# Variational principle & Emden's eqn

1. Oct 29, 2008

### Helios

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

$$M = - 4 \pi a^{3} \rho_{c} \xi^{2} \Theta'$$​

rewrite this as

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

but just let

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

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

$$\delta X = 0$$​

you have to use

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

this lead straight to

$$\delta X = ( \xi^{2} \Theta'' + 2 \xi \Theta' + \xi ^2 \Theta^{n} ) \delta \xi = 0$$​

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