Is there any proof of Galilean Transformation? Is there any proof of Galilean Transformation? Is it proved from experiment, theory or it simply is an axiom?
 A: The Galilean transformation is not proven from experiment.  The Galilean transformation should be used only when two frames of reference are moving at a constant speed (possibly zero) relative to each other.  But even with that limitation there is experimental evidence that it is incorrect and that the Lorentz transformation should be preferred.
All that said, the error in using the Galilean transformation for many every-day physical problems is small enough that it can often be ignored.
Here is a useful primer on the subject:
https://www.ck12.org/book/ck-12-physics---intermediate/section/22.1/
A: A Galilean transformation $G(R, \vec{v}, \vec{a}, s)$ can be uniquely expressed as a composition of -

*

*Translations: $(\vec{x}, t) \longmapsto (\vec{x} + \vec{a}, t + s),$

*Rotations: $(\vec{x}, t) \longmapsto (R\vec{x}, t),$

*Uniform Motion of Spacetime: $(\vec{x}, t) \longmapsto (\vec{x} + \vec{v}t, t),$
where, a point in spacetime is given by the ordered pair $(\vec{x}, t)$ with $\vec{x} \in \mathbb{R}^3$ and $t \in \mathbb{R}$ and $R:\mathbb{R}^3 \to \mathbb{R}^3$ is an orthogonal transformation. In particular, $R \in SO(3)$. One can prove that the set of all such transformations forms what is known as a group in mathematics. One can also show that the Galilean group is the group of motions of Galilean relativity, which stems from the Newtonian picture of absolute space and time which is physically valid only for $v = |\vec{v}| \ll c$, where $c$ is the speed of light.
One can indeed recover this Galilean group from the group of motions of special relativity (Poincaré group) under this limit via group contraction. The conceptual and empirical basis for Galilean transformations have been discussed at length by Galileo and in particular Newton in his Principia. Mathematically, the Galilean transformations simply embody the intuitive notion of addition and subtraction of velocities as vectors.
As far as the aforementioned proofs are concerned, I invite you to try writing them down :)
