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Variational principle & Emden's eqn

  1. Oct 29, 2008 #1
    I once tried to come up with a variational principle that would lead to Emden's equation. I think this is instructive. Start with the mass

    [tex]M = - 4 \pi a^{3} \rho_{c} \xi^{2} \Theta'[/tex]​

    rewrite this as

    [tex]M / 4 \pi a^{3} \rho_{c} + \xi^{2} \Theta' = 0[/tex]​

    but just let

    [tex]X = M / 4 \pi a^{3} \rho_{c} + \xi^{2} \Theta'[/tex]​

    the "variational principle" for Emden's eqn is just

    [tex]\delta X = 0[/tex]​

    you have to use

    [tex]\delta M = 4 \pi a^{3} \rho \xi^{2} \delta \xi [/tex] and [tex]\rho / \rho_{c} = \Theta^{n}[/tex]​

    this lead straight to

    [tex]\delta X = ( \xi^{2} \Theta'' + 2 \xi \Theta' + \xi ^2 \Theta^{n} ) \delta \xi = 0[/tex]​

    Voila! The stuff in the parenthesis is emden's eqn and must equal zero.
     
  2. jcsd
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