Solve Mercury's Perihelion Shift with Euler-Lagrange Equation

[LoopQG]

Homework Statement

I need to calculate S in order to calculate the perihelion shift of Mercury. I have found the Euler Lagrange equation to be:

$$G^{\nu\beta}(\partial_{\beta}S)(\partial_{\nu}S)-m^{2}=0$$

Which is a form of the Hamilton-Jacobi equation.

Which my professor tells me is correct. I am just not sure on how to start calculating S. I have been given $$G^{00},G^{11},G^{22},G^{33}$$ Do I just plug those into solve for S? I think I am just confused on how to start I think. Any help appreciated.

 

[Chris A]

space dynamics

I am working out a problem on orbital motion: Thus finding the minimum kinetic energy required for a comet to change the new orbit to a parabola after colliding with another planet of a bigger mass.

My idea is first setting the condition for a parabolic orbit: thus e=1
and using the equation E+Gm/r(1-ro/2r)=0
I am expand these equation further by making a few substitutions can i get help in continuing this? Thanks