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 stud y necessary and sufficient conditions for the (local) existence of a regula r Lagrangian associated with a real-analytic system of second order ordinar y differential equations. In Muzsnay's thesis this technique was applied to give a modern treatment of the 5-dimensional case, first studied in the cl assic paper of Douglas. In this paper we consider the case of arbitrary dim ension but we restrict ourselves to isotropic systems. Here isotropic means that the sectional curvature, which we define for a general Lagrangian (no t necessarily homogeneous), depends only on the tangent vector and is indep endent of the 2-plane containing the vector. In particular, in the homogene ous case, we characterize the connections which come from a Finsler structu re with isotropic curvature.