How are Galilean transformations logically connected to Newtonian mechanics?

Question

I used to think of them as separate independent rules. But after making the Why do we have to revise the definitions of momentum and force in special relativity? post, I realised that they're not separate.
How are they connected though? Does one imply the other? Why would some velocity transformation other than the Galilean transformations not make sense for Newtonian mechanics?
EDIT- The key equations in Newtonian mechanics are $F=ma$ and $F_{12}=-F_{21}$. And Galilean transformations are $x'=x-vt, t'=t$. Does one of these imply the other? How are they the "perfect" match for each other? Help me see how they are not independent things.

David Hammen · Accepted Answer

The Galilean transform is thoroughly baked into Newton's Principia, from his definitions, his laws, his corollaries, and his scholia. Newton mentioned Galileo's ship experiments multiple times in his Principia. Newton's definition of quantity of motion (what we now call linear momentum) makes it very clear that the Galilean transform is assumed to be true. His first two corollaries essentially state that forces are three-vectors.
The Galilean transform is even more thoroughly baked into modern teaching of Newtonian mechanics. Displacements, velocities, momenta, accelerations, and forces are taught as three-vectors. Most introductory physics texts start with one or more chapters that describe how three-vectors behave, and how one takes derivatives of those three-vectors with respect to time.
That those key quantities can be treated as three-vectors with time as the independent variable implicitly assumes that the Galilean transform is valid.

Vadim · Answer

Newtonian mechanics is valid only in inertial reference frames (existence of which is postulated by the Newton's first law), which are the reference frames that are related via the Galilean transformations (in non-relativistic mechanics). I don't really see how one can see Newtonian mechanics and Galilean transformations separately...
Update
To expand in a bit more details:

Inertial reference frames are references frame where a body, if acted upon by a zero net force, does not accelerate. Newton's first law is essentially a restatement of this definition: There exist such reference frames where ... (I assume that no one here confuses Newton's first law with a particular case of his third law.)
Galilean transformations allow to recalculate the coordinates and momenta from one reference frame to the other.
The first postulate of relativity is that the laws of physics are the same in all inertial frames of reference.
Lorentz transformations transform between different reference frames, accounting for the second postulate of relativity (constancy of the speed of light).

Thus, Galilean transformations can be viewed either as a Newtonian version of the Lorentz transformations (i.e. Lorentz transformations at $vll c$) or as a restatement of the first postulate of relativity for Newtonian mechanics.

How are Galilean transformations logically connected to Newtonian mechanics?

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