Beatification: Flattening the Poisson Bracket for Two-Dimensional Fluid and Plasma Theories

TL;DR

The paper introduces a perturbative method called beatification to transform Hamiltonian fluid and plasma systems into a form with fixed Poisson brackets, simplifying analysis and computation.

Contribution

It presents a novel transformation technique that removes variable dependence from Poisson brackets in two-dimensional fluid and plasma Hamiltonian systems.

Findings

Beatification simplifies Hamiltonian systems for fluid and plasma theories.

The method enables easier application of analytical and numerical techniques.

Transformations are demonstrated on key equations like Euler and Vlasov-Poisson.

Abstract

A perturbative method called beatification is presented for a class of two-dimensional fluid and plasma theories. The Hamiltonian systems considered, namely the Euler, Vlasov-Poisson, Hasegawa-Mima, and modified Hasegawa-Mima equations, are naturally described in terms of noncanonical variables. The beatification procedure amounts to finding the correct transformation that removes the explicit variable dependence from a noncanonical Poisson bracket and replaces it with a fixed dependence on a chosen state in phase space. As such, beatification is a major step toward casting the Hamiltonian system in its canonical form, thus enabling or facilitating the use of analytical and numerical techniques that require or favor a representation in terms of canonical, or beatified, Hamiltonian variables.