# Solving geodesic problems with Euler-Lagrange equation

This is the question:

Problem B.1 Two cities - Tel-Aviv, Israel and SanDiego, CA - have the same latitude 32 ◦ N, but, diﬀerent longitudes: Tel-Aviv is 34 ◦ E and San-Diego is 117 ◦ W. What is the maximal distance between the great circle arc and the 32 ◦ latitude line? By how much the path along the great circle (geodesic) arc will be shorter than the path along the latitude line? Hint: Spherical triangles (on a sphere with radius 1) satisfy a spherical law of cosines $$\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C)$$.

I don't want the solution. My main problem is I don't know where to begin and where I want to get to. Could you help me with figuring out how to get through such problems?

Euler-Lagrange equation: $$\frac{d}{dx}L_{y'}-L_y=0$$ while $$I[y(x)]=\int_{x_0}^{x_1}L(y',y,x)dx$$

Full description (as our lecturer assures us) is right here, page 2.

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Can you write the Euler-Lagrange equations for the length function? – Alexander Gruber Apr 23 '12 at 1:09
@AlexanderNikolasGruber I had edited the post. – Michael Sazonov Apr 23 '12 at 1:56
Is a 'latitude line' a straight line between the two cities? And if so, why bother with Euler Lagrange to solve the problem? – copper.hat Apr 23 '12 at 4:11
@copper.hat Yes. But the question is what is the maximal distance between geodesic and latitude lines. – Michael Sazonov Apr 23 '12 at 4:37
If I understand correctly, the geodesic, the latitude line and center of the earth all lie on the same 2-dim. plane. The end points of the geodesic define an angle $\alpha$ in this plane, the maximum separation will occur at $\frac{\alpha}{2}$, etc, etc... – copper.hat Apr 23 '12 at 6:03