From the Kinematics of Precession Motion to Generalized Rabi Cycles

We use both vector-parameter and quaternion techniques to provide a thorough description of several classes of rotations, starting with coaxial angular velocity Ω of varying magnitude. Then, we fix the magnitude and let Ω precess at constant rate about the z-axis, which yields a particular solution to the free Euler dynamical equations in the case of axially symmetric inertial ellipsoid. The latter appears also in the description of spin precessions in NMR and quantum computing. As we show below, this problem has analytic solutions for a much larger class of motions determined by a simple condition relating the polar angle and z-projection of Ω (expressed in cylindrical coordinates), which are both time-dependent in the generic case. Relevant physical examples are also provided.