Angular Quantities I

This is the first in a series of posts in which I want to share some hopefully interesting things about mathematical descriptions of rotational motion. This series was inspired by a talk given at the 2015 winter AAPT meeting in San Diego. The author claimed to have found a way to represent angular displacement as a vector (true, such an expression exists and is not widely used) and that angular displacements commute (false, in general they do not except when infinitesimal). The same author presented an updated poster on this topic at the recent winter meeting in Atlanta. In researching the arguments presented in these two talks, following up on the references therein, and in searching the undergraduate and graduate physics and mathematics teaching literature on descriptions of angular quantities, I stumbled onto some of the most interesting topics I’ve ever encountered. As you may have already guessed, I want to find ways of bringing these gems of understanding into the introductory courses so students won’t be so mystified when then encounter the in upper level courses. By the way, the papers from these talks aren’t availble online; I only have paper copies and I do not have the author’s permission to distribute them.

I am sure most of this will be trivial for many readers, so apoligies in advance. Even though I too studied out of Goldstein in grad school, it was not the case that all my existing conceptual mysteries were solved. As always, I tend to frame things from the point of view of that introductory physics student for whom we want to provide an unparalleled physics experience. I don’t want that student to ever say, “Well that was never pointed out to me in intro physics.” I want that student’s conceptual foundation to be better than mine was when I was that student.

In this initial post, I will list as many of the questions I can think of that arose as I researched this topic. I will not answer any of them in this post, but will attempt to do that in subsequent posts. I will put the questions into some preliminary order, but I can’t guarantee that order won’t change later. Some questions may change to more accurately reflect what I’m trying to explain.

1. What does it mean to be a vector?
2. What do vector dot products and vector cross products mean geometrically?
3. What is the physical significance of the double cross product (aka triple cross product)?
4. Is there a coordinate free expression for the total time derivative of a vector?
5. Is there a coordinate free expression for the time derivative of a unit vector (a direction)?
6. Can angular velocity be described as a vector?
7. Can angular displacement be described as a vector?
8. If work is calculated as the dot product of two vectors, then when calculating rotational work how can angular displacement not be a vector?
9. If angular velocity is a vector, shouldn’t its integral also be a vector and not a scalar?
10. Why does translational displacement commute?
11. How, if at all, are translation and rotation (revolution?) related?
12. Why do infinitesimal angular displacements commute?
13. Why do finite angular displacements not commute?
14. What is the distinction between rotation and orientation?
15. Is angular velocity the derivative of a rotation?
16. So then what is angular velocity the derivative of anyway?
17. Can angular velocity be integrated to get angular displacement?
18. Can these ideas be brought into the introductory calculus-based or algebra-based physics courses?

I think that’s all, at least for now. I don’t claim this list to be comprehensive. The number of questions isn’t significant either. Let’s see where this goes.