1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Classical Mechanics: Minimization of geodesic on a sphere

  1. Oct 23, 2012 #1


    User Avatar

    1. The problem statement, all variables and given/known data
    Use the result (6.41) of Problem 6.1 to prove that the geodesic (shortest path) between two given points on a sphere is a great circle. [Hint: The integrand f(ψ,ψ',θ) in (6.41) is independent of ψ, so the Euler-Lagrange equation reduces to ∂f/ψ' = c, a constant. This gives you ψ' as a function of θ. ou 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
    L = R \int_{\theta_{1} }^{\theta_{2}}\sqrt{1+\sin{\theta}\phi'^{2}}d\theta (6.41)

    \frac{\partial f}{\partial \phi}-\frac{\mathrm{d} }{\mathrm{d} \theta}\frac{\partial f}{\partial \phi'} = 0

    3. The attempt at a solution
    I am able to get down to \phi' = \frac{c}{\sin{\theta}\sqrt{\sin^{2}{\theta}-c^{2}}} by using the Euler-Lagrange equation and finding that ∂f/∂ψ' = c, but I am confused as to what the problem means by choosing the z axis to pass through point 1. The solution from (6.1), which is the L integral was performed in spherical polar coordinates, so where does this z axis come from? Any help getting past \phi' would be helpful. The integration of \phi' looks very difficult which is why I understand why the question recommends a trick; but I do not understand how to go about performing this 'trick'.
    Last edited: Oct 23, 2012
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted