On the inverse problem of the variational calculus: Existence of Lagrangians associated with a spray in the isotropic case

Using the Spencer-Goldschmidt version of the Cartan-Kahler theorem, we study necessary and sufficient conditions for the (local) existence of a regular Lagrangian associated with a real-analytic system of second order ordinary differential equations. In Muzsnay's thesis this technique was applied to give a modern treatment of the 5-dimensional case, first studied in the classic paper of Douglas. In this paper we consider the case of arbitrary dimension but we restrict ourselves to isotropic systems. Here isotropic means that the sectional curvature, which we define for a general Lagrangian (not necessarily homogeneous), depends only on the tangent vector and is independent of the 2-plane containing the vector. In particular, in the homogeneous case, we characterize the connections which come from a Finsler structure with isotropic curvature.