Several years ago, though two Galilean limits of electromagnetism were identified by Le Bellas and Levy-Leblond [LBLL] [Nuovo Cimento B 14, 217–230 (1973)] yet the existence of two well defined non-relativistic limits is not widely recognized so researchers as well as teachers utilize an incoherent mixture of both. However, recent works have shed new light on the choice of gauge conditions in classical electromagnetism.
In the paper under review, the authors recall the results of LBLL and use them to point out some prevailing confusion in the literature. Subsequently, they retrieve the results of LBLL first by examining the order of the magnitudes and then with a Lorentz-like manifestly covariant approach to Galilean covariance based on a 5-dimensional Minkowski manifold. Throughout the studies, they rely upon the Riemann-Lorentz approach based on the vector and scalar potentials rather than the Heavyside-Hertz formulation in terms of electromagnetic fields. Finally, while concluding the discussion, the authors go on to work out the applications taking some examples from the fluid dynamics and the physics of continuous media.