# Hamiltonian monodromy and Morse theory

## TL;DR

This paper demonstrates how Hamiltonian monodromy in integrable systems with two degrees of freedom can be computed using Morse theory, linking topological invariants to critical points of the Hamiltonian.

## Contribution

It introduces a novel approach to calculating Hamiltonian monodromy via Morse theory and Takens's index theorem, connecting different mathematical frameworks.

## Key findings

- Hamiltonian monodromy can be derived from Morse theory.
- Takens's index theorem relates energy-Chern number changes to critical points.
- The method provides a new perspective on existing approaches to monodromy.

## Abstract

We show that Hamiltonian monodromy of an integrable two degrees of freedom system with a global circle action can be computed by applying Morse theory to the Hamiltonian of the system. Our proof is based on Takens's index theorem, which specifies how the energy-h Chern number changes when h passes a non-degenerate critical value, and a choice of admissible cycles in Fomenko-Zieschang theory. Connections of our result to some of the existing approaches to monodromy are discussed.