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Ziwebach page 101

  1. Sep 16, 2007 #1
    1. The problem statement, all variables and given/known data

    I am confused about QC 6.4

    What is the difference between [tex] Xdot^{\mu} [/tex] and X. ? Is one just tensor notation and one just matrix notation?

    By the way how to you get the dot on top of the X?

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 16, 2007 #2


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    What do you know, pages 92 up to 115 are not part of the free preview. [tex] \dot{X}^{\mu} [/tex] is probably the tau or sigma derivative of the X^{\mu}, that's my guess.
  4. Sep 16, 2007 #3
    It is the tau derivative. And the X^mu' is the sigma derivative.
  5. Sep 16, 2007 #4
    Equations (6.49) and (6.50) on page 101 are straight forward partial derivatives of equation (6.46).
    [itex]X[/itex] is a 4-vector, as are [itex]\dot{X}[/itex] and [itex]X'[/itex] which are the [itex]\tau[/itex] and [itex]\sigma[/itex] derivatives of [itex]X[/itex] respectively. There are 4 [itex]X^{\mu}[/itex] and each one is one of the 4 components of [itex]X[/itex]. [itex]\dot{X}^{\mu}[/itex] and [itex]X^{\mu\prime}[/itex] are defined in equation (6.40) on page 100.
  6. Sep 16, 2007 #5
    What is confusing me is that [tex]L(\dot{X}^{\mu}, X^{\mu '})[/tex] in 6.46 implies that L is a function of only a single component of [tex]\dot{X}[/tex] and [tex] X^{'} [/tex], but then in the function definition all of the other components appear in the dot products. What is wrong with my thinking? Why is it not [tex]L(\dot{X}, X^{'})[/tex] ?
  7. Sep 16, 2007 #6


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    What do you mean by "simgle component"? The notation implies that L depends on all four [tex] \dot{X}^{\mu} [/tex] and all four [tex] X^{\mu '} [/tex].
  8. Sep 17, 2007 #7
    I agree with you. Zwiebach's notation here is a little 'funny'. He has for equation (6.46)

    [tex]\mathcal{L}(\dot{X}^{\mu},X^{\mu\prime}) = -\frac{T_0}{c}\sqrt{(\dot{X}\cdot{X}') - (\dot{X})^2(X')^2}[/tex]

    Now the rhs makes it clear that this is a function of all 4 components, but the notation on the lhs might be considered ambiguous. Certainly he does not mean to imply that [itex]\mathcal{L}[/itex] is a function of only one of the [itex]X^{\mu}[/itex], because then he would have to tell us which one. Do not let this confuse you, he means all 4. If, like me, you are marking up the margins of your book with notes, then simply cross out the lhs and rewrite it as


    and similarly for equation (6.45). If you do not mark up your book, then simply note that the rhs of (6.46) shows you what he had in mind.
    Last edited: Sep 17, 2007
  9. Sep 17, 2007 #8
    OK. But when he takes the partial derivative with respect to [tex] \dot{X}^{\mu} [/tex], this is really a partial with respect to the component not [tex] \dot{X} [/tex], right?
  10. Sep 17, 2007 #9
    Yes, in fact there are 4 such equations, one for each value of [itex]\mu[/itex]
  11. Sep 17, 2007 #10
    I finished the quick calculation. Thanks.
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