Fluid dynamics in the spirit of Cartan: A coordinate-free formulation of fluid dynamics for an inviscid fluid in inertial and non-inertial frames

TL;DR

This paper develops a coordinate-free formulation of inviscid fluid dynamics using Cartan's exterior calculus, highlighting invariance under Galilean and general coordinate transformations, and simplifying conservation law derivations.

Contribution

It introduces a novel exterior calculus framework for fluid dynamics that maintains symmetry invariance and simplifies derivations, advancing theoretical understanding.

Findings

Euler equations are invariant under Galilean transformations

Exterior calculus simplifies conservation law derivations

Formulation supports symmetry-preserving discretizations

Abstract

Using Cartan's exterior calculus, we derive a coordinate-free formulation of the Euler equations. These equations are invariant under Galileian transformations, which constitute a global symmetry. With the introduction of an appropriate generalized Coriolis force, these equations become symmetric under general coordinate transformations. We show how exterior calculus simplifies dramatically the derivation of conservation laws. We also discuss the advantage of an exterior calculus formulation with respect to symmetry-preserving discretizations of the equations.