Geodesic
On the potentiality of a class of operators relative to local bilinear forms
Ural mathematical journal, Tome 7 (2021) no. 1, pp. 26-37

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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.
Keywords: inverse problem of the calculus of variations, local bilinear form, potential operator, conditions of potentiality. 
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     title = {On the potentiality of a class of operators relative to local bilinear forms},
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Svetlana A. Budochkina; Ekaterina S. Dekhanova. On the potentiality of a class of operators relative to local bilinear forms. Ural mathematical journal, Tome 7 (2021) no. 1, pp. 26-37. http://geodesic.mathdoc.fr/item/UMJ_2021_7_1_a2/