Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

TL;DR

This paper introduces beatification, a method to efficiently derive weakly nonlinear Hamiltonian systems near equilibrium for fluids and plasmas with noncanonical Poisson brackets, facilitating analysis of their dynamics.

Contribution

The paper presents a novel beatification technique for extracting weakly nonlinear Hamiltonian systems in noncanonical settings, applicable to fluids and plasmas.

Findings

Applicable to finite and infinite-dimensional systems

Simplifies analysis of weakly nonlinear dynamics

Provides multiple illustrative examples

Abstract

A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria for systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.