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I apologize for the ambiguity in my title. It was rather difficult to figure out what is the most appropriate title for my questions.

My questions come from chapter 4 and chapter 5 of Goldstein, which are both about rotations.

First regards to the paragraph following equation (4.84), which states $dG_i = a_{ji}dG_j' +da_{ji}G_j'$ where prime coordinates are the body coordinate, and unprimed represents the space coordinate. $a$ represents the transformation matrix from space to body.

The paragraph states

It is no loss of generality to take the space and body axes as instantaneously coincident at the time t. Components in the two systems will then be the same instantaneously , but differentials will not be the same, since the two systems are revolving relative to each other. Thus, $G'_j=G_j$ but $a_{ji}dG_j' = dG_i'$.

I get its reasoning, except the conclusion. I agree that the differentials are different respect to the two coordinate system, but why does that imply $a_{ji}dG_j' = dG_i'$?

Second question I have comes from chapter 5, where it is stated (in section 1),

Any difference in the angular velocity vectors at two arbitrary points must lie along the line joining the two points.

Why is that true?

Qmechanic
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Quantization
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1 Answers1

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Answer to first question:

At the instant being considered, the space and body axes are identical, so at that moment the matrix $a$ that relates the two sets of axes is simply the identity matrix. $dG'$ is a vector, so $a_{ji}dG_j' = dG_i'$ is simply equivalent to the statement that with $I$ the identity matrix and $V$ an arbitrary vector, $IV=V$.

Answer to second question:

Immediately prior to the statement in question, the book has just derived that $(\omega_1 - \omega_2 ) \times R = 0$. Using one possible definition for the cross product, that equation is equivalent to $\left\| (\omega_1 - \omega_2 ) \right\| \left\| R \right\| \sin \theta \ n = 0$, where $n$ is a unit vector perpendicular to the plane containing $(\omega_1 - \omega_2 ) $ and $R$, and $\theta$ is the angle between $(\omega_1 - \omega_2 ) $ and $R$. $n$ is not the null vector, and $R$ is presumed to not be the null vector, so if $(\omega_1 - \omega_2 ) $ is also not the null vector, the only other possible way for that equation to hold is if $\sin \theta = 0$, i.e. $\theta = 0$. I.e. , $(\omega_1 - \omega_2 )$ must be parallel to $R$, which is what the book means by the sentence in question.

Red Act
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  • I have no idea why this answer got a down vote. The only thing I can think of is that perhaps the answer looks totally wrong because $a_{ji}dG_j'$ doesn't look anything like multiplying a matrix by a vector. So for the sake of people not following along in Goldstein, I'll point out that Goldstein uses the Einstein summation convention, which does make the above expression equivalent to multiplying a matrix by a vector. – Red Act Aug 21 '14 at 00:44
  • I do not know the reason for the down-vote. I have read the answer and I think it is helpful. Afterwards, I have checked the relevant parts of Goldstein's book and can acknowledge that the answer is appropriate. So I compensated the down-vote. – Tobias Aug 21 '14 at 07:05
  • Thank you Red Act, your answer was very helpful! I was not the one that gave you a down vote. I want to give you an up vote, but I apparently need more reputations. Thank you sincerely! – Quantization Aug 31 '14 at 22:34