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

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.