Geometry of integrable dynamical systems on 2-dimensional surfaces

TL;DR

This paper classifies smooth integrable vector fields on 2D surfaces using invariants like period functions and cohomology classes, and explores their Hamiltonianization via symplectic or Poisson structures.

Contribution

It provides a classification framework for integrable systems on surfaces based on invariants and investigates conditions for Hamiltonianization.

Findings

Classification using period and monodromy invariants

Expression of invariants via Puiseux series

Conditions for Hamiltonianization of integrable fields

Abstract

This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous invariants involved in this classification are the left equivalence classes of period or monodromy functions, and the cohomology classes of period cocycles, which can be expressed in terms of Puiseux series. We also study the problem of Hamiltonianization of these integrable vector fields by a compatible symplectic or Poisson structure.