Galilean Principle of Relativity

the principle of the physical equality of inertial systems in classical mechanics, which is manifested in the fact that the laws of mechanics are identical in all such systems. Hence, it follows that it is impossible to determine by any mechanical experiment conducted in any inertial system whether the given system is at rest or moving uniformly and rectilinearly. The principle was first established by Galileo in 1636. Galileo demonstrated the identical nature of the laws of mechanics for inertial systems by using as an example the phenomena occurring below the deck of a ship that is at rest or moving uniformly and rectilinearly (with respect to the earth, which may to a sufficient degree of accuracy be considered as an inertial system): “Have the ship proceed at any speed you like so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the efforts named, nor could you tell from any of them whether the ship was moving or standing still. ... In throwing something to your companion you will need no more force to get it to him whether he is in the direction of the bow or the stern with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern although while the drops are in the air the ship runs many spans” (Dialog o dvukh glavneishikh sistemakh mira ptolomeevoi i kopernikovoi [Dialogue Concerning the Two Chief World Systems]. Moscow-Leningrad, 1948, p. 147).

The motion of a material point is relative: its position, velocity, and type of trajectory depend on the frame of reference (reference object) with respect to which this motion is examined. At the same time, the laws of classical mechanics—that is, the relations that link the quantities describing the motions of material points and the interactions between them—are identical in all inertial systems. The relativity of mechanical motion and the identical nature (nonrelativity) of the laws of mechanics in different inertial systems constitute the content of the Galilean principle of relativity.

Mathematically, the Galilean principle of relativity expresses the invariance of the equations of mechanics with respect to transformations of the coordinates of moving points (and time) upon a transition from one inertial system to another (Galilean transformations).

Let there be two inertial frames of reference, one of which, Σ we shall agree to consider to be at rest; the second system, Σ’, moves with respect to Σ at a constant velocity u. Then the Galilean transformations for the coordinates of the material point in the systems Σ and Σ’ will have the form

(1)x′ = x - ut, y′ = y, z′ = z, t′ = t

The primed quantities refer to the system Σ’ and the unprimed quantities, to Σ. Thus, in classical mechanics time, like the distance between any fixed points, is considered identical in all frames of reference.

From Galilean transformations we can obtain the relations between the velocities of the motion of a point and its accelerations in both systems:

(2)v′ = v - u

a′ = a

In classical mechanics the motion of a particle is defined by Newton’s second law:

(3)F = ma

where m is the mass of a point and F is the resultant of all forces applied to it. Here the forces (and masses) are invariants in classical mechanics—that is, they are quantities that do not change on transition from one frame of reference to another. Therefore, during Galilean transformations, equation (3) does not change. This is the mathematical expression of the Galilean principle of relativity.

The Galilean principle of relativity is valid only in classical mechanics, in which motion at velocities far lower than the velocity of light are considered. At velocities close to the velocity of light, the motion of bodies conforms to the laws of Einstein’s relativistic mechanics, which are invariant with respect to other transformations of coordinates and time —Lorentz transformations (at low velocities they turn into Galilean transformations).