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This is an example of the kind of discussion I am interested in.

Imagine we have a large rotating turntable, kind of like a merry-go-round, but made of some transparent material. On its surface, as well as on the floor beneath we have some Ho train tracks and trains (things moving at constant speed relative to the platform or to the floor). As well as some troughs with marbles (dynamically moving objects) rolling along them, and some marbles rolling freely on the platform, as well as on the floor.

I want to talk about the specific coordinates of each of these objects at specific times, and over a period of time. I want a vocabulary for such things as the location of a particular object at an instant in time, but that location is not moving with the object. It is a fixed "mark" on the platform and/or floor. Then I want to talk about future locations of both the platform and floor marks for that point.

Of particular interest would be a development of the kinematics of relatively rotating reference frames which makes no appeal to dynamics. That is, all reference frames (inertial or not) are treated equally. This should result in a formal structure much like the Lorentz boosts of special relativity.

Before assuming this is a naive question, please see my disorganized personal notebook [deleted per SE policy]

To give an example of the vocabulary and notation I am already familiar with, this is the component form of the equations of rotational kinematics derived in my notes (based on Introduction to Vector and Tensor Analysis, By: Robert C. Wrede):

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And the standard terminology for those expressions:

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I'm not claiming expertise in the field. And I remain open to the possibility that I have overlooked the obvious. In fact, I am saying that I am dissatisfied with my treatment of the subject. But, I have invested considerable effort to learn about it. I should also share that the notation I use in this notebook is not even consistent with the standard I endeavor to follow elsewhere in my notes.

I feel as if I am reinventing the wheel by creating my own vocabulary and notation for discussing the relationship between a rotating reference frame and an inertial reference frame. Typically, authors introduce just enough notation and vocabulary to get to the end of the chapter discussing the topic. Either that, or they leave it to the reader to sort out what they really mean.

I would like to find a text which presents a coherent comprehensive framework for this topic.

An example of where problems arise is the treatment of velocity. Kinetically, a velocity vector is simply the derivative of the position coordinates of a point mass with respect to time. For an inertial observer, this lines up nicely with the law of inertia. For a rotating observer using a coordinate system fixed in the rotating reference frame, velocity is again simply the time derivative of position coordinates. But a point mass moving with constant velocity defined in terms of the rotating frame must be subjected to forces in order to produce that motion.

Both observers are justified in representing this kinematic velocity as a vector. Both are justified in treating the path of a point mass moving with such a constant velocity as a line segment in their own coordinate system. But we cannot simply rotate a velocity vector defined in the inertial frame in order to produce a vector which a rotating observer would consider the velocity vector of the point mass. Derivatives of the coordinate transformation come into play.

Steven Thomas Hatton
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    What exactly is unsatisfactory about the Wikipedia article on this topic? – G. Smith Apr 27 '21 at 03:54
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    with due respect... do you really expect people to read 20 pages of your notes? What’s wrong with the standard sources (v.g Landau, Goldstein etc?). – ZeroTheHero Apr 27 '21 at 05:03
  • Just glancing things over: "The time derivative of a position $\mathbf{r} ( t )$ in a rotating reference frame has two components, one from the explicit time dependence due to motion of the particle itself, and another from the frame's own rotation." This treats the position relative to the inertial frame as independent, and the representation in the rotating system as dependent. The discussion uses vector manipulation, but doesn't really talk about what is happening kinematically. I guess, if I'm stuck with it, I can use the term "apparent" for vectors living in the rotating system. – Steven Thomas Hatton Apr 27 '21 at 05:28
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    @ZeroTheHero My Gods, I hope no-one actually tries to read those notes (actually 50 pages). That wasn't my point. I have recently looked favorably at Goldstein, but I suspect the discussion will be more abstract than I want. I want a "rubber meets the road" account. The specific question motivating my post is: what would a pilot flying uniformly in a circle call the point on a road where a moving car is instantaneously located, in a way that distinguish the point from the car's position as a function of time? – Steven Thomas Hatton Apr 27 '21 at 05:41
  • there is a brief derivation of the rotating frame in my question here: https://physics.stackexchange.com/q/630781/226902 – Quillo Apr 27 '21 at 06:53
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    Hi Steven Thomas Hatton. Linking to private clouds, dropbox, etc, is for various reasons not acceptable on SE, cf. this meta post. – Qmechanic Apr 27 '21 at 08:00
  • https://www2.whoi.edu/staff/jprice/wp-content/uploads/sites/199/2018/12/aCt-P1-V8.pdf – Steven Thomas Hatton Apr 27 '21 at 09:13
  • Inconsistencies in notation have plagued the fields of kinematics, dynamics and robotics for years now. For example I find the excessive use of sub/superscripts tiresome and overwhelming. The most concise form of the kinematic relationship in the post is offered by screw theory and it looks like this $$ \boldsymbol{a}i = \boldsymbol{a}{i-1} + \boldsymbol{s}_i \ddot{q}_i + \boldsymbol{v}_i \times \boldsymbol{s}_i \dot{q}_i$$ – JAlex Apr 27 '21 at 15:16
  • @JAlex See Misner, Thorne and Wheeler, 8.3 regarding different representations of physical concepts. The index notation is very powerful, especially when leveraging the Einstein summation convention. I'm looking for a discussion of "applications". E.g., ants walking on turntables; fluid parcels in hurricanes. What I find when trying to express such ideas is that I need a vocabulary for things which are often treated as "obvious". I can write very concise formulations of the laws of rotational dynamics. But, for example, the cross product often not intuitively expressive. – Steven Thomas Hatton Apr 28 '21 at 09:08
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    Finding this was a result of my asking this question. https://ethz.ch/content/dam/ethz/special-interest/mavt/mechanical-systems/mm-dam/documents/Notes/Dynamics_LectureNotes.pdf Can't say it's an answer, but it's worth sharing. – Steven Thomas Hatton Apr 28 '21 at 09:23
  • @StevenThomasHatton - I should have clarified that the $\times$ used in screw theory isn't a cross product but an operator that differentiates on rotating frames. And the quantities aren't vectors in $\mathbb{R}^3$ but in $\mathbb{R}^6$. – JAlex Apr 28 '21 at 12:16

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