- #1
jostpuur
- 2,116
- 19
I'm trying to use an example to make sense out of the equation
[tex]
\mathcal{L}_X = d\circ i_X + i_X\circ d.
[/tex]
Some simple equations:
[tex]
\omega = \omega^1 dx_1 + \omega^2 dx_2
[/tex]
[tex]
i_X\omega = X_1\omega^1 + X_2\omega^2
[/tex]
[tex]
(d\omega)^{11} = (d\omega)^{22} = 0,\quad (d\omega)^{12} = \frac{1}{2}(\partial_1\omega^2 - \partial_2\omega^1) = - (d\omega)^{21}
[/tex]
[tex]
\mu = \mu^{12} dx_1\wedge dx_2 + \mu^{21} dx_2\wedge dx_1
[/tex]
[tex]
(i_X\mu)^1 = X_1 \mu^{11} + X_2 \mu^{21} = X_2 \mu^{21},\quad\quad (i_X\mu)^2 = X_1 \mu^{12}
[/tex]
[tex]
\eta = \eta^0
[/tex]
[tex]
(d\eta)^1 = \partial_1 \eta^0,\quad\quad (d\eta)^2 = \partial_2 \eta^0
[/tex]
Applying them:
[tex]
((d\circ i_X)\omega)^1 = \partial_1 (i_X\omega) = \partial_1(X_1\omega^1 + X_2\omega^2)
[/tex]
[tex]
((d\circ i_X)\omega)^2 = \partial_2 (i_X\omega) = \partial_2(X_1\omega^1 + X_2\omega^2)
[/tex]
[tex]
((i_X\circ d)\omega)^1 = X_2 (d\omega)^{21} = -\frac{X_2}{2}(\partial_1\omega^2 - \partial_2\omega^1)
[/tex]
[tex]
((i_X\circ d)\omega)^2 = X_1 (d\omega)^{12} = \frac{X_1}{2}(\partial_1\omega^2 - \partial_2\omega^1)
[/tex]
[tex]
((d\circ i_X \;+\; i_X\circ d)\omega)^1 = (\partial_1 X_1)\omega^1 \;+\; (\partial_1 X_2)\omega^2 \;+\; X_1\partial_1\omega^1 \;+\; \frac{X_2}{2}\partial_1\omega^2 \;+\; \frac{X_2}{2}\partial_2\omega^1
[/tex]
[tex]
((d\circ i_X \;+\; i_X\circ d)\omega)^2 = (\partial_2 X_1)\omega^1 \;+\; (\partial_2 X_2)\omega^2 \;+\; \frac{X_1}{2}\partial_2 \omega^1 \;+\; X_2\partial_2\omega^2 \;+\; \frac{X_1}{2}\partial_1 \omega^2
[/tex]
There exists a following formula for the Lie derivative:
[tex]
(\mathcal{L}_X\omega)^{i_1,\ldots ,i_k} = X\cdot \omega^{i_1,\ldots ,i_k} \;+\; \sum_{\alpha = 1}^k (\partial_{i_{\alpha}} X_j) \omega^{i_1,\ldots ,i_{\alpha - 1},j, i_{\alpha + 1}, \ldots , i_k}
[/tex]
In this example it becomes
[tex]
(\mathcal{L}_X\omega)^1 = X_1\partial_1 \omega^1 \;+\; X_2\partial_2\omega^1 \;+\; (\partial_1 X_1) \omega^1 \;+\; (\partial_1 X_2) \omega^2
[/tex]
[tex]
(\mathcal{L}_X\omega)^2 = X_1\partial_1 \omega^2 \;+\; X_2\partial_2 \omega^2 \;+\; (\partial_2 X_1)\omega^1 \;+\; (\partial_2 X_2) \omega^2
[/tex]
But this starts to look like
[tex]
\mathcal{L}_X \omega \neq (d\circ i_X + i_X\circ d)\omega
[/tex]
Where is this going wrong?
[tex]
\mathcal{L}_X = d\circ i_X + i_X\circ d.
[/tex]
Some simple equations:
[tex]
\omega = \omega^1 dx_1 + \omega^2 dx_2
[/tex]
[tex]
i_X\omega = X_1\omega^1 + X_2\omega^2
[/tex]
[tex]
(d\omega)^{11} = (d\omega)^{22} = 0,\quad (d\omega)^{12} = \frac{1}{2}(\partial_1\omega^2 - \partial_2\omega^1) = - (d\omega)^{21}
[/tex]
[tex]
\mu = \mu^{12} dx_1\wedge dx_2 + \mu^{21} dx_2\wedge dx_1
[/tex]
[tex]
(i_X\mu)^1 = X_1 \mu^{11} + X_2 \mu^{21} = X_2 \mu^{21},\quad\quad (i_X\mu)^2 = X_1 \mu^{12}
[/tex]
[tex]
\eta = \eta^0
[/tex]
[tex]
(d\eta)^1 = \partial_1 \eta^0,\quad\quad (d\eta)^2 = \partial_2 \eta^0
[/tex]
Applying them:
[tex]
((d\circ i_X)\omega)^1 = \partial_1 (i_X\omega) = \partial_1(X_1\omega^1 + X_2\omega^2)
[/tex]
[tex]
((d\circ i_X)\omega)^2 = \partial_2 (i_X\omega) = \partial_2(X_1\omega^1 + X_2\omega^2)
[/tex]
[tex]
((i_X\circ d)\omega)^1 = X_2 (d\omega)^{21} = -\frac{X_2}{2}(\partial_1\omega^2 - \partial_2\omega^1)
[/tex]
[tex]
((i_X\circ d)\omega)^2 = X_1 (d\omega)^{12} = \frac{X_1}{2}(\partial_1\omega^2 - \partial_2\omega^1)
[/tex]
[tex]
((d\circ i_X \;+\; i_X\circ d)\omega)^1 = (\partial_1 X_1)\omega^1 \;+\; (\partial_1 X_2)\omega^2 \;+\; X_1\partial_1\omega^1 \;+\; \frac{X_2}{2}\partial_1\omega^2 \;+\; \frac{X_2}{2}\partial_2\omega^1
[/tex]
[tex]
((d\circ i_X \;+\; i_X\circ d)\omega)^2 = (\partial_2 X_1)\omega^1 \;+\; (\partial_2 X_2)\omega^2 \;+\; \frac{X_1}{2}\partial_2 \omega^1 \;+\; X_2\partial_2\omega^2 \;+\; \frac{X_1}{2}\partial_1 \omega^2
[/tex]
There exists a following formula for the Lie derivative:
[tex]
(\mathcal{L}_X\omega)^{i_1,\ldots ,i_k} = X\cdot \omega^{i_1,\ldots ,i_k} \;+\; \sum_{\alpha = 1}^k (\partial_{i_{\alpha}} X_j) \omega^{i_1,\ldots ,i_{\alpha - 1},j, i_{\alpha + 1}, \ldots , i_k}
[/tex]
In this example it becomes
[tex]
(\mathcal{L}_X\omega)^1 = X_1\partial_1 \omega^1 \;+\; X_2\partial_2\omega^1 \;+\; (\partial_1 X_1) \omega^1 \;+\; (\partial_1 X_2) \omega^2
[/tex]
[tex]
(\mathcal{L}_X\omega)^2 = X_1\partial_1 \omega^2 \;+\; X_2\partial_2 \omega^2 \;+\; (\partial_2 X_1)\omega^1 \;+\; (\partial_2 X_2) \omega^2
[/tex]
But this starts to look like
[tex]
\mathcal{L}_X \omega \neq (d\circ i_X + i_X\circ d)\omega
[/tex]
Where is this going wrong?