  Classical Sturm non-oscillation and comparison theorems as well as the Sturm
theorem on zeros for solutions of second order differential equations have a
natural symplectic version, since they describe the rotation of a line in the
phase plane of the equation. In the higher dimensional symplectic version of
these theorems, lines are replaced by Lagrangian subspaces and intersections
with a given line are replaced by non-transversality instants with a
distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe
some properties of the Maslov index.
  Starting from the celebrated paper of Arnol'd on symplectic Sturm theory for
optical Hamiltonians, we provide a generalization of his results to general
Hamiltonians. We finally apply these results for detecting some geometrical
information about the distribution of conjugate and focal points on
semi-Riemannian manifolds and for studying the geometrical properties of the
solutions space of singular Lagrangian systems arising in Celestial Mechanics.
