Geodesic proof

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. $$\dot{\phi} = \frac{d\phi}{d\theta}$$

$$ds = \sqrt{r^{2}d\theta^{2} +r^{2}sin^{2}\theta d\phi}$$

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

$$f = \sqrt{1 +sin^{2}\theta \dot{\phi}^{2}}$$

$$\frac{d}{d\theta}\frac{\partial f}{\partial \dot{\phi}} = 0$$

$$\frac{\partial f}{\partial \dot{\phi}} = const = c$$

ok, let's rearrange...

$$\dot{\phi} = \frac{c}{\sqrt{r^{2} - c^{2}sin^{2}\theta}}$$

so let's substitute in s...

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

s = $$\int^{x_{1}}_{x_{2}} r^{2} \frac{d\theta}{r^{2} - c^{2}sin^{2}\theta}$$

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

Answers and Replies

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