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Geodesics and their linear equations

  1. Feb 16, 2008 #1
    Hi,

    So I am going over on how to find a geodesic from any metric, esp. on a 2-sphere. I have been looking at my lecture notes and am confused as to how my professor solves for the equation in terms of the variables, i.e. [tex] (\theta , \phi) [/tex].


    If i use the 2-sphere as an example here where the geodesic is given by:

    [tex] \ddot{\phi} = -2\cot \theta \dot{\phi} \dot{\theta} [/tex]

    [tex] \ddot{\theta} = \sin \theta \cos \theta \dot{\phi}^2 [/tex]

    So from the metric we get a first dervative

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

    My prof then goes on and simply states that

    [tex] \frac{1}{\sin^2 \theta} \frac{d}{d\tau} (\sin^2 \theta \dot{\phi}) = 0 [/tex]

    Where does the last equation come from... in lecture he simply stated it and moved on from there with

    [tex] (\sin^2 \theta \dot{\phi}) = l = constant [/tex]

    I really thinking i got a bit rusty on this since mechanics lies two years in the past.


    Thank you very much any help in advance.

    Cheers,

    Biest
     
    Last edited: Feb 16, 2008
  2. jcsd
  3. Feb 16, 2008 #2
    Multiply the first equation with [itex]\sin^2\theta[/itex] and you will arrive to the wanted result.
     
  4. Feb 16, 2008 #3
    Thanks.... I forgot to add one more question... What do we do when we have three geodesics? Just choose one and solve from there?
     
  5. Feb 16, 2008 #4
    Finding the geodesics is a really hard problem. In general you can not find a closed form solution for them. But if you have a killing field, i.e. [itex]\nabla_\alpha\,\xi_\beta+\nabla_\beta\,\xi_\alpha=0[/itex] then you have a constant along the geodesic, i.e. [itex]\xi_\alpha\,u^\alpha=C[/itex] where [itex]u^\alpha[/itex] is tangent to the geodesic.
     
  6. Feb 16, 2008 #5
    I know it is hard... i had to derive the conditions for homework and was stuck on the derivative of [itex] g_{\mu \nu} [/itex] anyway. At the moment we have just done polar coordinates and the 2-sphere. and now we just moved into particle orbits, which i have to work through as well cause i am trying how the Killing vector works there. I am starting to get it, but it is taking me a while already.
     
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