On the Finsler-metrizabilities of spray manifolds
Using the local coordinate apparatus, \textit{A. Rapcsák} [Publ. Math. (Debrecen) 8, 285-290 (1961; Zbl 0101.39901); Ann. Mat. Pura Appl. IV Ser. 57, (233-238 (1962; Zbl 0105.15901); Publ. Math. (Debrecen) 9, 164-167 (1962; Zbl 0129.36004)] gave necessary and sufficient conditions, expressed in second order PDEs for a Finsler space to be projectively related to a spray (path-space), and for the metrizability of a spray (path-space). In this selfcontained paper the authors first give a modern version and reformulation in terms of 0-homogeneous Hilbert 1-forms of Rapcsák's and other Finsler metrizability problems which include also the classical inverse problem of the calculus of variation, and then they reduce Rapcsák's equations to a first order PDE and an algebraic condition.
scientific article; zbMATH DE number 1822355