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Liouville's theorem

  1. Oct 4, 2009 #1
    Phase volume is constant.

    [tex]\int_{G_0}dx^0=\int_{G_t}dx^t[/tex]

    [tex]x=(x_1,...,x_{6N})[/tex]

    [tex]\int_{G_0}dx^0=\int_{G_t}dx^t=\int_{G_0}Jdx^0[/tex]

    We must prove that [tex]J=1[/tex]

    [tex]J=\frac{\partial (x_1^t,...,x_{6N}^t)}{\partial (x_1^0,...,x_{6N}^0)}[/tex]

    [tex]J[/tex] is determinant with elements

    [tex]a_{ik}=\frac{\partial x_i^t}{\partial x_k^0}[/tex]

    Minor for [tex]a_{ik}[/tex] is

    [tex]D_{ik}=\frac{\partial J}{\partial a_{ik}}[/tex]

    And now [tex]J[/tex] is define like

    [tex]J=\sum_{k}D_{ik}a_{ik}[/tex]

    Why is define like this? Why not

    [tex]J=\sum_{i,k}D_{ik}a_{ik}[/tex] ?
     
  2. jcsd
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