- #1
TadeusPrastowo
- 21
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Homework Statement
Proof the following:
\frac{\text{d}\boldsymbol\{\mathbf{I}\boldsymbol\}}{\text{d}t} \, \boldsymbol\omega = \boldsymbol\omega \times (\boldsymbol\{\mathbf{I}\boldsymbol\}\,\boldsymbol\omega)
where \boldsymbol\{\mathbf{I}\boldsymbol\} is a tensor: \boldsymbol\{\mathbf{I}\boldsymbol\} = \begin{bmatrix}<br /> I_{1,1} & I_{1,2} & I_{1,3} \\<br /> I_{2,1} & I_{2,2} & I_{2,3} \\<br /> I_{3,1} & I_{3,2} & I_{3,3} \\<br /> \end{bmatrix}
with \text{I}_{i,j} as the following:
\text{I}_{i,j} = \delta_{i,j} \sum_k r_k^2 - r_i\,r_j
and \boldsymbol\omega is a vector:
\begin{bmatrix}\omega_1 \\ \omega_2 \\ \omega_3\end{bmatrix}
Homework Equations
The relevant equation should be on how to perform \frac{\text{d}}{\text{d}t}\left(\begin{bmatrix}<br /> I_{1,1} & I_{1,2} & I_{1,3} \\<br /> I_{2,1} & I_{2,2} & I_{2,3} \\<br /> I_{3,1} & I_{3,2} & I_{3,3} \\<br /> \end{bmatrix}\right).
But, I haven't found an online resource that shows the way. Wikipedia on tensor derivative does not touch on how to perform tensor derivative element-by-element.
The Attempt at a Solution
\begin{align}<br /> \frac{\text{d}}{\text{d}t}\left(\begin{bmatrix}<br /> I_{1,1} & I_{1,2} & I_{1,3} \\<br /> I_{2,1} & I_{2,2} & I_{2,3} \\<br /> I_{3,1} & I_{3,2} & I_{3,3} \\<br /> \end{bmatrix}\right) &= \begin{bmatrix}<br /> \frac{\text{d}\,I_{1,1}}{\text{d}t} & \frac{\text{d}\,I_{1,2}}{\text{d}t} & \frac{\text{d}\,I_{1,3}}{\text{d}t} \\<br /> \frac{\text{d}\,I_{2,1}}{\text{d}t} & \frac{\text{d}\,I_{2,2}}{\text{d}t} & \frac{\text{d}\,I_{2,3}}{\text{d}t} \\<br /> \frac{\text{d}\,I_{3,1}}{\text{d}t} & \frac{\text{d}\,I_{3,2}}{\text{d}t} & \frac{\text{d}\,I_{3,3}}{\text{d}t} \\<br /> \end{bmatrix}<br /> \end{align}
Then, I calculate the above one as follows:
\begin{align}<br /> \frac{\text{d}\text{I}_{i,j}}{\text{d}t} &= \frac{\text{d}}{\text{d}t}\,(\delta_{i,j} \sum_k r_k^2 - r_i\,r_j) \\<br /> &= \frac{\text{d}}{\text{d}t}\,\delta_{i,j} \sum_k r_k^2 - \frac{\text{d}}{\text{d}t}\,r_i\,r_j \\<br /> &= \delta_{i,j} \frac{\text{d}}{\text{d}t}\, \sum_k r_k^2 - \frac{\text{d}}{\text{d}t}\,r_i\,r_j \\<br /> &= \delta_{i,j} \sum_k \frac{\text{d}}{\text{d}t}\,r_k^2 - \frac{\text{d}}{\text{d}t}\,r_i\,r_j \\<br /> &= \delta_{i,j} \sum_k \frac{\text{d}\,r_k^2}{r_k}\frac{\text{d}\,r_k}{\text{d}t} - \frac{\text{d}}{\text{d}t}\,r_i\,r_j \\<br /> &= \delta_{i,j} \sum_k 2\,r_k\,\omega_k - (\omega_i\,r_j + r_i\,\omega_j) \\<br /> \frac{\text{d}\text{I}_{i,j}}{\text{d}t} &= \begin{bmatrix}<br /> 2\,r_2\,\omega_2 + 2\,r_3\,\omega_3 & -(r_1\,\omega_2 + r_2\,\omega_1) & -(r_1\,\omega_3 + r_3\,\omega_1) \\<br /> -(r_2\,\omega_1 + r_1\,\omega_2) & 2\,r_1\,\omega_1 + 2\,r_3\,\omega_3 & -(r_2\,\omega_3 + r_3\,\omega_2) \\<br /> -(r_3\,\omega_1 + r_1\,\omega_3) & -(r_3\,\omega_2 + r_2\,\omega_3) & 2\,r_1\omega_1 + 2\,r_2\,\omega_2<br /> \end{bmatrix}<br /> \end{align}<br />
But, multiplying the result of the differentiation with \boldsymbol\omega does not yield \boldsymbol\omega \times (\boldsymbol\{\mathbf{I}\boldsymbol\}\,\boldsymbol\omega) = \begin{bmatrix}<br /> 0 & -\omega_3 & \omega_2 \\<br /> \omega_3 & 0 & -\omega_1 \\<br /> -\omega_2 & \omega_1 & 0<br /> \end{bmatrix}\left(\begin{bmatrix}<br /> r_2^2 + r_3^2 & -r_1 r_2 & -r_1 r_3 \\<br /> -r_2 r_1 & r_1^2 + r_3^2 & -r_2 r_3 \\<br /> -r_3 r_1 & -r_3 r_2 & r_1^2 + r_2^2<br /> \end{bmatrix}\begin{bmatrix}<br /> \omega_1 \\ \omega_2 \\ \omega_3<br /> \end{bmatrix}\right):
<br /> \begin{align}<br /> \frac{\text{d}\text{I}_{i,j}}{\text{d}t} \, \boldsymbol\omega &= \begin{bmatrix}<br /> 2\,r_2\,\omega_2 + 2\,r_3\,\omega_3 & -(r_1\,\omega_2 + r_2\,\omega_1) & -(r_1\,\omega_3 + r_3\,\omega_1) \\<br /> -(r_2\,\omega_1 + r_1\,\omega_2) & 2\,r_1\,\omega_1 + 2\,r_3\,\omega_3 & -(r_2\,\omega_3 + r_3\,\omega_2) \\<br /> -(r_3\,\omega_1 + r_1\,\omega_3) & -(r_3\,\omega_2 + r_2\,\omega_3) & 2\,r_1\omega_1 + 2\,r_2\,\omega_2<br /> \end{bmatrix}\begin{bmatrix}<br /> \omega_1 \\ \omega_2 \\ \omega_3<br /> \end{bmatrix} \\<br /> &= \begin{bmatrix}<br /> (2\,r_2\,\omega_2 + 2\,r_3\,\omega_3)\,\omega_1 - (r_1\,\omega_2 + r_2\,\omega_1)\,\omega_2 - (r_1\,\omega_3 + r_3\,\omega_1)\,\omega_3 \\<br /> -(r_2\,\omega_1 + r_1\,\omega_2)\,\omega_1 + (2\,r_1\,\omega_1 + 2\,r_3\,\omega_3)\,\omega_2 - (r_2\,\omega_3 + r_3\,\omega_2)\,\omega_3 \\<br /> -(r_3\,\omega_1 + r_1\,\omega_3)\,\omega_1 - (r_3\,\omega_2 + r_2\,\omega_3)\,\omega_2 + (2\,r_1\omega_1 + 2\,r_2\,\omega_2)\,\omega_3<br /> \end{bmatrix}<br /> \end{align}<br />
I also have tried several factorizations on paper but to no avail.
But, I may miss some wonderful factorization tricks.
Or, should a tensor be differentiated element-by-element in another way?
Thank you.
