where is the angular velocity vector in the body frame, and maps any vector into skew-symmetric matrix as follows:
Equation (1) follows immediately from the definition of the angular velocity. The angular velocity in the laboratory frame is defined by
But maps coordinates of vectors in the body frame into their coordinates in the laboratory frame. Thus and therefore
But from Eq. (3) we have therefore
and so Eq. (1) follows.
While description of rotations in terms of orthogonal matrices in principle suffices in classical mechanics, sometimes it is convenient to use the group of quaternions of unit norm, the group isomorphic to the group used in quantum mechanical description of half-integer spin particles. Quaternions have some advantages in numerical procedures (as for instance in 3D computer games), but they are also convenient for graphical representation of the intrinsic geometry of the rotation group. And this is what interests us, when we plot trajectories representing history of a spinning rigid body.
Quaternions of unit norm, representing rotations, form a 3-dimensional sphere in 4-dimensional Euclidean space, and we are projecting stereographically this sphere onto our familiar three-dimensional space, where we orient ourselves in a usual way known from everyday experience.
The question therefore arises: how the equation describing the time evolution looks like when represented in the quaternion setting?
To derive it we have to return to the fundamental relation between quaternions and rotations of vectors in space.
For every vector with components denote by the pure imaginary quaternion defined as:
Then to unit quaternion that is such that there corresponds rotation matrix such that for all the following identity holds
where is the commutator. The property follows directly form the definitions and from the quaternion multiplication rules. Every pure imaginary quaternion is of the form for some
for all and all We now differentiate both sides with respect to On the left we use the standard product rule for differentiation, but we pay attention so as to preserve the order, because multiplication of quaternions is non commutative. On the right we enter with the differentiation under the “hat”, because it is a linear operation. We obtain:
Assume now that is a solution of Eq. (1), so that We have
We now use Eq. (12) on the right to obtain
Putting this into Eq. (15) and multiplying both sides on the right by we obtain
From Eq. (16) it follows that the quaternion is pure imaginary: Therefore there exists vector such that
Eq. (19) can now be written as
Since the above holds for any we deduce that
Using (20) we finally obtain:
Note: It may be that the final formula can be derived in a shorter way, but I do not know how. It is this last formula that I was using when verifying that the algorithm provided in Meeting with remarkable circles gives indeed a solution of the evolution equation.