Galilean non-invariance of classical electromagnetism
If Galilean transformations were invariant for not only mechanics but also electromagnetism, Newtonian relativity would hold for the whole of the physics. However, we know from Maxwell's equation that formula_1, which is the velocity of the propagation of electromagnetic waves in vacuum. Hence, it is important to check if Maxwell's equation is invariant under Galilean relativity. For this, we have to find the difference (if any), in the observed force of charge when it is moving at a certain velocity and observed by two reference frames formula_2 and formula_3 in such a  way that the velocity of formula_2 is formula_5 more than formula_3 (which is at absolute rest). Electric and magnetic field under Galilean relativity. In order to check whether Maxwell's equation is invariant under Galilean transformation, we have to check how the electric and magnetic field transforms under Galilean transformation.Let a charged particle/s or body is moving at a velocity formula_7 with respect to S frame. So, we know that formula_8 in formula_3 frame and formula_10 informula_11 frame from Lorentz Force. Now, we assume that Galilean invariance holds. That is, formula_12 and formula_13(from observation). formula_14 formula_15 This equation  is valid for all formula_7. Let, formula_17 By using equation (a) in (1), we get Transformation of formula_18 and formula_19. Now, we have to find the transformation(if any) of charge and current densities under Galilean transformation. Let, formula_18 and formula_19 be charge and current densities with respective to S frame respectively. Then, formula_22 and formula_23 be the charge and current densities in formula_11 frame respectively. We know, formula_25 Again, we know that formula_26 Thus, formula_27 formula_28 Thus, we have Transformation of formula_29, formula_30 and formula_31. We know that formula_32. Here, formula_33. Since q'=q, formula_34 and t'=t(Galilean principle), we get Now , Let formula_35 t'=t formula_36 formula_37 As, formula_38 Similarly, formula_39 and formula_40 Thus, we get Transformation of Maxwell's equation. Now by using equations (a) to (g) we can easily see that Gauss's law and Ampère's circuital law doesn't preserve its form. That is, it non-invariant under Galilean transformation. Whereas, Gauss's law for magnetism and Faraday's law preserve its form under Galilean transformation. Thus, we can see that Maxwell's equation does not preserve its form under Galilean transformation, i.e., it is not invariant under Galilean transformation.