The functional formulation of second-order ordinary differential equations

Summary.

Let $D \subset \mathbb{R} \times \mathbb{R}$ be the diagonal, $F:\mathbb{R} \times (\mathbb{R} \times \mathbb{R}\backslash D) \times \mathbb{R}^{n} \times \mathbb{R}^{n} \to \mathbb{R}^{n}$ be a smooth mapping. Necessary and sufficient conditions for the existence of a differential equation $\ifmmode\expandafter\ddot\else\expandafter\"\fi{x} = f(\tau ,x,\ifmmode\expandafter\dot\else\expandafter\.\fi{x})$ with complete solutions x satisfying $F(\tau,\alpha,\beta,x(\alpha),x(\beta))=x(\tau)$ are deduced. These necessary and sufficient conditions consist of functional equations for F and of a smooth extensibility condition. Illustrative examples are presented to demonstrate this result. In these examples, the mentioned functional equations for F are related to the functional equations for geodesics, to Jensen’s equation, to the functional equations for conic sections and to Neuman’s result for linear ordinary differential equations.