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Rotation
Let us try to define a rotation vector
whose magnitude
is the angle of the rotation,
, 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.105 shows the effect of applying two successive
rotations,
one about
, and the other about the
, to a standard six-sided die. In the
left-hand case, the
-rotation is applied before the
-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
-rotation plus the
-rotation does not equal
the
-rotation plus the
-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.105:
Effect of successive rotations about perpendicular axes on a six-sided die.
![\begin{figure}
\epsfysize =3in
\centerline{\epsffile{AppendixA/figA.08.eps}}
\end{figure}](img3326.png) |
But, this is not quite the end of the story. Suppose that we take a general vector
and rotate it about
by a small angle
.
This is equivalent to rotating the coordinate axes about the
by
.
According to Equations (A.1280)-(A.1282), we have
![\begin{displaymath}
{\bf a}' \simeq {\bf a} +\delta\theta_z \,{\bf e}_z\times {\bf a},
\end{displaymath}](img3329.png) |
(1305) |
where use has been made of the small angle approximations
and
. The above equation can easily be generalized to allow
small rotations about
and
by
and
,
respectively. We find that
![\begin{displaymath}
{\bf a}' \simeq {\bf a} + \delta \mbox{\boldmath$\theta$}\times {\bf a},
\end{displaymath}](img3333.png) |
(1306) |
where
![\begin{displaymath}
\delta\mbox{\boldmath$\theta$} = \delta\theta_x \,{\bf e}_x + \delta\theta_y \,{\bf e}_y +
\delta\theta_z \,{\bf e}_z.
\end{displaymath}](img3334.png) |
(1307) |
Clearly, we can define a rotation vector, ![$\delta$](img3335.png)
, 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 above equation,
a small
-rotation plus a small
-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,
![\begin{displaymath}
\mbox{\boldmath$\omega$} = \lim_{\delta t\rightarrow 0} \frac{\delta
\mbox{\boldmath$\theta$} }{\delta t},
\end{displaymath}](img3336.png) |
(1308) |
must be a vector as well. Also, if
is interpreted as
in Equation (A.1306) then it follows that the equation of motion of a vector
which precesses about the origin with some angular velocity
is
![\begin{displaymath}
\frac{d {\bf a}}{dt} = \mbox{\boldmath$\omega$}\times {\bf a}.
\end{displaymath}](img3338.png) |
(1309) |
Next: Scalar Triple Product
Up: Vector Algebra and Vector
Previous: Vector Product
Richard Fitzpatrick
2011-03-31