Canonical description of incompressible fluid -- Dirac brackets approach

TL;DR

This paper introduces a new canonical framework for incompressible fluid dynamics using Dirac brackets, transforming the traditional Poisson brackets into a form that leads to nonlinear, non-local equations for velocity.

Contribution

It develops a novel canonical description employing Dirac brackets to incorporate incompressibility constraints into fluid dynamics.

Findings

Derives nonlinear, non-local equations for fluid velocity

Replaces Poisson brackets with Dirac brackets in the canonical formulation

Provides a new mathematical framework for incompressible fluids

Abstract

We present a novel canonical description of the incompressible fluid dynamics. This description uses the dynamical constraints, in our case reflecting "incompressibility" assumption, and leads to replacement of usual hydrodynamical Poisson brackets for density and velocity fields with Dirac brackets. The resulting equations are then known nonlinear, and non-local in space, equations for incompressible fluid velocity.