# I Geodesics on a sphere and the christoffel symbols

1. Apr 29, 2016

### acegikmoqsuwy

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 $$\dfrac{d^2 x^a}{dt^2}+\Gamma^a_{bc} \dfrac{dx^b}{dt}\dfrac{dx^c}{dt}=0$$ for the geodesic equation, with the metric $$ds^2=d\theta^2+\sin^2\theta d\phi^2$$. After solving for the Christoffel symbols and plugging in, I got the system of differential equations $$\dfrac{d^2\theta}{dt^2}=\sin\theta\cos\theta \left(\dfrac{d\phi}{dt}\right)^2$$ and $$\dfrac{d^2\phi}{dt^2}=-2\cot\theta\left(\dfrac{d\phi}{dt}\dfrac{d\theta}{dt}\right)$$, but when I plug in the formula for a great circle, $$\tan \theta\cos\phi=1$$ by making the parametrization $$t=\cot\theta=\cos\phi$$, it does not satisfy the differential equations. Can anyone explain to me where I've gone wrong?

2. Apr 29, 2016

### Orodruin

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