  Perturbed geodesics are trajectories of particles moving on a semi-Riemannian
manifold in the presence of a potential. Our purpose here is to extend to
perturbed geodesics on semi-Riemannian manifolds the well known Morse Index
Theorem. When the metric is indefinite, the Morse index of the energy
functional becomes infinite and hence, in order to obtain a meaningful
statement, we substitute the Morse index by its relative form, given by the
spectral flow of an associated family of index forms. We also introduce a new
counting for conjugate points, which need not to be isolated in this context,
and prove that the relative Morse index equals the total number of conjugate
points. Finally we study the relation with the Maslov index of the flow induced
on the Lagrangian Grassmannian.
