In every inertial frame, we expect the laws of physics to stay the same. To the best of our knowledge, inertial frames can be mapped to one another by transformations. Here's an example - drop a ball and observe its motion. You should find that it obeys F = ma. We know that if we rotate, translate, or drop the same ball on a moving train, the laws of physics stay the same - the ball obeys F = ma. But is that all? What transformations on our space time preserve the laws of physics? If we do something funky like spin around on a spot, will the laws of physics look the same to us? Try this yourself. If you start spinning and drop a ball in front of you, this doesn't work - the ball doesn't drop in a straight line. This mini essay is about which **transformations** of frame of reference keep classical physics the same. For now, let's call this set of transformations *Galilean transformations*. What's the size / dimension of this group of symmetries, and what kinds of things are allowed in them? I came across this problem in Vladimir Arnold's book on classical mechanics. The first part is to show **that** **any** Galilean transformation can be written as the composition of a boost (moving in a reference frame at different velocity), rotation, and translation. Here's some ideas of a proof of that result. I found this problem extremely challenging. The key strategy is to prove that Galilean transformations in general, can be written as affine maps. --- Arnold defines a Galilean structure on a space with 1 time dimension and 3 space dimensions as - a space that is affine, - a space that has *time*, which is a linear map T that restricts the 4-point to its first coordinate. The kernel of this map is a subspace known as simultaneous events. - the subspace of simultaneous events has the Euclidean norm, and Galilean transformations need to preserve such a norm. CLAIM: galilean transformations are a group that acts on affine space PROOF: by definition. CLAIM: galilean transformations need to preserve all distances between all points at a given time. PROOF: by definition / axiom / experiment CLAIM: galilean transformations are automorphisms PROOF: unknown / why should this be true - except maybe because of the first idea, of structure preservation, and so these should be CLAIM: automorphisms on A^4 are affine maps PROOF: known result, which can be found on wikipedia. CLAIM: galilean transformations are affine maps PROOF: if we convince ourselves why galilean transformations need to be automorphisms, then this result is true by the lemma above Now, if we are convinced at Galilean transformations are affine maps, we can then write ( x, t ) -> A * ( x , t )  + B At which point, the isometry and time restriction conditions make it clear that such a function is a composition on rotation, translation and boosting.