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Math Help - Lie derivative

  1. #1
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    Lie derivative

    Hello,

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

     <br />
L_X (i_Y \alpha)=i_{[X,Y]} \alpha+i_Y (L_X \alpha)<br />

    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
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  2. #2
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    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))
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