Geodesic proof

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Homework Statement


proof that shortest path between two points on a sphere is a great circle.

Homework Equations



Euler-Lagrange and variational calculus

The Attempt at a Solution



in sphereical coords:

N.B. [tex]\dot{\phi} = \frac{d\phi}{d\theta}[/tex]

[tex]ds = \sqrt{r^{2}d\theta^{2} +r^{2}sin^{2}\theta d\phi}[/tex]

s = [tex]\int^{x_{1}}_{x_{2}} ds = \int^{x_{1}}_{x_{2}} r \sqrt{1 +sin^{2}\theta \dot{\phi}} d\theta[/tex]

[tex]f = \sqrt{1 +sin^{2}\theta \dot{\phi}^{2}}[/tex]

[tex]\frac{d}{d\theta}\frac{\partial f}{\partial \dot{\phi}} = 0[/tex]

[tex]\frac{\partial f}{\partial \dot{\phi}} = const = c[/tex]

ok, let's rearrange...

[tex]\dot{\phi} = \frac{c}{\sqrt{r^{2} - c^{2}sin^{2}\theta}}[/tex]

so let's substitute in s...

s = [tex] \int^{x_{1}}_{x_{2}} r \sqrt{1 +sin^{2}\theta \frac{c^2}{r^{2} - c^{2}sin^{2}\theta} d\theta[/tex]

s = [tex] \int^{x_{1}}_{x_{2}} r^{2} \frac{d\theta}{r^{2} - c^{2}sin^{2}\theta} [/tex]

but i can't integrate that, so what to do?
 

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