Graduate Computing the Thomas Precession: Help from PFers Needed!

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The discussion focuses on deriving the expression for Thomas precession from the textbook "Classical Mechanics" by Goldstein/Poole/Safko. The user has successfully computed the matrix L'' but encounters an issue when approximating it, resulting in γ^2 instead of γ in the calculations. A suggestion is made that, to first order in β, γ^2 can be approximated as γ, which is considered a valid but informal approach. The user seeks clarification on this approximation and any potential mistakes in their derivation. The conversation emphasizes the nuances of mathematical approximations in classical mechanics.
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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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