Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton’s laws hold in all frames related to one another by a Galilean transformation. In other words, all frames related to one another by such a transformation are inertial (meaning, Newton's equation of motion is valid in these frames). In this context it is sometimes called Newtonian relativity.
Among the axioms from Newton's theory are:
Galilean relativity can be shown as follows. Consider two inertial frames S and S' . A physical event in S will have position coordinates r = (x, y, z) and time t in S, and r' = (x' , y' , z' ) and time t' in S' . By the second axiom above, one can synchronize the clock in the two frames and assume t = t' . Suppose S' is in relative uniform motion to S with velocity v. Consider a point object whose position is given by functions r' (t) in S' and r(t) in S. We see that
The velocity of the particle is given by the time derivative of the position:
Another differentiation gives the acceleration in the two frames: