Metrizability of 1-form projective deformation of sprays

In this paper, we consider a special case of the inverse problem of calculus of variations. For a given spray $S$ on a manifold $M$, we investigate the projective deformation of $S$ by a 1-form $\beta \in {\mathrm{\Lambda }}^1(M)$ on $M$, precisely, $S_{{}_{\beta }}=S-2\beta \mathcal{C}$. We show that, in general, the spray $S_{{}_{\beta }}$ is not Finsler metrizable. Moreover, the metrizability of $S_{{}_{\beta }}$, when the background spray is flat, is characterized. In this case, we establish an explicit formula for the Finsler function whose geodesic spray is $S_{{}_{\beta }}$. We conclude that this metric is a projectively flat metric of nonzero constant flag curvature; that is, the obtained metric is a solution for Hilbert's fourth problem.