2016: 0.45

Online
ISSN
1615-7168
See all formats and pricing
More options …
Volume 18, Issue 2

Issues

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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Sandra Germani / Andrea Spiro
  • Corresponding author
  • Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri 9, 62032 Camerino (Macerata), Italy
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2018-03-26 | DOI: https://doi.org/10.1515/advgeom-2018-0004

Abstract

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

References

  • [1]

    S. C. Anco, J. Pohjanpelto, Classification of local conservation laws of Maxwell’s equations. Acta Appl. Math. 69 (2001), 285–327. MR1885280 Zbl 0989.35126CrossrefGoogle Scholar

  • [2]

    I. M. Anderson, The variational bicomplex. Utah State University preprint (1989), http://math.uni.lu/~jubin/seminar/bicomplex.pdf.

  • [3]

    A. V. Bocharov, V. N. Chetverikov, S. V. Duzhin, N. G. Khor’ kova, I. S. Krasil’ shchik, A. V. Samokhin, Y. N. Torkhov, A. M. Verbovetsky, A. M. Vinogradov, Symmetries and conservation laws for differential equations of mathematical physics, volume 182 of Translations of Mathematical Monographs. Amer. Math. Soc. 1999. MR1670044 Zbl 0911.00032Google Scholar

  • [4]

    E. Fiorani, A. Spiro, Lie algebras of conservation laws of variational ordinary differential equations. J. Geom. Phys. 88 (2015), 56–75. MR3293396 Zbl 1308.70050CrossrefWeb of ScienceGoogle Scholar

  • [5]

    S. Germani, Leggi di conservazione e simmetrie: un approccio geometrico al Teorema di Noether. Tesi di Laurea Magistrale, Università di Camerino, Camerino, 2012.Google Scholar

  • [6]

    M. Gromov, Partial differential relations. Springer 1986. MR864505 Zbl 0651.53001Google Scholar

  • [7]

    D. Joyce, On manifolds with corners. In: Advances in geometric analysis, volume 21 of Adv. Lect. Math. (ALM), 225–258, Int. Press, Somerville, MA 2012. MR3077259 Zbl 1317.58001Google Scholar

  • [8]

    Y. Kosmann-Schwarzbach, The Noether theorems. Springer 2011. MR2761345 Zbl 1216.01011Google Scholar

  • [9]

    I. S. Krasil’ shchik, V. V. Lychagin, A. M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, volume 1 of Advanced Studies in Contemporary Mathematics. Gordon and Breach Science Publishers, New York 1986. MR861121 Zbl 0722.35001Google Scholar

  • [10]

    B. A. Kupershmidt, Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms. In: Geometric methods in mathematical physics (Proc. NSF-CBMS Conf., Univ. Lowell, Lowell, Mass., 1979), volume 775 of Lecture Notes in Math., 162–218, Springer 1980. MR569303 Zbl 0439.58016Google Scholar

  • [11]

    A. Kushner, V. Lychagin, V. Rubtsov, Contact geometry and non-linear differential equations, volume 101 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 2007. MR2352610 Zbl 1122.53044Google Scholar

  • [12]

    J. M. Lee, Introduction to smooth manifolds. Springer 2013. MR2954043 Zbl 1258.53002Google Scholar

  • [13]

    V. V. Lyčagin, Contact geometry and second-order nonlinear differential equations (Russian). Uspekhi Mat. Nauk 34 (1979), no. 1(205), 137–165. MR525652 Zbl 0427.58002Google Scholar

  • [14]

    J. Milnor, Morse theory. Princeton Univ. Press 1963. MR0163331 Zbl 0108.10401Google Scholar

  • [15]

    E. Noether, Invariante Variationsprobleme. Nachr. Königl. Ges. Wissensch. Göttingen, Math.-Phys. Klasse (1918), 235–257. Zbl 46.0770.01Google Scholar

  • [16]

    P. J. Olver, Applications of Lie groups to differential equations. Springer 1986. MR836734 Zbl 0588.22001Google Scholar

  • [17]

    P. J. Olver, Noether’s theorems and systems of Cauchy–Kovalevskaya type. In: Nonlinear systems of partial differential equations in applied mathematics, Part 2 (Santa Fe, N.M., 1984), volume 23 of Lectures in Appl. Math., 81–104, Amer. Math. Soc. 1986. MR837699 Zbl 0656.58039Google Scholar

  • [18]

    P. J. Olver, Book review of [8], Bull. Amer. Math. Soc. (N.S.) 50 (2013), 161–167. MR2994999Google Scholar

  • [19]

    A. Spiro, Cohomology of Lagrange complexes invariant under pseudogroups of local transformations. Int. J. Geom. Methods Mod. Phys. 4 (2007), 669–705. MR2343431 Zbl 1153.70009CrossrefWeb of ScienceGoogle Scholar

  • [20]

    F. Takens, A global version of the inverse problem of the calculus of variations. J. Differential Geom. 14 (1979), 543–562 (1981). MR600611 Zbl 0463.58015CrossrefGoogle Scholar

  • [21]

    K. Yamaguchi, Contact geometry of higher order. Japan. J. Math. (N.S.) 8 (1982), 109–176. MR722524 Zbl 0548.58002CrossrefGoogle Scholar

  • [22]

    K. Yamaguchi, Geometrization of jet bundles. Hokkaido Math. J. 12 (1983), 27–40. MR689254 Zbl 0561.58002CrossrefGoogle Scholar

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.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston. Copyright Clearance Center

Comments (0)

Please log in or register to comment.
Log in