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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 $z=f(\sqrt{x^2+y^2})$

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In your question there are two questions. What is the one which gives you troubles? – Giovanni De Gaetano Nov 23 '12 at 10:00
Well for the first part I have managed to find the Euler-Lagrange and then the respective candidate for the extremal, but I am unable to solve the second part which is asking for the Geodesics for the surface of revolution of $$z=f(\sqrt{x^2+y^2})$$. – M Pat Nov 24 '12 at 20:54

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