A Classical View of the Euler Angles and the Euler Kinematic Equations in NMR

Abstract

The link between spin kinematics and rigid-body kinematics made evident in the recently proposed rotation-operator approach is considered here using a classical picture of spin precession. This link is discussed using both the rotation angle/axis {Φ,n} and the Euler angle {α, β, γ} parametrizations of rotations. These parametrizations are first compared by using the rotation-operator approach to derive the kinematic relations for the Euler–Rodrigues (ER) parameters {cos Φ/2,nsin Φ/2} via the rotation angle/axis {Φ,n} parametrization of the rotation operator. Then, from a classical point of view, a comparison of the rotation angle/axis {Φ,n} and the Euler angle {α, β, γ} parametrizations of the rotation implicit in the Bloch equations is used: (i) to rederive the same kinematic relations obtained via the rotation-operator approach for both the ER parameters and for the Euler angles {α, β, γ} (Euler's kinematic equations), and (ii) to solve Euler's kinematic equations for the time-dependent Euler angles directly without using quadratures, in the case of a time-independent effective field.