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Lie Bracket question

  1. Aug 31, 2011 #1

    WannabeNewton

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    1. The problem statement, all variables and given/known data
    Show that if the vector fields [itex]X[/itex] and [itex]Y[/itex] are linear combinations (not necessarily with constant coefficients) of [itex]m[/itex] vector fields that all commute with one another, then the lie bracket of [itex]X[/itex] and [itex]Y[/itex] is a linear combination of the same [itex]m[/itex] vector fields.
    3. The attempt at a solution
    I started off by denoting the [itex]m[/itex] vector fields by [itex]V_{(a)}[/itex] so that [itex][V_{(c)}, V_{(d)}] = 0[/itex] for all [itex]V_{(a)}[/itex]. I wrote the vector fields [itex]X[/itex] and [itex]Y[/itex] as [itex]X = \alpha ^{c}V_{(c)}[/itex] and [itex]Y = \beta ^{d}V_{(d)}[/itex] where [itex]\alpha ,\beta [/itex] are scalar functions. Then, [itex][X, Y] = [\alpha ^{c}V_{(c)}, \beta ^{d}V_{(d)}][/itex] right? I worked out the lie derivative in component form and put it back in abstract form to get [tex][X, Y]= [\alpha ^{c}V_{(c)}, \beta ^{d}V_{(d)}] = \alpha ^{c}V_{(d)}(V_{(c)}\cdot \triangledown \beta ^{d}) - \beta ^{d}V_{(c)}(V_{(d)}\cdot \triangledown \alpha ^{c})[/tex] but I don't see how this helps me at all in showing that [itex][X, Y][/itex] can be written as a linear combination of the [itex]V_{(a)}[/itex]'s (the m vector fields). Help please =D.

    EDIT: I forgot to mention that I am using the Einstein summation convention here so that any repeated letters with one on top and one on bottom indicates summation.
     
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  3. Aug 31, 2011 #2

    WannabeNewton

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    Apparently, [tex][X, Y]= \alpha ^{c}V_{(d)}(V_{(c)}\cdot \triangledown \beta ^{d}) - \beta ^{d}V_{(c)}(V_{(d)}\cdot \triangledown \alpha ^{c})[/tex] is sufficient to conclude that the lie bracket is a linear combination of the m vector fields. I'm not convinced only because, while the expression does contain linear combinations of the m vector fields, the expression doesn't look neat in the slightest. If anyone wants to weigh in I would be very glad but I will assume that the conclusion is correct for now as it agrees with what the text has.
     
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