Svetlana A. Budochkina     (Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya str., 117198 Moscow, Russian Federation)
Ekaterina S. Dekhanova     (Peoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya str., 117198 Moscow, Russian Federation)


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.


Inverse problem of the calculus of variations, Local bilinear form, Potential operator, Conditions of potentiality

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DOI: http://dx.doi.org/10.15826/umj.2021.1.003

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