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- TL;DR Summary
- While computing the transformation matrix associated to Thomas precession - as described by Goldstein (7.3) - I cannot reproduce a step in the derivation. This is a step later than the one described in my previous post "Computing the Thomas precession".

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.

Based on the text, the transformation S_3 -> S_1 should be the composition of a boost on the x" axis

and a boost on the y" axis

which, when composed assuming γ'=1, yield

.

The differences with (7.20) are:

Thanks,

Francesco

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.

Based on the text, the transformation S_3 -> S_1 should be the composition of a boost on the x" axis

and a boost on the y" axis

which, when composed assuming γ'=1, yield

The differences with (7.20) are:

- the '-' sign in elements [1,2], [2,2], [2,1], [3,1]
- the '0' in position [3,2]
- the element in position [2,3], which can be approximated as (dropping the " for readability):

β_{x}β_{y}γ = γ(β_{y}/β_{x})β_{x}^{2}≈ γ(β_{y}/β_{x})β^{2}= γ(β_{y}/β_{x})(1-1/γ^{2}) = (γ-1/γ)β_{y}/β_{x}. Close, but not identical.

Thanks,

Francesco