Relation between vectors in body coordinates and space coordinates

In summary, the equation ##a_{ji}dG_j'=dG_i'## is correct and is used to show that ##dG_i=dG_i'##, but this leads to a contradiction. The mistake lies in the assumption that ##a_{ji}(G_{j2}'-G_{j1}')=G_{i2}-G_{i1}##, which is incorrect.
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Happiness
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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.

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
 
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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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  • #3
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.)
 

Related to Relation between vectors in body coordinates and space coordinates

1. What are vectors in body coordinates and space coordinates?

Vectors in body coordinates and space coordinates are mathematical quantities that have both magnitude and direction. In body coordinates, they represent the position, orientation, and movement of an object relative to its own frame of reference. In space coordinates, they represent the position, orientation, and movement of an object relative to a fixed or global frame of reference.

2. How are vectors in body coordinates and space coordinates related?

Vectors in body coordinates and space coordinates are related through a transformation matrix. This matrix converts a vector from one coordinate system to another, allowing us to describe the same physical quantity in different frames of reference.

3. What is the significance of understanding the relation between vectors in body coordinates and space coordinates?

Understanding the relation between vectors in body coordinates and space coordinates is crucial in many scientific fields, including physics, engineering, and robotics. It allows us to accurately describe and analyze the motion of objects in different frames of reference, as well as design and control complex systems.

4. How do we calculate the transformation matrix for converting vectors between body coordinates and space coordinates?

The transformation matrix for converting vectors between body coordinates and space coordinates can be calculated using a combination of rotation and translation matrices. These matrices depend on the relative orientation and position of the two coordinate systems.

5. Can vectors in body coordinates and space coordinates be used interchangeably?

No, vectors in body coordinates and space coordinates cannot be used interchangeably. They represent different physical quantities in different frames of reference and therefore cannot be directly compared or combined. However, they can be transformed into each other using the appropriate transformation matrix.

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