Let $X$ a smooth finite-dimensional manifold and $W_\Gamma(X)$ the manifold of geodesic arcs of a symmetric linear connection $\Gamma$ on $X$. In a previous paper [Differential Geometry and Applications (Brno, 1995) 603-610 (1996; Zbl 0859.58011)] the author introduces and studies the Poisson manifolds of geodesic arcs, i.e. manifolds of geodesic arcs equipped with certain Poisson structure. In this paper the author obtains necessary and sufficient conditions for that a given Lagrange function generates a Poisson manifold of geodesic arcs. These conditions are formulated in terms of tangent Frobenius algebras.