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Helmholtz conditions and their generalizations

The result's identifiers

  • Result code in IS VaVaI

    <a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61988987%3A17310%2F10%3AA13016BX" target="_blank" >RIV/61988987:17310/10:A13016BX - isvavai.cz</a>

  • Result on the web

  • DOI - Digital Object Identifier

Alternative languages

  • Result language

    angličtina

  • Original language name

    Helmholtz conditions and their generalizations

  • Original language description

    The Euler-Lagrange morphism E1 is well-understood. It is also known that the kernel of the Helmholtz morphism E2 consists of locally variational dynamical forms, and is characterized by Helmholtz conditions. We study the image of E2 and the kernel of thenext morphism E3, and solve the corresponding local and global inverse problem when a three-form comes from a dynamical form, i.e., corresponds to a system of diferential equations. We find identities, that are a generalization of the Helmholtz conditions to this situation, and show that the problem is closely related to the question on existence of a closed three-form. The obtained results extend known results on Lagrangians and locally variational dynamical forms to general dynamical forms, and opena new possibility to study non-variational equations by means of closed three-forms, as a parallel to extremal problems (variational equations) that are studied by means of closed two-forms (Cartan forms, symplectic geometry).

  • Czech name

  • Czech description

Classification

  • Type

    J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)

  • CEP classification

    BA - General mathematics

  • OECD FORD branch

Result continuities

  • Project

  • Continuities

    S - Specificky vyzkum na vysokych skolach

Others

  • Publication year

    2010

  • Confidentiality

    S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Data specific for result type

  • Name of the periodical

    Balkan Journal of Geometry and Its Applications

  • ISSN

    1224-2780

  • e-ISSN

  • Volume of the periodical

    15

  • Issue of the periodical within the volume

    1

  • Country of publishing house

    RO - ROMANIA

  • Number of pages

    10

  • Pages from-to

    80-89

  • UT code for WoS article

  • EID of the result in the Scopus database