I am hoping to dig a little deeper into what Taylor says on page 9 of Classical Mechanics. I've provided an excerpt just below:
A more important difference arises when two frames are in relative motion; that is, when one origin is moving relative to the other. In Section 1.4 we shall find that not all such frames are physically equivalent. In certain special frames, called inertial frames, the basic laws hold true in their standard, simple form. (It is because one of these basic laws is Newton's first law, the law of inertia, that these frames are called inertial.) If a second frame is accelerating or rotating relative to an inertial frame, then this second frame is noninertial, and the basic laws — in particular, Newton's laws — do not hold in their standard form in this second frame.>
The emphasis on origin is mine.
My question is:
The sentence including origin seems to imply that, when classifying a reference frame, our procedure begins with determining whether the origin of a given reference frame is moving relative to an inertial frame. If it is not, then the frame is inertial. Otherwise (if the origin of this other frame is moving with respect to the original), we must examine whether the two frames are physically equivalent (presumably evaluating the derivatives of the basis vectors of the other frame in terms of the inertial frame). Is this implication true? It seems to fly in the face of what he says two sentences later - "If a second frame is accelerating or rotating relative to an inertial frame, then this second frame is noninertial." Essentially, is a frame that has a coincidental origin with an inertial frame but which is rotating with respect to that frame inertial (does the introduction of the Coriolis acceleration count as a "fudge factor")?