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I Geodesics on a sphere and the christoffel symbols

  1. Apr 29, 2016 #1
    Hi, I recently tried to derive the equations for a geodesic path on a sphere of radius 1 (which are supposed to come out to be a great circle) using the formula [tex]\dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0[/tex] for the geodesic equation, with the metric [tex]ds^2=d\theta^2+\sin^2\theta d\phi^2[/tex]. After solving for the Christoffel symbols and plugging in, I got the system of differential equations [tex]\dfrac{d^2\theta}{dt^2}=\sin\theta\cos\theta \left(\dfrac{d\phi}{dt}\right)^2[/tex] and [tex]\dfrac{d^2\phi}{dt^2}=-2\cot\theta\left(\dfrac{d\phi}{dt}\dfrac{d\theta}{dt}\right)[/tex], but when I plug in the formula for a great circle, [tex]\tan \theta\cos\phi=1[/tex] by making the parametrization [tex]t=\cot\theta=\cos\phi[/tex], it does not satisfy the differential equations. Can anyone explain to me where I've gone wrong?
  2. jcsd
  3. Apr 29, 2016 #2


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    You cannot pick just any parametrisation to satisfy the geodesic equations. You need a parametrisation which fixes the length of the tangent vector.
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