Quaternion kinematic and dynamic differential equations

Many useful identities pertaining to quaternion multiplications are generalized. Among them multiplicative commutativity is the most powerful. Since quaternion space includes the 3D vector space, the physical quantities related to rotations, such as angular displacement, velocity, acceleration, and momentum, are shown to be vector quaternions, and their expressions in quaternion space are derived. These kinematic and dynamic differential equations are further shown to be invertible due to the fact that they are written in quaternion space, and the highest order term of the rotation parameters can be expressed explicitly in closed form.>