On the existence of a variational principle for an operator equation with second derivative with respect to “ time”

On a representation of an operator equation with first time derivative in the form of a B u -Hamiltonian equation

“…In the present paper, we use the notation and terminology in [3][4][5]. We will assume that the bilinear form…”

On indirect variational formulations for operator equations1

“…Consider the operator N : 1]. For N to be potential on the convex open set D(N ) with respect to Φ it is necessary and sufficient to have (6)…”

“…If a = 0, c = 0 then the equations (31) can not be represented in the form of the Euler-Lagrange equations on the set D(N ) (32) with respect to (33) because the operator [6]…”

On indirect representability of fourth order ordinary differential equation in form of Hamilton - Ostrogradsky equations

“…These issues are closely related to the inverse problem of the calculus of variations (IPCV) in the following 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. There is a large number of works devoted to inverse problems of the calculus of variations: for ordinary differential equations and partial differential equations [3], [4], [7], [9], [19], [20], [26], [27], operator equations [6], [21], [22], differential-difference equations [8], [17], [18], stochastic differential equations [23], [24], [25], fractional differential equations [1], [10], [14], [28]. In these works, nonlocal bilinear forms were mainly used to solve the IPCV.…”