HallsofIvy said:
And, simply put, physical rotations aren't commutative.
Take a book (a math book, of course!) and hold it in front of you with its front cover toward you. Rotate around the vertical axis to your left (its right side moves toward you, its left side away). Now rotate it around the axis going directly away from you, through the center of the book, clockwise (its top goes down, bottom up). You should now have the book lying on it back with front cover up.
Now, start again with the front cover of the book facing you and do exactly those two rotations in the opposite order. After the first rotation, counter clockwise, you will still have the front cover facing you but with its top to your right. After the second rotation, the front cover will be facing to your left, the top of the book toward you. A completely different position that the first example.
I (and most physicists and engineers, and Euler) look at a sequence of rotations from a slightly different perspective, that of rotations about axes defined in terms of the body being rotated. So, defining some axes, let the x and y axes be on the plane defined by the cover of the book. The +x axis points from the top to the bottom while the +y points from left to right. The +z axis completes a right hand system, so it is coming out of the front cover; in the orientation described by Halls, toward your eyes. The first rotation sequence starts with a +90 degree right hand rule rotation about the book's +x axis followed by a -90 degree rotation about the book's (rotated) +y axis. Reversing that order, a -90 degree rotation about the +y axis followed by a +90 degree rotation about the rotated +x axis leaves the book with the binding down such that if you flipped the book open you could read it (the text would be oriented "the right way").
Regardless of how you look at rotations, rotations in three space are not commutative.
s_mr66 said:
furthermore why we can write angular velocity as vector?
Short answer: Two reasons.
- In the limit of infinitesimally small rotations, rotations in three space are commutative (rotation A followed by rotation B is the same as rotation B followed by rotation A).
- Any sequence of rotations in three space can described in terms of a single rotation about a single axis. In other words, one can always find a single axis rotation that results in that exact same final orientation as does the sequence of rotations. (Note: This concept does not apply in other dimensions; it is special to three space.)
Given two rotations, rotation 1 a rotation of angle θ
1 about axis u
1 and rotation 2 a rotation of angle θ
2 about axis u
2, the single axis rotation that describes rotation 1 followed by rotation 2 will be about the axis
<br />
\vec u_{12} = \cos\frac{\theta_2}2 \sin\frac{\theta_1}2 \hat u_1<br />
+ \cos\frac{\theta_1}2 \sin\frac{\theta_2}2 \hat u_2<br />
+ \sin\frac{\theta_1}2 \sin\frac{\theta_2}2 \hat u_1\times\hat u_2
The single axis rotation corresponding to rotation 2 followed by rotation 1 will be about the axis
<br />
\vec u_{21} = \cos\frac{\theta_2}2 \sin\frac{\theta_1}2 \hat u_1<br />
+ \cos\frac{\theta_1}2 \sin\frac{\theta_2}2 \hat u_2<br />
- \sin\frac{\theta_1}2 \sin\frac{\theta_2}2 \hat u_1\times\hat u_2
Note that the cross product terms are of opposite sign in the two rotation sequences. The single axis rotations will be about different axes. Note also that the other two terms are identical and that this cross product term involves the product of the sines of the two half angles. As both rotation angles θ
1 and θ
2 tend to zero, this cross product term will tend to zero much faster than will the other terms. The two sequences look more and more like one another as the rotation angles get small. The two sequences become identical in the limit of infinitesimally small rotations.
