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

In the paper, we obtain the Birkhoffian structure (in a generalized sense) of a third-order ordinary differential equation. For this purpose, the given equation is reduced to a system of first-order ordinary differential equations, the indirect representability (in a generalized sense) of the considered system of equations in the form of Euler–Lagrange equations is investigated and the corresponding functional (Hamilton action) is constructed.