Isaac Scientific Publishing

Advances in Analysis

AAN > Volume 3, Issue 1, January 2018

The Incomplete Exact Inverse Problem of the Calculus of Variations

Download PDF (526.9 KB) PP. 52 - 65 Pub. Date: January 4, 2018

DOI: 10.22606/aan.2018.31006

Author(s)

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

Abstract

In the common theory of the inverse problem, a system of differential equations is given and we ask whether this system is identical with the Lagrange system of an appropriate variational integral. In this article, only a small part of the Euler–Lagrange system may be prescribed in advance.The lack of information does not affect the results. The classical Helmholz solvability conditions and the Tonti resolving formula are adapted for this incomplete problem. Elementary and self–contained algorithmical approach is applied.

Keywords

Euler–Lagrange expression; divergence; Helmholz condition; exact inverse problem, differential complex. MSC 2010: 49N45

References

[1] V. Chrastinová and V. Tryhuk. On the exact inverse problem of the calculus of variations. Advances in Analysis, 2(3):196–218, 2017.

[2] Jan Chrastina. Inverse problem of the classical calculus of variations. Arch. Math. (Brno), 18(1):9–14, 1982.

[3] D. Krupka and D. Saunders, editors. Handbook of global analysis. Elsevier Science B.V., Amsterdam, 2008.

[4] D. Krupka. The Sonin-Douglas problem. In The inverse problem of the calculus of variations, Atlantis Stud. Var. Geom., chapter 2, pages 31–73. Atlantis Press, Paris, 2015.