Galileo's principle of relativity states that the laws of mechanics are invariant in every inertial frame of reference.
This is well illustrated by Galileo’s ship. What is meant here by "laws of mechanics"? Are these Newton's laws of motion, conservation laws, Lagrange/Hamilton equations, or something else?
Galileo's principle of relativity states: "It is impossible by mechanical means to say whether we are moving or staying at rest". This predates but underpins Newton's laws of motion. It states that the basic principles governing the motion of objects (which would later be formalized in Newton's laws) apply equally in all inertial frames (frames that are either at rest or move with a constant velocity).
Galileo did not specifically mention Newton's laws, conservation laws, or the formulations of mechanics by Lagrange and Hamilton as he preceded all of these developments, however, his principle implies that these fundamental laws of mechanics are invariant across different inertial frames.
This principle is significant because it introduces the idea that the laws governing physical phenomena are consistent and universal, regardless of the observer's state of motion, laying the groundwork for classical mechanics and influencing future developments in physics, including Einstein's theories of relativity.
By "laws of Physics," we should mean the equations describing physical phenomena. Their invariance does not mean that the same phenomenon is described in the same way in different reference frames. This is clearly wrong. It means that if we prepare our experiments in the same way in different reference frames the invariance of the equations implies the same solutions and the practical impossibility of deciding only on the basis of the phenomena what the motion of the reference frame is.
From the historical side, one should distinguish between what we call Galilean invariance and what Galileo meant by his ship argument.
Galilean invariance is limited to the classical mechanics formalism. Galileo's ship argument has a much wider meaning. Even if mechanics dominated the physics of Galileo's time, the ship argument was not limited to mechanical phenomena. When Galileo wrote (translation as in this Wikipedia page )
... The fish in their water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship's motion is common to all the things contained in it, and to the air also.
He claimed that no phenomenon allows the distinction of the motion of two reference frames in uniform relative motion. Flies, butterflies, and fish are not exclusively mechanical systems. Still, they behave the same way.
Galilean invariance means that in going between different frames of reference, velocities are additive. For example, the velocity of a baseball thrown from a moving car is the sum of the ball's velocity relative to the car and the car's velocity relative to the ground. This formalism works for velocities much smaller than the speed of light.
It fails for very high velocities and is replaced by relativistic invariance, in which velocities are not purely additive. In this case, the velocity of a beam of light emitted by the headlight of a moving car is not added to that of the car; it is always c, no matter how fast the car is moving.
