# Math Help - Lie derivative

1. ## Lie derivative

Hello,

I want to show this equation of the Lie-derivative:

$
L_X (i_Y \alpha)=i_{[X,Y]} \alpha+i_Y (L_X \alpha)
$

whereas L_X is the Lie derivative and X,Y are vector fields.

I try to understand the equation, but i'm hopelessly.

Can you help me by my task.

Regards

2. How do you define the Lie derivative? For any of the (equivalent) definitions, Lie derivative has the property that it commutes with contraction. And $i_Y(\alpha)$ is the contraction of Y and $\alpha$.

Let $C(Y \otimes \alpha)$ denote the contraction $i_Y(\alpha)$, we have
$L_X(i_Y(\alpha))$
= $L_X(C(Y \otimes \alpha))$
commutes with contraction = $C(L_X(Y \otimes \alpha))$
Leibniz rule = $C(L_X(Y) \otimes \alpha + Y \otimes L_X(\alpha))$
linearity = $C(L_X(Y) \otimes \alpha) + C(Y \otimes L_X(\alpha))$
perform contraction = $i_{L_X(Y)}(\alpha) + i_Y (L_X(\alpha))$
= $i_{[X,Y]}(\alpha) + i_Y (L_X(\alpha))$