Index theorems for symplectic systems

The Jacobi equation along a geodesic gamma in a semi-Riemannian manifold (M -n,g) produces, by a parallel trivialization of the tangent bundle TM along gamma, a Morse-Sturm equation in R-n. More generally, the linearized Hamil ton equation along a solution Gamma of a Hamiltonian vector field H-over-ar row in a symplectic manifold (M-2n, omega) produces a first order linear di fferential system in R-n circle plus R-n* whose flow preserves the canonica l symplectic form; such systems are called symplectic differential systems. By "index theorems" for symplectic. systems we mean those results that rel ate two or more of the following objects: (1) the conjugate (or focal) poin ts of the system, (2) the index or the co-index. of (suitable restrictions of) the so called index form associated to the system, (3) the spectrum of the second order linear differential operator associated to the system. In this paper we present a collection of index theorems that were proven recen tly (References [5], [6], [8], [9], [10], [11]).