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(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Variational principle & Emden's eqn

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