Perfect Fluid Theory and its Extensions

TL;DR

The paper provides a comprehensive overview of perfect-fluid theory, unifying the Lagrangian and Eulerian descriptions through a canonical Hamiltonian framework and Clebsch-type parameterizations. It highlights how volume-preserving diffeomorphisms and Chern-Simons-like quantities arise as structural features, and extends the formalism to relativistic settings via a current-based Lagrangian. It introduces two central IRrotational models, the Galilean Chaplygin gas and its Lorentz-invariant Born–Infeld extension, both derivable from Nambu–Goto brane actions and connected by hodographic transformations, with rich symmetry structures and integrability in low dimensions. The work further discusses supersymmetric and non-Abelian extensions, and noncommutative generalizations, underscoring deep connections between fluid mechanics and higher-dimensional field theories.

Abstract

We review the canonical theory for perfect fluids, in Eulerian and Lagrangian formulations. The theory is related to a description of extended structures in higher dimensions. Internal symmetry and supersymmetry degrees of freedom are incorporated. Additional miscellaneous subjects that are covered include physical topics concerning quantization, as well as mathematical issues of volume preserving diffeomorphisms and representations of Chern-Simons terms (= vortex or magnetic helicity).