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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

$(\frac{d\phi}{dt})^2=\frac{M}{r^3}$

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!

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I'm not sure if you are coming from a math or physics background but, for the orbit to be stable you need the force to be pulling you back if you step off the orbit which comes from considering the second derivative of the potential. This collection of results might be useful for you: en.wikipedia.org/wiki/Schwarzschild_geodesics –  Alex R. May 22 '12 at 1:39
    
Thanks - I think I worked it out now! –  Edward Hughes May 22 '12 at 16:05

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