I'm looking at geodesics in the Schwarzschild geometry, and have come up against something I cannot prove. I've shown that for a particle moving on a geodesic with $r$ constant and $\theta=\pi/2$ we must have
I'm now trying to show that such an orbit is stable for $r>6M$. I'm pretty sure that I need another equation to show this. Should I go back to the geodesic equation and derive something else? I'd imagine I need to take a perturbative approach, but simply taking $\phi(t)=\phi_0+\epsilon(t)$ is obviously no use to get a condition on $r$!
Any help would be much appreciated!