Revealing Noncanonical Hamiltonian Structures in Relativistic Fluid Dynamics

TL;DR

This paper uncovers the noncanonical Hamiltonian framework of relativistic fluid dynamics, revealing conserved quantities like helicity and enstrophy, and extends classical fluid invariants to relativistic regimes.

Contribution

It identifies the noncanonical Poisson structure and Casimir invariants of the relativistic Euler equations, generalizing classical fluid invariants to relativistic fluids.

Findings

Relativistic fluid flows preserve helicity and enstrophy as conserved quantities.

The conserved quantities are Casimir invariants of the noncanonical Poisson structure.

The framework applies to fluids with a relativistic $$-barotropic equation of state.

Abstract

We present the noncanonical Hamiltonian structure of the relativistic Euler equations for a perfect fluid in Minkowski spacetime. By identifying the system's noncanonical Poisson bracket and Hamiltonian, we show that relativistic fluid flows preserve helicity and enstrophy as conserved quantities in three-dimensional and two-dimensional cases, respectively. This holds when the fluid follows a relativistic $\gamma$-barotropic equation of state, which generalizes the classical barotropic condition. Furthermore, we demonstrate that these conserved quantities are Casimir invariants associated with the noncanonical Poisson structure. These findings open new avenues for applying Hamiltonian theory to the study of astrophysical fluids and relativistic plasmas.