Dj. Siminovitch, A CLASSICAL VIEW OF THE EULER ANGLES AND THE EULER KINEMATIC EQUATIONS IN NMR, Journal of magnetic resonance. Series A, 117(2), 1995, pp. 235-245
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.