The attitudes of constant angular velocity motions

Abstract

A rotation can be represented by a tensor which in turn can be parameterized by an angle of rotation and an axis of rotation. This paper presents the first systematic study of the class of rotation tensors which possess constant angular velocities. In addition to the expected rotations featuring constant axes of rotation and constant angular rates, the existence of two other types of rotation are shown. For these rotations, the time variation of the angle and axis of rotation are coupled by an integrable dynamical system. Although the temporal behavior of the axis and angle of rotation can be complex, it also demonstrated how the rotations can be considered as motions about the angular velocity vector. One of the methods used to show this result is based on an analogue to the geometry of a circular helix. The presence of the class of rotations in rigid body dynamics and twistless helical deformations of rods is also discussed.