# Classification of Casimirs in 2D hydrodynamics

## TL;DR

This paper provides a comprehensive list of Casimir invariants for 2D Euler hydrodynamics on boundaryless surfaces, introducing generalized enstrophies and discussing potential extensions to surfaces with boundaries.

## Contribution

It offers a complete classification of Casimirs for 2D Euler equations and introduces generalized enstrophies as new invariants for coadjoint orbits.

## Key findings

- Complete set of Casimirs for 2D Euler hydrodynamics
- Introduction of generalized enstrophies as invariants
- Discussion on extension to surfaces with boundaries

## Abstract

We describe a complete list of Casimirs for 2D Euler hydrodynamics on a surface without boundary: we define generalized enstrophies which, along with circulations, form a complete set of invariants for coadjoint orbits of area-preserving diffeomorphisms on a surface. We also outline a possible extension of main notions to the boundary case and formulate several open questions in that setting.