Affine Lie-Poisson Reduction, Yang-Mills magnetohydrodynamics, and superfluids

TL;DR

This paper develops affine Lie-Poisson reduction theory and applies it to derive Poisson brackets and circulation theorems for Yang-Mills, Hall magnetohydrodynamics, and superfluids, advancing the geometric understanding of these systems.

Contribution

It introduces a new affine Lie-Poisson reduction framework and applies it to complex fluid models, deriving their Poisson structures and circulation theorems.

Findings

Derived Poisson brackets via reduction of canonical cotangent bundles.

Established Kelvin-Noether circulation theorems for the studied systems.

Unified geometric approach to magnetohydrodynamics and superfluids.

Abstract

This paper develops the theory of affine Lie-Poisson reduction and applies this process to Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids. As a consequence of this approach, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples.