Deriving the gravity equation from the action S using calculus of variations

[zhushualbert]

could you please tell me how to obtain gravity equation from gravity field Action S as in detail as possible, especially the derivation of calculus of variations?
or you could point out some papers for me,
and thanks very very very very very very very very very much!
p.s. as in detail as possible! thanks!

 

[dx]

The action is given by the integral S = ∫ L(q,v) dt, whre L is a function of coordinates and velocities called the 'Lagrangian'. Hamilton's principle states that this integral is stationary for the actual motion of the system (or minimum for a sufficiently small segment of the motion). Using the calculus of variations, one can show that this leads to the differential equation

(d/dt)(∂L/∂v) = ∂L/∂q (Euler-Lagrange)

(for each coordinate q and and it's time derivative v).

For two particles of mass m and M, the lagrangian is

$$\mathcal{L} = \frac{mv_1^2}{2} + \frac{Mv_2^2}{2} + \frac{GmM}{|r_1 - r_2|}$$

Use this in the Euler-Lagrange equations to get the differential equations of motion.

 

[zhushualbert]

I kown what you said, what's important for me is the derivation from the gravity field action(Einstain-hiber action) to gravity field equation(einstain eqution).
Thank you all the same!