- #1
Helios
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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
rewrite this as
but just let
the "variational principle" for Emden's eqn is just
you have to use
this lead straight to
Voila! The stuff in the parenthesis is emden's eqn and must equal zero.
[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.