- #1

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- 148

*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

**ω**dt = d

**Ω**. 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 d

**e**

_{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.