Lecture of Bence Racskó

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Published on 2022. máj. 31. kedd, 12:15

The inverse problem to the calculus of variations

The Department of Geometry is pleased to announce that

Bence Racskó

(University of Szeged, Hungary)

gives a talk at the Kerékjártó Seminar with the title

Date and place:

Abstract:
The inverse problem to the calculus of variations is concerned with determining which differential equations are variational, i.e. arise as the Euler-Lagrange equations associated to a Lagrangian, and to specify all such Lagrangians. We distinguish the weak and strong inverse problems. The weak inverse problem involves differential equations in concrete form, while the aim of the strong inverse problem is to characterize those equations which may be transformed into variational differential equations by the means of some appropriate equivalence transformation. The inverse problem originates in the 19th century works of Helmholtz and Sonin and the local aspects of the weak problem have been solved in the mid-20th century by Vainberg and Tonti by methods of potential theory. Uncovering the global aspects of the weak inverse problem necessitated the introduction of novel mathematical methods - such as the variational bicomplex, the C spectral sequence or finite order variational sequences in the late 70s and early 80s. These are differential complexes defined on jet bundles over fibered manifolds, whose elements correspond to objects from the classical calculus of variations and their differentials to usual variational operations such as total divergences, the Euler-Lagrange mapping or the Helmholtz operator. Through the mentioned techniques, it becomes possible to study the global aspects of the inverse problem. Cohomologies of the complexes correspond to global topological obstructions to the solution of the inverse problem, and their local homotopy operators provide systematic, computable solutions to the local inverse problem. The strong inverse problem does not have a general solution, even to this day, however certain special subcases, especially those related to ordinary differential equations are amenable to solution. The purpose of this talk is to articulate on the solution of the weak inverse problem as well as the mathematical background necessary for it, and to overview a number of important results related to the strong inverse problem.

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Tájékoztatás:
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A Polgári Törvénykönyvről szóló 2013. évi V. törvény 2:48. § (2) bekezdése alapján a tömegfelvételek, valamint a nyilvános közéleti szereplés esetén nincs szükség a résztvevők hozzájárulására sem a felvétel elkészítéséhez, sem annak felhasználásához, de az érintetteket erről előzetesen tájékoztatni kell.
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