Laws of dynamics and reference frame

Question

I was having trouble understanding the following text given in my mechanics textbook:-

If we choose an arbitrary set of frames of reference which are all in uniform rectilinear motion relative to each other and if. in addition, it is known that the laws of dynamics hold for one of these frames then the first and the second laws of dynamics are stated in the same manner for all the frames of reference we have chosen. All such frames are referred to as inertial(Galilean) frames of reference and the Galilean inertial law is valid only for such frames. This is the proposition we call Galileo's relativity principle.

According to the above definition of inertial frames of reference the frames of reference need to be moving uniformly w.r.t each other, this doesn't limit them from moving with some acceleration given that all have the same acceleration. Then, there is the part where the book mentions that the third law of dynamics need not be stated in the same way for all the frames of reference, indicating that they can be accelerated w.r.t an inertial frame of reference . Also, according to the definition of inertial frame in the book which is as follows:-

The investigation of motions whose velocities are small in comparison with the velocity of light indicates that coordinate system whose origin is connected with the center of mass of the bodies forming the Solar System and whose axes have invariable direction relative to the fixed stars can be taken as an inertial frame of reference.

I find the two definitions to be rather conflicting as one says that no external force should be acting on the inertial frame or rather it should be in a position of rest, while the other says that it can have some acceleration. So, which one is correct, or is it me interpreting the whole thing incorrectly.

Answer 1

First of all the definition of inertial frame is actually very fuzzy throughout the literature. The main definition can be equivalently expressed in many ways. I presonally prefer this simple one:

An inertial frame is where Newton's Second law is valid.

i.e. Unless an external force is acting on a body, it doesn't accelerate. Now remember this is one of the many way of defining it. But this definition leads to lesser ambiguity.

Now once you've established a frame to be inertial, all the frames moving in a constant velocity w.r.t. it is also inertial. This is used in both Galilean relativity and Special Relativity by Einstein.

The second definition you've put is merely an example of what can be called "approximately inertial frame". It's not a physical definition. Rather it's a example. Of course that frame is not inertial. But for practical physics of everyday life, you can take it to be so. But while studying galactic dynamics, certainly it's no good.

It's when Einstein gave his General relativity we found out that the idea of a globally inertial frame is fictitious and must be abandoned.