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We give here a self contained and elementary introduction to theConley-Zehnder index for a path of symplectic matrices. We start from thedefinition of the index as the degree of a map into the circle for a pathstarting at the identity and ending at a matrix for which 1 is not aneigenvalue. We prove some properties which characterize this index using normalforms for symplectic matrices obtained from geometrical considerations. Weexplore the relations to Robbin-Salamon index for paths of Lagrangians. We givean axiomatic characterization of the generalization of the Conley-Zehnder indexfor any continuous path of symplectic matrices defined by Robbin and Salamon.

The Conley-Zehnder index for a path of symplectic matrices