ABSTRACT
In this chapter, we will study the properties of an important invariant of linear periodic Hamiltonian systems called Maslov index. Such an invariant will play an important role in connection with a suitably defined Morse index of periodic solutions of non-linear systems. The Maslov index is, roughly speaking, the number of half windings made by the fundamental solution of a linear Hamiltonian system in the symplectic group. Both non-degenerate systems (systems with no non-trivial periodic solutions) and degenerate ones will be examined. We point out that some authors prefer to call this invariant Conley-Zehnder index, using the term Maslov index for an integer which labels paths of Lagrangian subspaces: with this language, the invariant we study here is a special instance of the Maslov index, adapted to the case of periodic orbits. Our terminology agrees with the one used by Salamon and Zehnder in [SZ92] and by Long (see [Lon97] and the literature cited later on).
