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Phil's short draft (dated about 1.11.15) from an appendix on Euler angles and frame computations. It differentiates the basis vectors e_n(t)=R(t)e'_n(t), showing the equation holds in any frame, then sets the time derivative of e'_n to zero and asks why evaluating in S instead of S' is not valid. It ends by defining A' = (dR'/dt)R^-1. Text is a rough template draft with equation references to other sections.
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Footnote. A vector equation valid only in Frame S' so cannot evaluate in Frame S.
Write
en(t) = R(t)e'n(t)
at time t which may be evaluated in any Frame. Then
den(t) ≡ en(t+dt) - en(t)
= R(t+dt)e'n(t+dt) - R(t)e'n(t)
≈ [ R(t) (∂te'n(t)) + (∂tR(t))e'n(t) ] dt . // neglect order (dt)2 etc .
so
(den(t)/dt) = R(t) [∂te'n(t)] + [∂tR(t)] e'n(t) .
This vector equation is valid in any frame. For example.
[(den(t)/dt)]i = R(t)ij[(∂te'n(t))]j + [∂tR(t)]ij (e'n(t))j // Frame S
[(den(t)/dt)]'i = R'(t)ij[(∂te'n(t))]'j+ [∂tR(t)]'ij (e'n(t))'j . // Frame S'
If we set ∂te'n(t) = 0 for all t, then we are assuming we are evaluating in Frame S' and then (G.7.22) becomes
(den(t)/dt)S' = (∂tR(t))e'n(t) // vector equation valid only in Frame S'
[(den(t)/dt)S']'i = [∂tR(t)]'ij (e'n(t))'j . // OK, evaluated in Frame S'
[(den(t)/dt)S']i = [∂tR(t)]ij (e'n(t))j . // NOT OK, evaluated in Frame S [ why not OK? ]
Finally,
[∂tR(t)]' = R'(t+dt) - R'(t) = R(t+dt) - R(t) = dR = dR'
since for example R'(t) = R(t)R(t)R-1(t) = R(t) as in (1.1.35). We end up then with
(den/dt)S'= (dR'/dt) e'n(t) where dR' = R(t+dt) - R(t) // compare (G.7.5)
In Frame S' evaluation this says
[(den/dt)S']'i= (dR'/dt)ij (e'n(t))j
Later we define. A' ≡ [(dR'/dt) R-1(t) ] with A'ij = (dR'/dt)ik [R-1(t)]kj