Dispersive deformations of the Hamiltonian structure of Euler's equations

TL;DR

This paper investigates how dispersive deformations affect the Hamiltonian structure of 2D Euler's equations, showing that up to second order, such deformations are trivial, thus preserving the original structure.

Contribution

It derives the Poisson brackets for 2D hydrodynamics from the full algebra of vector fields and analyzes the triviality of dispersive deformations up to second order.

Findings

Dispersive deformations of Euler's Poisson brackets are trivial up to second order.

Poisson brackets for 2D hydrodynamics can be obtained via reduction from the full vector field algebra.

The Hamiltonian structure of Euler's equations remains stable under certain dispersive deformations.

Abstract

Euler's equations for a two-dimensional system can be written in Hamiltonian form, where the Poisson bracket is the Lie-Poisson bracket associated to the Lie algebra of divergence free vector fields. We show how to derive the Poisson brackets of 2d hydrodynamics of ideal fluids as a reduction from the one associated to the full algebra of vector fields. Motivated by some recent results about the deformations of Lie-Poisson brackets of vector fields, we study the dispersive deformations of the Poisson brackets of Euler's equation and show that, up to the second order, they are trivial.