I've had a look around online, but I haven't been able to find something which answers this in a way I understand. Essentially, I'm trying to figure out why force is invariant under a Galilean transformation.
The answer to this this question wasn't really what I'm looking for. I think the framework of the answer is too advanced, as I am looking for something in a more simple framework than metrics on a four dimensional space. It's also not clear to me why the fact that force is coordinate free means it is invariant under a Galilean transformation. Could one not argue that since it is coordinate free, it's invariant under any coordinate transformation? But I know this is wrong, since force isn't invariant under a transformation to a non-inertial frame of reference?
Here is another similar question which has been pointed out to me, which I also saw when I was looking around. The actual question is written in a way which I do not understand - I think the question is posed at a more advanced level than I am considering, judging by the notation. The answer agrees that the topic is complicated, and suggests reading about affine tensors in shell theory. Since this question is relevant to some sub-honours undergraduate university material, I shouldn't think it is so advanced to require reading that.
I've seen an argument in Resnick's Introduction to Special Relativity, but it seems to assume that the force is conservative, then differentiating the resulting potential, whereas I am looking for something more general. I'm also convinced that there must be a simpler way to see this.
I've also seen something along the lines of "force is a vector, and so transforms nicely in this way" - however, I didn't really understand why this argument should hold.
The kind of level of reasoning I'm looking for is touched on in the first question I linked to above. I'm looking for a more intuitive argument, rather than a more rigorous mathematical proof. The author writes
It seems reasonable to assume that a given force acting on a body shouldn't change depending on the frame of reference, i.e. that F' = F, since physical laws should be observer independent and hence the equations describing them should also be observer independent.
However, I don't want to just assume that a force shouldn't change depending of the frame of reference of the observer. Again, if we changed to an accelerating reference frame, this wouldn't be true, and so I'm convinced there must be a way to see this more clearly from the fact Galilean transformations relate coordinates in frames of reference moving at constant velocity relative to each other.
Can anyone shed any light on this?