A celebrated result of Poincaré asserts that a closed minimizing oriented geodesic on a Riemannian two dimensional manifold is linear unstable. In the last decades, many attempts and new results were obtained, in order to generalize this result to higher dimensional (maybe nonoriented) Riemannian manifolds. Starting from these classical results, we provide a general instability criterion for closed semi-Riemannian geodesics in terms of the so-called spectral index (being the Morse index truly infinite). As by product, we recover all the existing results on Riemannian manifold by proving new proofs.This lecture is based on joint work with Alessandro Portaluri and Ran Yang.