I am having a little trouble with the following question on geodesics, all that i have done so far is found the Euler-Lagrange equations, but have not been able to go any further. I was hoping maybe someone can help?
Question :
Show that the problem of finding geodesics on a surface $g(x,y,z) = 0$ joining points $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ can be found by obtaining the minimum of $\int_{t_1}^{t_2} \! \sqrt{(\dot x)^2+(\dot y)^2+(\dot z)^2} +\lambda(t)g(x,y,z)\ dt$
Hence find the Geodesics of the following:
The surface of revolution $z=f(\sqrt{x^2+y^2})$