I am trying to solve the geodesic equations on a unit sphere:
\begin{align} \frac{d^2 \theta}{d \lambda^2} - \sin\theta\cos\theta \frac{d \phi}{d \lambda} \frac{d \phi}{d \lambda} = 0 \\ \frac{d^2 \phi}{d \lambda^2} + 2 \cot\theta \frac{d \theta}{d \lambda} \frac{d \phi}{d \lambda} = 0 \\ \end{align}
The tangent vector I wish to transport is given by $(0,0,1)$ in the $(r,\theta, \phi)$ coordinate system at the initial point $(1,\pi/4, 0)$.
I'm not sure how to go about the problem. Since a great circle isn't traversered in the $\theta = \pi/4$ plane, the circle isn't a geodesic, as can be easily be easily checked. Does this mean I have to solve the two coupled differential equations to find the most general solution of the geodesic? Or is there a simple way to go about the problem given the initial point and the tangent vector?