Given a one-parameter family $\{g_{\lambda }:\lambda \in [a,b]\}$ of semi Riemannian metrics on an n-dimensional manifold M, a family of time-dependent potentials $\{V_{\lambda }:\lambda \in [a,b]\}$ and a family $\{{\sigma }_{\lambda }:\lambda \in [a,b]\}$ of trajectories connecting two points of the mechanical system defined by $(g_{\lambda },V_{\lambda })$, we show that there are trajectories bifurcating from the trivial branch ${\sigma }_{\lambda }$ if the generalized Morse indices $\mu \left({\sigma }_a\right)$ and $\mu \left({\sigma }_b\right)$ are different. If the data are analytic we obtain estimates for the number of bifurcation points on the branch and, in particular, for the number of strictly conjugate...