Metric formulation of Galilean invariance in five dimensions

This is a summary of recent publications and works in progress. We describe a five-dimensional metric formulation of Galilean covariance and illustrate it with various examples in field theory. As a first illustration, we recover the two Galilean limits of electromagnetism first examined by Le Bellac and Levy-Leblond. Then we describe the field theoretical formulation of some fluids and superfluids models : Navier-Stokes equation, Takahashi Lagrangian for irrotational fluids and Thellung-Ziman Lagrangian for Helium II. Our last examples treat non-relativistic Bhabha equations for spin 0 and I particles, and Dirac equation for spin 1/2.