Isaac Scientific Publishing

Advances in Analysis

On the Exact Inverse Problem of the Calculus of Variations

Download PDF (690.7 KB) PP. 196 - 218 Pub. Date: May 3, 2017

DOI: 10.22606/aan.2017.23005

Author(s)

  • Veronika Chrastinová*
    Brno University of Technology, Faculty of Civil Engineering, Department of Mathematics, Veverí 331/95, 602 00 Brno, Czech Republic
  • Václav Tryhuk*

    Brno University of Technology, Faculty of Civil Engineering, AdMaS centre, Purkynova 139, 612 00 Brno, Czech Republic

Abstract

The article is devoted to the problem of whether or not a given system of differential equations is identical with the Euler–Lagrange system of an appropriate variational integral. The actual theories which rest on the Helmholz solvability condition and the local Tonti formula are revised. Quite elementary approach is applied. Then the Helmholz condition turns into an easy matter together with unexpected consequence, the solution of incomplete inverse problem. Since the Tonti formula does not give the economical solution, new direct and even global approach is proposed for the determination of all first–order variational integrals related to the second–order Euler–Lagrange system. It employs the fibered de Rham theory where the multiple–valued (ramified) solutions are included as well. The article is of a certain interest also for nonspecialists.

Keywords

Euler–Lagrange expression, divergence, Helmholz condition, exact inverse problem, de Rham theory.

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