I'm trying to solve the following problem without success.
Let $V$ be a smooth function on a pseudo-Riemannian manifold $(M, g)$, which is either bounded from above or from below. Show that there exists a function f related to V and a pseudo Riemannian metric: $$g_N=g+fdu^2,\; N=M\times\mathbb{R}$$ Where $u$ is the coordinate on the $\mathbb{R}$-factor, such that the solutions of the Lagrangian system defined by $\mathcal{L}(v)=\frac{1}{2}g(v,v)-V(\pi v)$ and $v\in TM$, correspond precisely to the geodesics on the manifold $(N, g_N)$.
From my point of view the system should be decoupled because the metric is split in two terms. Probably I totally misunderstand the concept of geodesic equations.