Computing the Thomas Precession: Help from PFers Needed!

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SUMMARY

The forum discussion centers on the derivation of the Thomas precession as outlined in "Classical Mechanics" by Goldstein, Poole, and Safko. The user, Francesco, successfully computed the matrix L'' (equation 7.18) but encountered discrepancies when approximating it according to equation 7.19 with γ' set to 1. Specifically, Francesco's calculations yielded γ^2 instead of γ in the third row, prompting a request for clarification on this approximation. Another user suggested that to first order in β, γ^2 can be approximated as γ, which is a valid approach in this context.

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Coelum
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TL;DR
While computing the transformation matrix associated to Thomas precession - as described by Goldstein (7.3) - I cannot reproduce a step in the derivation.
Dear PFer's,
I am reproducing the steps to derive the expression for the Thomas precession, as described in Goldstein/Poole/Safko "Classical Mechanics". Hereafter an excerpt from the book describing the step I am currently working on.
Screenshot from 2022-08-29 17-35-19.png

I have been able to compute the matrix L'' (eqn. 7.18). However, computing its approximation as described above (eqn 7.19 and γ' = 1), I get a different result:

Screenshot from 2022-08-29 17-38-37.png
.
As you can see, I get γ^2 rather than γ in the third row. I cannot find my mistake and I cannot see how γ can approximate γ^2. Any hint?
Thanks a lot in advance,

Francesco
 
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Just an idea: to first order in β, γ^2 ≈ γ since γ is quadratic in β. Sort of dirty trick, but formally correct?
 

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