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I Lagrange multiplier

  1. Nov 29, 2016 #1
    Hi, I have a question about lagrange multiplier

    Let's say we are given with the following constraints

    Σ{Ni}=N and Σ{NiEi}=total energy. N and total energy are constants by definition.
    if we take the derivative with respect to Nj,

    ∂Σ{Ni}/∂Nj=∂N/∂Nj
    where i=j, ∂Σ{Ni}/∂Nj=1 and ∂N/∂Nj = 0 because N is constant.

    On slide 14, it says ∂N/∂Nj = 0 while ∂Σ{Ni}/∂Nj=1 with the preceding constraint Σ{Ni}=N.
    Then, we can conclude

    ∂Σ{Ni}/∂Nj= ∂N/∂Nj=0.

    This is quite ambiguous to me.
    if we assume we have a constraint x1+x2+x3..+xn=const.

    partial of this constraint with respect to xj will be 1=0 therefore it is contradiction.
    how shoud I interpret this?
     
  2. jcsd
  3. Nov 29, 2016 #2

    ShayanJ

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    Gold Member

    A constraint means that your variables are not independent from each other. So when I have ## x_1+x_2+\dots+x_n=const ##, it means when I decrease one of the ##x##s, another increases to maintain the constant. But the way you differentiate it, you assume they're independent variables which is a wrong assumption.
     
  4. Nov 29, 2016 #3
    hmm I get it thanks .
     
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