ON THE POTENTIALITY OF A CLASS OF OPERATORS RELATIVE TO LOCAL BILINEAR FORMS1

Abstract: 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.


Introduction
In the modern calculus framework, the classical inverse problem of the calculus of variations (IPCV) is a problem of constructing an integral functional such that its equations of extremals coincide with given equations. The issues considered in the paper are closely related to the following statement of the IPCV generalizing its classical statement. For a given equation, one needs to construct a functional such that its set of stationary points coincides with the set of solutions to this equation. 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. Methods of investigating operators for the potentiality relative to local bilinear forms were developed in [6,13,20].
The main aim of the paper is to find a solution to an IPCV for a second-order ordinary differential equation. Local bilinear forms will play a significant role in the investigation.
Below, we use the notation and terminology of [2,3,13,15]. Assume that U and V are linear normed spaces over R.
The following definition and theorem will be needed for the sequel.
Definition 1 [13]. An operator N : D(N ) ⊂ U → V is called potential on the set D(N ) relative to a local bilinear form Φ(u; ·, ·) : Theorem 1 [13]. Consider a Gâteaux differentiable operator N : D(N ) ⊂ U → V and a local bilinear form Φ(u; ·, ·) : V × V → R such that, for any fixed elements u ∈ D(N ) and g, h ∈ D(N ′ u ), the function ψ(ε) = Φ(u + εh; N (u + εh), g) belongs to the class C 1 [0,1]. For N to be potential on the convex set D(N ) relative to Φ, it is necessary and sufficient to have Under this condition, the potential F N is given as where u 0 is a fixed element of D(N ).
Note that N ′ u and Φ ′ u are the Gâteaux derivatives of N and Φ at the point u.

Conditions of potentiality
Consider an ordinary differential equation of the second order Here, u = u(t) is an unknown function, a ∈ C 2 ([t 0 , t 1 ] × T ) and b, c, d ∈ C 1 ([t 0 , t 1 ] × T ) are given functions, and T ⊆ R.
is a nonlocal bilinear form and conditions (2.4) and (2.5) are represented in the form
In this case, functional (5.2) becomes Note that functional (5.3) was obtained in another way in [8].
Example 2. Consider the following equation: In this case, The operator N (5.4) is not potential on D(N ) (2.2) relative to bilinear forms (2.12) and (2.17) because c(t) = 0.
We find M = M (u(t)) such that the operator N (5.4) is potential on D(N ) (2.2) relative to a bilinear form of type (2.3).
We find M = M (u(t)) such that the operator N (5.5) is potential on D(N ) (2.2) relative to a bilinear form of type (2.3).

Conclusion
In the paper, we obtained the following results: the potentiality of the operator of a secondorder ordinary differential equation relative to a local bilinear form was investigated, a formula for constructing the functional was given, and the structure of the corresponding Euler-Lagrange equation was defined. In particular, applications and extensions of the work consist in the possibility to establish connections between the invariance of the functional, the given equation, and its first integrals and to spread the proposed scheme of investigation to higher-order equations.