Maslov indices for periodic orbits

TL;DR

This paper generalizes the Conley-Zehnder index to arbitrary symplectic manifolds, linking it to the Maslov phase in trace formulas, thus broadening its applicability in Hamiltonian dynamics.

Contribution

It introduces a new generalization of the Conley-Zehnder index applicable to all symplectic manifolds, connecting it to the Maslov phase in trace formulas.

Findings

Generalized Maslov index for arbitrary symplectic manifolds

Connection between the index and Maslov phase in trace formulas

Extension of index theory in Hamiltonian systems

Abstract

It is shown that there is a generalization of the Conley-Zehnder index for periodic trajectories of a classical Hamiltonian system $(Q, \omega, H)$ from the case $Q = T^*R^n$ to arbitrary symplectic manifolds. As it turns out, it is precisely this index which appears as a Maslov phase in the trace formulas by Gutzwiller and Duistermaat-Guillemin. Contribution presented at the XIX ICGTMP Salamanca June 92.