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I've been going thru the book by Hodges and Roithmayr regarding dynamics theory and I can't seem to prove that: enter image description here

I have tried using multiple substitutions, including the fact that every component of the direction cosine matrix is equal to its cofactor, in multiple, different ways. But I feel like I've been walking in circles. The last attempt I made was, when using the definition of angular velocity:

enter image description here

And then writing every derivative of the coordinate system by substituting its direction cosines and substituting the cofactors inside. This led to another interesting thing:

enter image description here

Which may or may not be related to the proof I'm looking for, but I cannot work out how. My guess is that this would be a pretty standard equation to use, since angular velocity is such a big thing in dynamics, but I could not find it anywhere.

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  • $\begingroup$ I think the "Maxwell–Betti reciprocal work theorem" may help. en.wikipedia.org/wiki/Betti%27s_theorem $\endgroup$
    – r13
    Apr 9 at 23:19
  • $\begingroup$ Could you please explain the $^A\vec\omega^B$ notation? I've never came across it. $\endgroup$
    – NMech
    Apr 10 at 5:39
  • $\begingroup$ @NMech is the angular velocity of a body B in a reference frame A. $\endgroup$ Apr 10 at 13:42
  • $\begingroup$ @r13 I have found the answer, should I post it myself? $\endgroup$ Apr 10 at 13:43
  • $\begingroup$ I don't know if the rule permits that (answer the question of yourself) or not. But since nobody has provided a response with a pointed answer, I think it is considered a good service to the readers who have the same question. Also, it might draw further responses with great insight. So why not, I support posting your findings. $\endgroup$
    – r13
    Apr 10 at 13:52
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Knowing that the derivative of a vector in frame A can be written as:

enter image description here

And similarly, the derivative of the same vector written in the B frame can be written as:

enter image description here

Substituting the second equation in the first one we have:

enter image description here

enter image description here

For the above to be true, either the vector being differentiated is a null vector (which is a trivial solution) or:

enter image description here

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