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A transformation that in classical mechanics defines the transition from one inertial coordinate system to another such system that executes a rectilinear motion at a constant velocity with respect to the first. The coordinate system is understood to be four-dimensional with three space coordinates and one time coordinate. Let $(x,y,z,t)$ be a given inertial coordinate system; then the coordinates $(x',y',z',t')$ of any other inertial system that is moving with respect to the first system rectilinearly and at a uniform velocity are connected (up to a displacement of the coordinate origin and rotation of the axes) with the coordinates $(x,y,z,t)$ by the Galilean transformation:

$$x'=x-v_xt,\quad y'=y-v_yt,\quad z'=z-v_zt,\quad t'=t,$$

where $v_x,v_y,v_z$ are the components of the velocity of the motion of the system $(x',y',z',t')$ with respect to the system $(x,y,z,t)$.

The fundamental laws of classical mechanics are invariant with respect to Galilean transformations, but the equation of the propagation of the front of a light wave (an electromagnetic effect), for example, is not. This is why the Galilean transformation was generalized by H.A. Lorentz (cf. [[Lorentz transformation|Lorentz transformation]]). These transformations form the foundations of the special theory of relativity. The Lorentz transformations become Galilean transformations for $v\ll c$.

The Galilean transformations form a group, which is a subgroup of the group of non-homogeneous (general) Galilean transformations. The latter group, known as the Galilean group, is obtained from the group of (proper) Galilean transformations by the addition of the transformations consisting in displacement of the coordinate origin and of the zero point in time.



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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V.I. Arnol'd,   "Mathematical methods of classical mechanics" , Springer  (1978)  (Translated from Russian)  {{MR|}} {{ZBL|0692.70003}} {{ZBL|0572.70001}} {{ZBL|0647.70001}} </TD></TR></table>