Larmor precession

From Physics wiki

A particle that possesses a magnetic dipole moment $\boldsymbol{\mu}\,$ experiences torque when placed in a constant magnetic field $\mathbf{B}\,$ , given by

$\boldsymbol{\tau} = \frac{d\mathbf{L}}{dt} = \boldsymbol{\mu}\times\mathbf{B}\,$ .

For a large class of problems, $\boldsymbol{\mu}\,$ is proportional to the angular momentum $\mathbf{L}\,$ via

$\boldsymbol{\mu} = \gamma \mathbf{L}\,$ ,

where $\gamma\,$ is the gyromagnetic ratio, and is classically $\tfrac{q}{2m}\,$ . Thus

$\frac{d\mathbf{L}}{dt} = - \gamma \mathbf{B} \times \mathbf{L}\,$ .

However, from the study of rigid bodies we know that if a body is rotating with an angular velocity $\boldsymbol{\Omega}\,$ , then a constant vector $\mathbf{r}\,$ in the body-fixed frame rotates according to