Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics

TL;DR

This paper classifies generic coadjoint orbits and simple Morse functions for symplectomorphism groups on surfaces, providing a solution to Arnold's problem on invariants of 2D ideal fluid flows.

Contribution

It introduces a classification framework for coadjoint orbits and Morse functions, and defines anti-derivatives on measured Reeb graphs to describe invariants.

Findings

Classification of generic coadjoint orbits for surface symplectomorphism groups

Classification of simple Morse functions under group actions

Introduction of anti-derivatives on measured Reeb graphs

Abstract

We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V.Arnold's problem on describing all invariants of generic isovorticed fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties.