(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

[tex]L = R \int_{\theta_1}^{\theta_2} \sqrt{1 + sin^2(\theta ) \phi ' ^ 2} d\theta [/tex]

Use the result to prove that the geodesic (shortest path) between two given points on a sphere is a great circle. [Hint: The integrand f(phi,phi_prime,theta) in the result is independent of phi so the Euler-Lagrange equation reduces to partial_f/partial_phi_prime = c, a constant. This gives you phi_prime as a function of theta. You can avoid doing the final integral by the following trick: There is no loss of generality in choosing your z axis to pass through the point 1. Show that with this choice the constant c is necessarily zero and describe the corresponding geodesics.

2. Relevant equations

[tex]\partial f / \partial x = d/du \partial f / \partial x'[/tex]

3. The attempt at a solution

I am having a bit of difficulty interpreting this problem. Using the Euler-Lagrange equation I get the following:

[tex]\phi ' ^ 2 = C^2 / ( sin^4 (\theta ) - C^2 sin^2(\theta ) [/tex]

The hint kind of confusing me. How can 'c' be zero? If I were to have the z-axis at point 1, I would think that [tex]\theta_1[/tex] = 0 if anything.

Can anyone help guide me on the right path for this problem?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Geodesic Sphere

**Physics Forums | Science Articles, Homework Help, Discussion**