Galilean invariance

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames. Galileo Galilei first described this principle in 1632 in his Dialogue Concerning the Two Chief World Systems using the example of a ship travelling at constant velocity, without rocking, on a smooth sea; any observer doing experiments below the deck would not be able to tell whether the ship was moving or stationary.

Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws hold in all frames related to one another by a Galilean transformation. In other words, all frames related to one another by such a transformation is inertial (meaning, Newton's equation of motion is valid in this frame). In this context it is sometimes called Newtonian relativity.

Among the axioms from Newton's theory are:

Galilean relativity can be shown as follows. Consider two inertial frames S and S' . A physical event in S will have position coordinates r = (x, y, z) and time t; similarly for S' . By the second axiom above, one can synchronize the clock in the two frames and assume t = t' . Suppose S' is in relative uniform motion to S with velocity v. Consider a point object whose position is given by r' (t) = r(t) in S. We see that

The velocity of the particle is given by the time derivative of the position:

Another differentiation gives the acceleration in the two frames:

It is this simple but crucial result that implies Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows Newton's laws of mechanics, if valid in one frame, must hold for all frames. But it is assumed to hold in absolute space, therefore Galilean relativity holds.

A comparison can be made between Newtonian relativity and special relativity.

Some of the assumptions and properties of Newton's theory are:

In comparison, the corresponding statements from special relativity are as follows:

Notice both theories assume the existence of inertial frames. In practice, the size of the frames in which they remain valid differ greatly, depending on gravitational tidal forces.

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