When studying a physical phenomenon that can be described by classical mechanics, we generally work with Hamiltonian differential equations.
Among Hamiltonian systems, there is a special class of systems called completely integrable systems which have a very interesting topological structure on the phase space.
This structure gives a local change of coordinates, called action-angle coordinates, which transforms the system's flow into a linear flow on invariant tori.
Hamiltonian monodromy is the simplest topological obstruction to the existence of global action-angle coordinates.
In this talk, I will introduce, in R^4 , all the concepts mentioned in the previous paragraph from a geometric perspective.
I will then explain how, using spectral Lax pairs, one can introduce a Riemann surface such that the computation of Hamiltonian monodromy reduces to
calculating a residue at infinity of a meromorphic form defined on this Riemann surface. Finally, I will conclude with an overview of some perspectives on this work.
