which, by the definition of the cross product, can be written:
In general, the angular velocity in an n-dimensional space is the time derivative of the angular displacement tensor which is a second rank skew-symmetric tensor. This tensor will have n(n-1)/2 independent components and this number is the dimension of the Lie algebra of the Lie group of rotations of an n-dimensional inner product space. It turns out that in three dimensional space angular velocity can be represented by vector because number of independent components is equal to number of dimensions of space.
Angular velocity of a rigid body
In order to deal with the motion of a rigid body, it is best to consider a coordinate system that is fixed with respect to the rigid body, and to study the coordinate transformations between this coordinate and the fixed "laboratory" system. As shown in the figure on the right, the lab system's origin is at point O, the rigid body system origin is at O' and the vector from O to O' is R. A particle (i) in the rigid body is located at point P and the vector position of this particle is Ri in the lab frame, and at position ri in the body frame. It is seen that the position of the particle can be written:
The defining characteristic of a rigid body is that the distance between any two points in a rigid body is unchanging in time. This means that the length of the vector is unchanging. By Euler's rotation theorem, we may replace the vector with where is a rotation matrix and is the position of the particle at some fixed point in time, say t=0. This replacement is useful, because now it is only the rotation matrix which is changing in time and not the reference vector , as the rigid body rotates about point O'. The position of the particle is now written as:
Taking the time derivative yields the velocity of the particle:
where Vi is the velocity of the particle (in the lab frame) and V is the velocity of O' (the origin of the rigid body frame). Since is a rotation matrix its inverse is its transpose. So we substitute :
Continue by taking the time derivitve of :
Applying the formula (AB)T = BTAT:
is the negative of its transpose. Therefore it is a skew symmetric 3x3 matrix. We can therefore take its dual to get a 3 dimensional vector. is called the angular velocity tensor. If we take the dual of this tensor, matrix multiplication is replaced by the cross product. Its dual is called the angular velocity pseudovector, ω.
Substituting ω into the above velocity expression:
It can be seen that the velocity of a point in a rigid body can be divided into two terms - the velocity of a reference point fixed in the rigid body plus the cross product term involving the angular velocity of the particle with respect to the reference point. This angular velocity is the "spin" angular velocity of the rigid body as opposed to the angular velocity of the reference point O' about the origin O.
It is an important point that the spin angular velocity of every particle in the rigid body is the same, and that the spin angular velocity is independent of the choice of the origin of the rigid body system or of the lab system. In other words, it is a physically real quantity which is a property of the rigid body, independent of one's choice of coordinate system. The angular velocity of the reference point about the origin of the lab frame will, however, depend on these choices of coordinate system. It is often convenient to choose the center of mass of the rigid body as the origin of the rigid body system, since a considerable mathematical simplification occurs in the expression for the angular momentum of the rigid body.