A CLASSICAL VIEW OF THE EULER ANGLES AND THE EULER KINEMATIC EQUATIONS IN NMR

The link between spin kinematics and rigid-body kinematics made eviden t in the recently proposed rotation-operator approach is considered he re using a classical picture of spin precession. This link is discusse d using both the rotation angle/axis {Phi, (n) over cap} and the Euler angle {alpha, beta, gamma} parametrizations of rotations. These param etrizations are first compared by using the rotation-operator approach to derive the kinematic relations for the Euler-Rodrigues (ER) parame ters {cos Phi/2, (n) over cap sin Phi/2} via the rotation angle/axis { Phi, (n) over cap} parametrization of the rotation operator. Then, fro m a classical point of view, a comparison of the rotation angle/axis { Phi, (n) over cap} and the Euler angle {alpha, beta, gamma} parametriz ations of the rotation implicit in the Bloch equations is used: (i) to rederive the same kinematic relations obtained via the rotation-opera tor approach for both the ER parameters and for the Euler angles {alph a, beta, gamma} (Euler's kinematic equations), and (ii) to solve Euler 's kinematic equations for the time-dependent Euler angles directly wi thout using quadratures, in the case of a time-independent effective f ield. (C) 1995 Academic Press, Inc.