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\begin{document}

Thomas precession is a consequence of the fact that infinitesimal
lorentz transformations don't commute.

coordinate systems:

\begin{itemize}

\item x-lab

\item x'=\( L(\beta _{i})x \) = particle rest frame at time t

\item \( x''=L(\beta _{i}+\delta \beta _{i}) \)= particle rest frame at time \( t+\delta t \).

\end{itemize}

Now, \( L(\beta _{i}+\delta \beta _{i})L^{-1}(\beta _{i})\neq L(\triangle \beta _{i}) \). It is actually, \( R(\delta \omega )L(\triangle \beta _{i}) \) where \( \delta \omega =\frac{\gamma ^{2}}{\gamma +1}\epsilon _{ijk}\beta _{j}\delta \beta _{k} \) and \( \triangle \beta _{i}=\gamma ^{2}\delta \beta _{parallel}+\gamma \delta \beta _{perp} \). The rate of change of \( \delta \omega  \) is
\( \frac{\gamma ^{2}}{1+\gamma }\frac{\epsilon _{ijk}a_{j}v_{j}}{c^{2}}=-\frac{1}{2}\frac{1}{m^{2}c^{2}r}\frac{dV}{dr}L_{i} \) using \( F_{i}=ma_{i}=\frac{-r_{i}}{r}\frac{dV}{dr} \). The normal precession is:\( \frac{g}{2}\frac{1}{m^{2}c^{2}r}\frac{dV}{dr}L_{i} \) By adding these up, you get
the total angular momentum. Since g=2 for spin, the zeeman effect
is explained.

\end{document}