An Indirect Variational Formulation of a Third-order Ordinary Differential Equation and Birkhoff’s Equations

Abstract: In the paper, we obtain the Birkhoffian structure (in ageneralized sense) of a third-order ordinary differentialequation. For this purpose, the given equation is reduced to asystem of first-order ordinary differential equations, theindirect representability (in a generalized sense) of theconsidered system of equations in the form of Euler–Lagrangeequations is investigated and the corresponding functional(Hamilton action) is constructed. © Pleiades Publishing, Ltd. 2024.