On the inverse problem of the calculus of variations for a class of coupled dynamical systems

The inverse problem of the calculus of variations for a class of coupled dynamical systems-the so-called driven second-order differential equations-are analysed. A family of necessary and sufficient conditions are found that assure the existence of a local Lagrangian function for a given driven second-order differential equation. These conditions are stated geometrically and some of their consequences and examples are discussed.