On the Hamiltonian formulation of incompressible ideal fluids and magnetohydrodynamics via Dirac's theory of constraints

TL;DR

This paper explores the Hamiltonian formulation of incompressible ideal fluids and magnetohydrodynamics using Dirac's theory of constraints, deriving a Dirac bracket that enforces incompressibility and projects onto solenoidal velocity fields.

Contribution

It introduces a Dirac bracket framework for incompressible fluids and MHD, incorporating primary constraints of constant density within Hamiltonian formalism.

Findings

Derived a Dirac bracket that enforces incompressibility

Projected Hamiltonian structure onto solenoidal velocity fields

Unified treatment of fluid and MHD Hamiltonian systems

Abstract

The Hamiltonian structures of the incompressible ideal fluid, including entropy advection, and magnetohydrodynamics are investigated by making use of Dirac's theory of constrained Hamiltonian systems. A Dirac bracket for these systems is constructed by assuming a primary constraint of constant density. The resulting bracket is seen to naturally project onto solenoidal velocity fields.