Two errors about rotations in Goldstein's "Classical Mechanics"

[Erland]

Classical Mechanics by Herbert Goldstein is one of the most used textbooks on this subject, perhaps the most used one.
However, I found a couple of errors in Section 4.9 (in 3rd ed, written with Charles Poole and John Safko) about rotations.

First, at p. 172, the angular velocity vector ω is defined by the relation ωdt = dΩ. The authors write in a footnote: "Note that ω is not the derivative of any vector." (by which they mean a time derivative).
But this is wrong. If ω is a (vector valued) continuous function of time (even weaker conditions suffice), then it follows from the Fundamental Theorem of Calculus, applied to each of its components, that it has a (vector) antiderivative w.r.t time.
Presumably, the authors meant that ω is not the derivative of any physically interesting vector.

Second, at p. 173, the authors try to express the angular velocity vector in terms of the Euler angles and their time derivatives, and they write the following about infinitesimal rotations: "The general infinitesimal rotation associated with ω can be considered as consisting of three successive infintesimal rotations [associated with the Euler angles] ... "
But these rotations associated with the Euler angles will not always be all infinitesimal, even if the resulting rotation is. For example, if the new z-axis (the ς-axis), after the resulting infinitesimal rotation, lies in the old xz-plane, but differs slightly from the old z-axis, then the first Euler angle, φ, will be 90° (or -90°), since the node line must be the old y-axis, and the third Euler angle, ψ, will be approximately -φ. These Euler angles, and the roatations associated with them, are certainly not infinitesimal, although they almost cancel each other out, making the resulting rotation infintesimal.
Only if a unit vector e_z along the z-axis is moved by the resulting infinitesimal rotation by a vector de_z, whose angle to the old yz-plane is infinitesimal, all three Euler angles are infinitesimal.

In general, I think Goldstein et. al. talk too much about infinitesimal roatations. I think it is simpler to look at time derivatives of rotation matrices instead.

 

[D H]

CFirst, at p. 172, the angular velocity vector ω is defined by the relation ωdt = dΩ. The authors write in a footnote: "Note that ω is not the derivative of any vector." (by which they mean a time derivative).
But this is wrong. If ω is a (vector valued) continuous function of time (even weaker conditions suffice), then it follows from the Fundamental Theorem of Calculus, applied to each of its components, that it has a (vector) antiderivative w.r.t time.
Presumably, the authors meant that ω is not the derivative of any physically interesting vector.

I agree with Goldstein here. It's just not a vector. In fact, even ω is not quite a vector. It's a two-form, and by happy coincidence, the one and only finite cartesian space where a NxN two-form has N independent elements is three dimensional space. Angular velocity and momentum look kinda-sorta like 3-vectors in three dimensional space, but that's the only space this happens. As a skew symmetric tensor, angular velocity and momentum look exactly like a NxN two form in any positive integer N.

In a way, an even worse problem arises because in order to integrate you need a representation. The standard representation of angular velocity is to express it in the rotating frame. With this, angular velocity at time t versus at time t+Δt are not commensurate. They are in different frames. You can't add them, so integration doesn't make a bit of sense.

Second, at p. 173, the authors try to express the angular velocity vector in terms of the Euler angles and their time derivatives, and they write the following about infinitesimal rotations: "The general infinitesimal rotation associated with ω can be considered as consisting of three successive infintesimal rotations [associated with the Euler angles] ... "
But these rotations associated with the Euler angles will not always be all infinitesimal, even if the resulting rotation is. For example, if the new z-axis (the ς-axis), after the resulting infinitesimal rotation, lies in the old xz-plane, but differs slightly from the old z-axis, then the first Euler angle, φ, will be 90° (or -90°), since the node line must be the old y-axis, and the third Euler angle, ψ, will be approximately -φ. These Euler angles, and the roatations associated with them, are certainly not infinitesimal, although they almost cancel each other out, making the resulting rotation infinitesimal.

You're looking at it wrong. The bottom paragraph of page 173 and the top half of page 174 are trying to find a mapping from Euler angle rates to angular velocity. This mapping is always well-defined, even in the case of gimbal lock, which is what you are trying to describe. The mapping from angular velocity to Euler rates is ill-defined at gimbal lock conditions, but that's not what Goldstein is trying to describe.

Besides, everyone knows that Euler angles are an ill-defined concept at gimbal lock. Euler angles are evil through and through, but everyone knows that (except the people who insist on using that centuries-old convention).

In general, I think Goldstein et. al. talk too much about infinitesimal roatations. I think it is simpler to look at time derivatives of rotation matrices instead.

That can be made more rigorous with Lie theory. Unfortunately, Goldstein hints at Lie theory multiple times but doesn't seem to want to touch it. Even more unfortunately, the mathematicians who write books and papers on Lie theory completely scoff at the idea of writing a "physics math" treatise dedicated especially to rotations in three space. There's no happy middle ground! I don't think I truly understood rotation until I learned Lie theory, but getting there was not an easy journey. I would have loved a graduate level "physics math" version of Lie theory for SO(3). (In case you didn't catch it, I'm using the term "physics math" pejoratively.)