On the inverse variational problem for one class of quasilinear equations

Tunitsky, Dmitry Vasilievich

doi:10.1016/j.geomphys.2019.103568

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2021

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“…These problems are also related to the mechanics of finite-and infinite-dimensional systems [7,8,[11][12][13]. There is a large number of works devoted to IPCVs for different types of equations and their systems: in particular, for ordinary differential equations and differential equations with partial derivatives [4,6,13,18,19,21], operator equations [2,3,14,15], differentialdifference equations [5,9,10], and stochastic differential equations [16,17]. In these works, nonlocal bilinear forms were mainly used to solve an IPCV.…”

Section: Introductionmentioning

confidence: 99%

On the Potentiality of a Class of Operators Relative to Local Bilinear Forms

Budochkina

Dekhanova

2021

UMJ

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The inverse problem of the calculus of variations (IPCV) is solved for a second-order ordinary differential equation with the use of a local bilinear form. We apply methods of analytical dynamics, nonlinear functional analysis, and modern methods for solving the IPCV. In the paper, we obtain necessary and sufficient conditions for a given operator to be potential relative to a local bilinear form, construct the corresponding functional, i.e., found a solution to the IPCV, and define the structure of the considered equation with the potential operator. As a consequence, similar results are obtained when using a nonlocal bilinear form. Theoretical results are illustrated with some examples.

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