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The authors review some recent results concerning a class of differential equations, the so-called symplectic differential systems. They are linear systems in R n ⊕ R n ∗ whose flow preserves the canonical symplectic form and which appear naturally in many areas of mathematics and physics. For instance, in the study of the Jacobi equation along a semi-Riemannian geodesic we obtain special cases of symplectic systems. Some index theorems are presented, that is, results relating the following objects: the conjugate (or focal) points of the system, the index or the co-index of the index form associated to the system, and the spectrum of the second-order linear differential operator associated to the system. In the case of symplectic systems whose initial data belong to a fixed Lagrangian subspace of R n ⊕ (R n) ∗ (as in the case of Jacobi fields in semi-Riemannian geometry) it is possible to define the Maslov index of the system. The notions of spectral index and index form are discussed and two main results are presented, giving equalities of the Maslov index respectively with the spectral index of the system and with the difference between the inde

Year: 2013

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oai:CiteSeerX.psu:10.1.1.363.2492

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