Relation between vectors in body coordinates and space coordinates

Happiness · Apr 7, 2016

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.

Printed textbook page with equations 4.81 to 4.84 on rate of change of a vector in space and body coordinates, including d G/dt (space) = d G/dt (body) + omega x G and omega dt = d Omega.

Two 3D coordinate diagrams showing a passive transformation from axes (x, y, z) to rotated axes (x', y', z') with angle Φ, and an active rotation of a rectangular block through angle Φ' with the body's axes labelled.

Printed page of text in a physics textbook introducing the infinitesimal rotation of a rigid body and comparing vector components measured in body axes and space axes.

Charles Link · Apr 7, 2016

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.

Happiness · Apr 7, 2016

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}##.

Textbook page with equations: a_ji = (epsilon-bar)_ij = -epsilon_ij; -epsilon_ij = -epsilon_ijk dOmega_k = epsilon_ikj dOmega_k; dG_i = dG'_i + epsilon_ikj dOmega_k G_j; dG_i = dG'_i + (dOmega x G)_i, labeled (4.85).

Happiness · Apr 8, 2016

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.

Charles Link · Apr 8, 2016

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.)

Relation between vectors in body coordinates and space coordinates

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