Relation between vectors in body coordinates and space coordinates

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Discussion Overview

The discussion revolves around the relationship between vectors in body coordinates and space coordinates, specifically focusing on the transformation equations involving rotation matrices and their implications. The scope includes mathematical reasoning and conceptual clarification regarding the validity of certain equations in the context of vector transformations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant questions the validity of the equation ##a_{ji}dG_j'=dG_i'##, suggesting it does not make sense in the context of vector transformations.
  • Another participant proposes an alternative equation, ##dG_i=dG_i '+ da_{ji} G_j '##, arguing that the original equation may contain an error and that the author intended to convey that ##dG_i## is not equal to ##dG_i'##.
  • Some participants assert that the equation is not a mistake, noting that it was used correctly in subsequent equations (4.84 and 4.85), and suggest that ##a_{ji}=\delta_{ji}## might clarify the situation.
  • One participant presents a contradiction derived from the equation, leading to a reevaluation of the transformation and identifying a mistake in the application of the equation.

Areas of Agreement / Disagreement

Participants express differing views on the correctness of the equations presented, with some asserting that the original equation is valid while others believe it contains errors. The discussion remains unresolved regarding the implications of the transformations and the validity of the equations.

Contextual Notes

There are limitations in the assumptions made about the transformation matrices and their derivatives, as well as the specific conditions under which the equations hold true. The discussion highlights the dependence on definitions and the potential for misinterpretation of the mathematical expressions involved.

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Why is ##a_{ji}dG_j'=dG_i'## ?
from the third last line below.

##G_i=a_{ji}G_j'## because a vector labelled by the space axes is related to the same vector labelled by the body axes via a rotation transformation.

If ##a_{ji}dG_j'=dG_i'##, then we are saying a vector ##dG'## labelled by the body axes is related to the same vector labelled similarly via a rotation transformation. This doesn't make sense.

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They are doing the case where instantaneously the ## a_{ij} ## is the identity matrix, but it will still have a derivative. I think the equation should read, (in this case), ## dG_i=dG_i '+ da_{ji} G_j ' ##. The author was trying to say that ## dG_i ## is not equal to ## dG_i ' ## , but it appeared he might have written down something that isn't correct.
 
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Charles Link said:
They are doing the case where instantaneously the ## a_{ij} ## is the identity matrix, but it will still have a derivative. I think the equation should read, (in this case), ## dG_i=dG_i '+ da_{ji} G_j ' ##. The author was trying to say that ## dG_i ## is not equal to ## dG_i ' ## , but it appeared he might have written down something that isn't correct.

It doesn't seem like the equation is a mistake because he substituted it into (4.84) to get the equation you wrote and (4.85).

I think it's just because ##a_{ji}=\delta_{ji}##.

Screen Shot 2016-04-08 at 12.44.06 pm.png
 
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It seems like using ##a_{ji}dG_j'=dG_i'## I can prove that ##dG_i=dG_i'##, a contradiction.

Let ##G_{j1}'## and ##G_{j2}'## be the vector ##G## at time ##t=0## and ##t=dt## respectively, labelled using the body axes.

Then, ##dG_j'=G_{j2}'-G_{j1}'##.

##a_{ji}dG_j'=a_{ji}(G_{j2}'-G_{j1}')=G_{i2}-G_{i1}=dG_i##.

Thus, ##dG_i'=dG_i##, a contradiction.

What's wrong?

EDIT: I found the mistake. ##a_{ji}(G_{j2}'-G_{j1}')=G_{i2}-G_{i1}## is wrong.
 
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Happiness said:
It doesn't seem like the equation is a mistake because he substituted it into (4.84) to get the equation you wrote and (4.85).

I think it's just because ##a_{ji}=\delta_{ji}##.

View attachment 98771
Yes, you are correct, and what he wrote is correct. (And yes, like you said, it can even be used to get the equation I wrote.)
 

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