Dynamics of symplectic fluids and point vortices

TL;DR

This paper develops a Hamiltonian framework for symplectic fluid dynamics, introduces symplectic vorticity, and analyzes the behavior and integrability of symplectic point vortices, extending existing mathematical results.

Contribution

It extends Ebin's long-time existence results to broader metrics and provides a detailed study of symplectic point vortex dynamics and symmetries.

Findings

Extended Ebin's long-time existence results to non-compatible metrics.

Described symmetry groups of symplectic point vortices.

Analyzed integrability of symplectic vortex dynamics.

Abstract

We present the Hamiltonian formalism for the Euler equation of symplectic fluids, introduce symplectic vorticity, and study related invariants. In particular, this allows one to extend D.Ebin's long-time existence result for geodesics on the symplectomorphism group to metrics not necessarily compatible with the symplectic structure. We also study the dynamics of symplectic point vortices, describe their symmetry groups and integrability.