The search for a potential function S allowing us to reconstruct a given metric tensor g and a given symmetric covariant tensor T on a manifold $\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with a canonically defined Lagrangian on $T\mathcal{M}$ . The connection between this problem, the geometric structure of the space of pure states of quantum mechanics, and the theory of contrast functions of classical information geometry are outlined.