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

In the paper we solve the problem on the representability of a fourth order ordinary differential equation in the form of Hamilton-Ostrogradsky equations. Local bilinear forms play an essential role in the investigation of the potentiality property of the considered equation. It is well known that the problem of representing differential equations in the form of Hamilton-Ostrogradsky equations is closely related to the existence of a solution to the inverse problem of the calculus of variations, that is, for a given equation one needs to construct a functional-variational principle. To solve this problem, we first obtain necessary and sufficient conditions for the given equation to admit an indirect variational formulation relative to a local bilinear form and then construct the corresponding Hamilton-Ostrogradsky action. Note that the found conditions are analogous to the Helmholtz potentiality conditions for the given ordinary differential equation. We also define the structure of the considered equation with the potential operator and use the Ostrogradsky scheme to reduce the given equation to the form of Hamilton-Ostrogradsky equations.
It should be noted that applications and extensions of the work are the possibility to establish connections between invariance of the functional and first integrals of the given equation and to extend the proposed scheme to partial differential equations and systems of such equations.