Galilean Relativity and Newton's Laws

[leo.]

I'm a little bit confused about the relationship between Galileo's Principle of Relativity and Newton's Laws. Indeed, as I understand, the Galilean Principle of Relativity is what Galileo presented with Salviatti's ship discussion. The discussion seems to lead to a simple idea: "if one performs mechanics experiments and makes observations at rest it will be the same as if it he were in uniform motion along a straight line". As a consequence, observing the phenomenon would not allow one to detect the movement since it wouldn't be able to tell the difference.

Nowadays, however, the principle is stated as "the laws of mechanics are invariant in every inertial reference frame". This, however, requires an understanding of what one inertial reference frame is supposed to be. As far as I know, the idea of inertial reference frame is provided by Newton's laws. Indeed as I know one inertial reference frame is considered as one in which Newton's laws holds. But given this idea, Galileo's principle written as "the laws of mechanics are invariant in every inertial reference frame" seems meaningless: it should be like that by the definition of an inertial frame.

I also know that Newton's Laws came just after Galileo, so his discussion was made before anyone knew about his laws of motion. In that case, first of all, how Galileo's discussion of Salviatti's ship relates to the statement that "the laws of mechanics are invariant in every inertial reference frame"? And also, how his principle relates to Newton's laws? Is it already included in the definition of an inertial frame?

 

[Dale]

Nowadays, however, the principle is stated as "the laws of mechanics are invariant in every inertial reference frame".

Well, you have to be careful with that formulation. You can write the laws of mechanics in a manifestly coordinate independent fashion so that the laws of mechanics are the same in every frame, regardless of whether the frame is inertial or not.

I am not sure what the official modern formulation is, but I don't think that it matters too much. If you start with the laws of motion then you can derive inertial frames as a class of frames that are equivalent under those laws. If you start with the equivalence of inertial frames then you can constrain the form of the laws of mechanics. Different texts will pick different ones to start from.

 

[HallsofIvy]

First, one of the things that occurred to me when I first read of Galileo's "ship" was that this couldn't really be true because boats are always pitching and rocking with the waves! Of course that was not what Galileo was talking about- pitching and rocking require accelerations and Galileo's "inertial reference frames" required moving with no acceleration. Now, the problem is that today we recognize that an external force is, locally, the same as an accelerated reference frame so the question of exactly what an "inertial reference frame" is is more complicated!