Rotation

Let us try to define a rotation vector $\theta$ whose magnitude is the angle of the rotation, $\theta$, and whose direction is parallel to the axis of rotation, in the sense determined by a right-hand circulation rule. Unfortunately, this is not a good vector. The problem is that the addition of rotations is not commutative, whereas vector addition is commuative. Figure A.11 shows the effect of applying two successive $90^\circ$ rotations, one about $Ox$, and the other about the $Oz$, to a standard six-sided die. In the left-hand case, the $z$-rotation is applied before the $x$-rotation, and vice versa in the right-hand case. It can be seen that the die ends up in two completely different states. In other words, the $z$-rotation plus the $x$-rotation does not equal the $x$-rotation plus the $z$-rotation. This non-commuting algebra cannot be represented by vectors. So, although rotations have a well-defined magnitude and direction, they are not vector quantities.

Figure A.11: Effect of successive rotations about perpendicular axes on a six-sided die.

But, this is not quite the end of the story. Suppose that we take a general vector $\bf a$ and rotate it about $Oz$ by a small angle $\delta \theta_z$. This is equivalent to rotating the coordinate axes about the $Oz$ by $-\delta\theta_z$. According to Equations (A.20)–(A.22), we have

$${\bf a}' \simeq {\bf a} +\delta\theta_z \,{\bf e}_z\times {\bf a},$$ (1.48)

where use has been made of the small angle approximations $\sin\theta \simeq \theta$ and $\cos\theta\simeq 1$. The previous equation can easily be generalized to allow small rotations about $Ox$ and $Oy$ by $\delta \theta_x$ and $\delta\theta_y$, respectively. We find that

$${\bf a}' \simeq {\bf a} + \delta \mbox{\boldmath$\theta$} \times {\bf a},$$ (1.49)

where

$$\delta \mbox{\boldmath$\theta$} = \delta\theta_x \,{\bf e}_x + \delta\theta_y \,{\bf e}_y + \delta\theta_z \,{\bf e}_z.$$ (1.50)

Clearly, we can define a rotation vector, $\delta$$\theta$, but it only works for small angle rotations (i.e., sufficiently small that the small-angle approximations of sine and cosine are good). According to the previous equation, a small $z$-rotation plus a small $x$-rotation is (approximately) equal to the two rotations applied in the opposite order. The fact that infinitesimal rotation is a vector implies that angular velocity,

$$\mbox{\boldmath$\omega$} = \lim_{\delta t\rightarrow 0} \frac{\delta \mbox{\boldmath$\theta$} }{\delta t},$$ (1.51)

must be a vector as well. Also, if ${\bf a}'$ is interpreted as ${\bf a}(t+\delta t)$ in Equation (A.49) then it follows that the equation of motion of a vector that precesses about the origin with some angular velocity $\omega$ is

$$\frac{d {\bf a}}{dt} = \mbox{\boldmath$\omega$} \times {\bf a}.$$ (1.52)