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The Galilean principle of relativity states that:

The laws of mechanics are invariant in all inertial reference frames

That means that if we have two inertial frames of reference $S$ and $S'$ then Newton's laws are valid in both of them. In principle, I didn't understand the importance of such principle.

At first it seems unnecessary since mathematical objects by definition don't depend on the coordinate system chosen: if we want to overkill, a vector at a point is a derivation on the ring of germs of functions at the point. This defines a vector $v\in T_p M$ without reference to coordinates, and we could represent $v$ in many different coordinate basis and even in some non-coordinate basis.

Because of that, since $F = ma$ is an equality between vectors, and since they are defined in such abstract way, if we change the coordinate system, the vectors as geometrical objects themselves would still obey the equation.

But this is Math, not Physics, and the principle says that the laws of mechanics are invariant in all reference frames. Thinking more I thought about it in the following way, which I wan't to know is the right way to get it:

Newton's laws of motion are laws, they are proposed based on observation and tested by experiment. So when we say that $F = ma$, this cannot be proven using logic, but rather it's to be tested. To test it, or use it in some way, we must observe a particle, or system of particles under the influence of some net force.

This observation, in the real world, must be done with some chosen coordinate system. The observation is made from some particular viewpoint which to observe the phenomenon uses coordinates $(x,y,z)$ for instance. The question then if the law is valid in another reference frame is not a question about the mathematical objects, but about the law itself.

So the question is: from another viewpoint, the law still holds? And thus, we are interesting in knowing if some other observation is made from another viewpoint using different coordinates $(x',y',z')$ the law will stand after we test it.

This is the importance of the principle? It's a statement not about mathematical objects themselves (which are of course independent of coordinates) but about the behavior of nature? So in that sense, it is a way to guarantee that the same laws that hold good on reference frames on the surface of Earth can still be used on the Sun or whatever other place in the universe, as long as the reference frame is inertial?

Why this principle is really important and not just a consequence of the mathematical model we use in Newtonian Mechanics?

Gold
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  • I think the word inertial here is crucial. Because $F\neq ma$ in a non-inertial frame of reference. – Ali Moh Mar 08 '15 at 18:49
  • I know that, inertial frames are those exactly on which $F = ma$ is valid. My doubt is, since vectors are themselves indepedent of the reference frame, why a principle saying that the laws are invariant in all inertial frames is important? – Gold Mar 08 '15 at 18:51
  • Then consider this hypothetical universe where there is a pervading small uniform electric field pointing in some specified direction. Then different inertial observers can tell whether they are moving or not by locally performing some experiment. Then Galilean invariance is broken in such universe. – Ali Moh Mar 08 '15 at 19:14
  • As far as I know that's the point with special relativity isn't it? Galilean principle talks about the "laws of mechanics" while Einstein extended to include electromagnetism changing the principle to "the laws of physics". – Gold Mar 08 '15 at 20:09
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    No, Galilean relativity principle (invariance with respect to the Galilei transformation) is a modern term and refers to all laws just as the Einstein relativity principle does. However, it was assumed in 18-19th century that it is valid only for phenomena of mechanics and not for phenomena of optics or electromagnetism. So EM was thought to break Galilei relativity principle even before Einstein. – Ján Lalinský Mar 09 '15 at 05:19
  • Are you aware of Galileo's allegory of Salviati's Ship? It's pretty clear that Galileo meant for his principle to be valid for all laws of physics from this allegory. The doubts about EM only came well after Galileo. In keeping with Ján Lalinský's answer, you can think of Einstein's SR as a Galileo-consistent theory as I discuss here and here. – Selene Routley Mar 09 '15 at 09:43

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So in that sense, it is a way to guarantee that the same laws that hold good on reference frames on the surface of Earth can still be used on the Sun or whatever other place in the universe, as long as the reference frame is inertial?

Exactly.

Why this principle is really important and not just a consequence of the mathematical model we use in Newtonian Mechanics?

It is more general statement than Newtonian mechanics. Newtonian mechanics is consistent with Galilei principle, but Galilei principle admits different theories than Newtonian mechanics.

Ján Lalinský
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