lie derivative problem/error

[jostpuur]

I'm trying to use an example to make sense out of the equation


\mathcal{L}_X = d\circ i_X + i_X\circ d.


Some simple equations:


\omega = \omega^1 dx_1 + \omega^2 dx_2



i_X\omega = X_1\omega^1 + X_2\omega^2



(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}




\mu = \mu^{12} dx_1\wedge dx_2 + \mu^{21} dx_2\wedge dx_1



(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}




\eta = \eta^0



(d\eta)^1 = \partial_1 \eta^0,\quad\quad (d\eta)^2 = \partial_2 \eta^0


Applying them:


((d\circ i_X)\omega)^1 = \partial_1 (i_X\omega) = \partial_1(X_1\omega^1 + X_2\omega^2)



((d\circ i_X)\omega)^2 = \partial_2 (i_X\omega) = \partial_2(X_1\omega^1 + X_2\omega^2)



((i_X\circ d)\omega)^1 = X_2 (d\omega)^{21} = -\frac{X_2}{2}(\partial_1\omega^2 - \partial_2\omega^1)



((i_X\circ d)\omega)^2 = X_1 (d\omega)^{12} = \frac{X_1}{2}(\partial_1\omega^2 - \partial_2\omega^1)



((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



((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


There exists a following formula for the Lie derivative:


(\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}


In this example it becomes


(\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



(\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


But this starts to look like


\mathcal{L}_X \omega \neq (d\circ i_X + i_X\circ d)\omega


Where is this going wrong?

[haushofer]

For your particular example of \omega being a one-form and X the vector field, I get



[di_{X}+i_{X}d ]\omega = d[\omega(X)] + d\omega (X) \\
= d(\omega_{\mu}X^{\mu}) + 2[\partial_{[\mu}\omega_{\nu]}dx^{\mu}\wedge dx^{\nu}] (X)



This becomes


[\omega_{\mu}\partial_{\nu}X^{\mu} + X^{\mu}\partial_{\nu}\omega_{\mu}]dx^{\nu} + X^{\mu}[\partial_{\mu}\omega_{\nu} - \partial_{\nu}\omega_{\mu}]dx^{\nu}


We see that two terms cancel and we obtain



[\omega_{\mu}\partial_{\nu}X^{\mu} + X^{\mu}\partial_{\mu}\omega_{\nu}]dx^{\nu}



This looks like the Lie-derivative:



\mathcal{L}_{X}\omega = [X^{\nu}\partial_{\nu}\omega_{\mu} + \partial_{\mu}X^{\nu}\omega_{\nu}]dx^{\mu}



If this doesn't clarify your problem I can look at all those explicit index-stuff you wrote down, but hopefully this helps :P

[jostpuur]

(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}



The mistake was here. That was the same thing as


i_X\mu = i_X\big(\mu^{ij} dx_i\wedge dx_j\big) = X_i \mu^{ij} dx_j


when it should have been


i_X\mu = \frac{1}{2}i_X\big((\mu^{ij} - \mu^{ji})dx_i\otimes dx_j\big) = \frac{1}{2} X_i(\mu^{ij} - \mu^{ji})dx_j


I found that when following your calculation, and looking where it started going differently.