Variational principle & Emden's eqn

[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

[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.

 

[AndyW]

Thank you for sharing your attempt at deriving Emden's equation using a variational principle. I appreciate your efforts and would like to provide some feedback on your approach.

Firstly, it is important to note that Emden's equation is a differential equation that describes the density profile of a self-gravitating, spherically symmetric gas cloud. It is derived from the equations of hydrostatic equilibrium and mass conservation. Therefore, any variational principle that leads to Emden's equation should also be based on these fundamental principles.

In your derivation, you start with the expression for the mass of the cloud, which is written in terms of the density and velocity profile (M = -4πa^3ρ_cξ^2Θ'). However, this expression does not take into account the effects of gravity and pressure, which are crucial in the formation of the density profile described by Emden's equation.

Next, you rewrite the expression as X = M/4πa^3ρ_c + ξ^2Θ', where X is a new variable. This step is not very clear and it is not clear how this new variable relates to the original mass expression.

Finally, you propose the variational principle \delta X = 0, where X is a function of ξ and Θ. However, this principle does not take into account the physical constraints of the system, such as mass conservation and hydrostatic equilibrium. Therefore, it is not a valid approach for deriving Emden's equation.

In order to derive Emden's equation using a variational principle, one would need to start with the fundamental principles of hydrostatic equilibrium and mass conservation, and then use variational techniques to find the density profile that satisfies these principles. This approach has been successfully used by several scientists in the past to derive Emden's equation.

In conclusion, while your attempt at deriving Emden's equation using a variational principle is interesting, it is not a valid approach. I would encourage you to continue exploring different methods and techniques in your scientific pursuits.