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I Relation between vectors in body coordinates and space coordinates

  1. Apr 7, 2016 #1
    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.

    Screen Shot 2016-04-08 at 1.40.36 am.png

    Screen Shot 2016-04-08 at 2.05.47 am.png

    Screen Shot 2016-04-08 at 2.07.51 am.png
     
    Last edited: Apr 7, 2016
  2. jcsd
  3. Apr 7, 2016 #2

    Charles Link

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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.
     
  4. Apr 7, 2016 #3
    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
     
    Last edited: Apr 8, 2016
  5. Apr 8, 2016 #4
    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.
     
    Last edited: Apr 8, 2016
  6. Apr 8, 2016 #5

    Charles Link

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    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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