Hamiltonian analysis of interacting fluids

Rabin Banerjee,Subir Ghosh,Arpan Krishna Mitra

doi:10.1140/epjc/s10052-015-3412-y

Abstract

Ideal fluid dynamics is studied as a relativistic field theory with particular importance on its hamiltonian structure. The Schwinger condition, whose integrated version yields the stress tensor conservation, is explicitly verified both in equal-time and light-cone coordinate systems. We also consider the hamiltonian formulation of fluids interacting with an external gauge field. The complementary roles of the canonical(Noether) stress tensor and the symmetric one obtained by metric variation are discussed. Finally, a non-relativistic reduction of the system in light-cone coordinates has been carried out which reproduces results found earlier in the literature.

Highlights

Fluid dynamics as an applied science has a long history, but its generalization as a relativistic theory and its subsequent analysis as a relativistic field theory is a relatively recent development
The hydrodynamic equations are essentially the local conservation laws supplemented by the constitutive relations that express the stress tensor in terms of the fluid variables
A lagrangian version of fluid dynamics is plagued with obstructions due to the presence of a Casimir operator, the vortex helicity

Summary

Introduction

Fluid dynamics as an applied science has a long history, but its generalization as a relativistic theory and its subsequent analysis as a relativistic field theory is a relatively recent development. In the presence of interaction, Tμν and μν do not match The former generates the correct equations of motion for all the dynamical variables but does not yield the correct conservation law of the stress tensor. The mutual exchange of ideas can work both ways in the fluid–gravity correspondence: fluid systems can yield results relevant in e.g. black hole physics, Hawking radiation [17], while gravitational physics can provide new ideas in the context of viscous fluids and turbulence, to name a few All these considerations require a systematic study of the fluid system as a field theory in the Euler scheme, which is essentially a hamiltonian framework.

Relativistic fluid mechanics in equal-time coordinates

Conservation laws in the hamiltonian formulation and Schwinger condition

Conservation laws in hamiltonian formulation and Schwinger condition

Interacting fluid model

Conclusion and future prospects