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?(adsbygoogle = window.adsbygoogle || []).push({});

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

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!

# I Geodesics on a sphere and the christoffel symbols

Have something to add?

Draft saved
Draft deleted

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