A Formula for the Maslov Index of Linear Autonomous Hamiltonian Systems

The Maslov index of a Lagrangian path under a certain transversality assumption is given by an algebraic count of the intersections of the path with a co-oriented subvariety of the Lagrangian Grassmannian called the Maslov cycle. In these notes with the aim of the well known results for computing the Maslov index given by Robbin and Salamon in [16] and by using the Hörmander [8, Section 3.3] and Kashiwara signature we explicitly compute this index in the case of linear autonomous Hamiltonian systems. We will include an application in order to compute conjugate points along semi-Riemannian geodesics. Dedicated to the memory of my friend Erich Monteleone