If no forces act on a material point it will not be accelerated.

Or

If no forces act on a material point, it can always be
find a reference frame
in which that has no acceleration.

The Newton’s first law is sometimes called the law of inertia, and the reference frames that it defines are called inertial reference frames or just inertial frame.

Any frame of reference which is at rest or moves uniformly in
a straight line with respect to some inertial system is itself an inertial system.
On the contrary, any system being in accelerated motion with
respect to an inertial system is noninertial

The number of inertial system of reference is incalculable. At the same time different inertial systems are immovable relative to each other or they move with constant velocity.

order to deduce this statement let us consider coordinates relationship for different inertial system.

Galileo’s Transformations

A practical question arises: what is a reference body with which the inertial frame of coordinates will be bound up? The method to choose it consists in taking different physical bodies and analysing whether a body is in a state of rest or uniform straight-line motion if there is no resultant force acting on it. If one of these states takes place, the inertial frame of reference will be bound up with such a body.

Since we have come to Newton’s Laws of Motion through laboratory experiments on the Earth, we know that a frame of reference rigidly attached to the Earth is a good one - at least within the accuracy of our measurements. Having learnt the secondary school physics course, you used to apply this frame of reference and solved a lot of physical problems. But we know that the Earth rotates about its axis, thus it has its centripetal acceleration. Moreover, the Earth moves around the Sun, and the centripetal acceleration caused by this motion exists. If the mentioned above accelerations are not significant for a physical problem under solution, we shall depreciate them and deem the Earth to be a centre of an inertial system (geocentric system of coordinates). Otherwise it is necessary to look for another centre of the inertial system. A better one is related to the Sun - this is the heliocentric system, though it is not the perfect inertial system because the Solar system also rotates.

There is a great number of inertial frames of reference in the Universe. In fact each frame moving uniformly and in a straight-line direction relative to a certain original inertial frame will be inertial too. The question to be solved is: how can we describe the motion of an object in any other inertial frame of reference if we know how this object moves relative to the original inertial frame?

Let K and be two frames of reference. Frame is in the state of rest and frame moves at constant velocity relative to frame . Initially the origins of both frames ( and ) coincide and at the time the position vector of the frame is . Fig 1.17. The position of an object, , is defined by position vector with respect to the frame , and position vector with respect to the frame .

On the basis of the rule of vectors addition, we write:

As the velocity of a system in relation to other one is constant that

Then

Instead of this vector equation we can write three scalar equations:

These equations enable us to find the coordinates of any material object in system if its coordinates in system are known.

Another important equation should be added:

,

або

it means that time in classical physics is deemed an absolute physical quantity, it is the same in all inertial frames of reference (with one limitation: speeds have to be much smaller than a speed of light).

Last four equations are called Galileo’s transformations.

We get Galileo’s Law of velocities addition, differentiating equation for position vector with respect to time:

,

or

It is a well-known rule of velocity addition.

If a body moves at constant velocity in one inertial frame of reference
it will move at constant velocity in any other inertial frame of reference.
Therefore, the first law of dynamics is valid in both frames of reference

Differentiating this equation with respect to time we get:

As constant therefore

This expression proves that

the acceleration is the same in both frames of reference,
i.e. acceleration is an absolute quantity or an invariant

Or

Thus, the same forces act on the point m in both inertial systems of reference. It means:

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