= particle rest frame at time t
= particle rest frame at time 
.
coordinate systems:
= particle rest frame at time t = particle rest frame at time .
Now, 
. It is actually, 
where 
and 
. The rate of change of 
is
using 
. The normal precession is: 
By adding these up, you get
the total angular momentum. Since g=2 for spin, the zeeman effect
is explained.