Galilean transformations x Galilean Group

[kent davidge]

It seems that there is a difference between Galilean transformations and (the transformations of the) Galilean group, for one thing: rotations.

The former is usually defined as the transformations $\{\vec{x'} = \vec x - \vec v t, \ t' = t \}$, where $\vec v$ is the primed frame velocity relative to the first frame. On the other hand, rotations are also a possibility in going from one inertial frame to another, and they seem to be included in the Galilean Group.

So when people refer to Galilean Transformations do they mean the transformations that leave Newtons second law invariant? And when they are considering all transformations that leave the second law covariant then they are talking about the Galilean group?

 

[Nugatory]

It depends on the context.
Elementary treatments generally consider only motion along one spatial dimension called $x$ (ignoring the other two completely or only mentioning $y'=y$, $z'=z$ as afterthoughts subsequently ignored). In this case, the discussion is going to be limited to the subgroup of boosts. The author won't even mention that there is such a thing as a "Galilean Group" and that what they're doing is equivalent to considering a "subgroup" of that group; this approach has the pedagogical advantage of working for an audience that has never heard of group theory.

More sophisticated treatments will allow for the more general set of transformations, although will often adopt the simpler picture if no rotations are involved

You are expected to be able to figure out for yourself which kind of treatment you're dealing with.