2. TRANSFORMATION OF COORDINATES, THE GALILEAN TRANSFORMATION, INERTIAL SYSTEMS
 
   The position of some point in space can be defined by the coordinates of a coordinate system, which is connected to some other coordinate system, as a reference. For example, in Fig. 2.1 two coordinate systems are shown in a plane - system and , whose axes are parallel.

Fig. 2.1	Fig. 2.2

   The system is marked by , and the system by . The origin of the system (point ) is given in the system with coordinates , . It can be seen in Fig. 2.1 that the coordinates , of the point in the system can be presented as a function of coordinates , of the system by the following relation

(2.1)

   Coordinates , can also be presented as a function of the coordinates ,

(2.2)

   A similar transformation can also be derived when the axes of these two systems are at a certain angle, that is when they are not parallel.
   The above mentioned transformation is used in cases when the systems have no relative motion.
   Let us assume that the system is moving translatory and at constant speed relatively to the system (Fig. 2.2). In that case the coordinates of the origin are and , where and are the corresponding speed components , and is time. The coordinates of a point in the system, can be expressed in terms of the coordinates , of the system in the following way

(2.3)

   As in the previous case when the systems had no mutual motion, converse transformation may be used

(2.4)

   The same relations are valid for two three - dimensional systems which mutually have translatory motion at constant speed

(2.5)

and

(2.6)

   At this transformation time is the same for both coordinate systems. In classical physics time is the absolute magnitude. It passes evenly and it does not depend upon space, the body of reference, the coordinate system or anything else from the outside.
   The above mentioned transformation is called Galilean transformation in honor of the founder of mechanics. It is used for all inertial systems. The inertial system is the system of coordinates, in which inertial law retains its original shape. In connection with that, Newtonian relativity principle says: "There is an infinite number of equivalent systems known to us as an inertial, which have an uniform and rectilinear motion in relation to one another, where the laws of mechanics are fulfilled in the classical form." This means that if one system is inertial so is any other system inertial if, in relation to the first, it moves uniformly and rectilinearly.
   Now we examine the case in Fig. 2.2. Let the speed be constant in the first system, which means that the acceleration is equal to zero, so inertial law is valid for it, and therefore we say that the system is inertial. We can see from Eq. (2.5) that the moving system, which moves rectilinearly and uniformly relatively to is also inertial because

(2.7)