  These notes are based on the mini-course given in June 2004 in Cetraro,
Italy, in the frame of a C.I.M.E. school. Of course, they contain much more
material that I could present in the 6 hours course. The main goal is to give
an idea of the general variational and dynamical nature of nice and powerful
concepts and results mainly known in the narrow framework of Riemannian
Geometry. This concerns Jacobi fields, Morse's index formula, Levi Civita
connection, Riemannian curvature and related topics.
  I tried to make the presentation as light as possible: gave more details in
smooth regular situations and referred to the literature in more complicated
cases.
