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The text I am reading now defined geodesics to be those curves that satisfy the following differential equation:

$\ddot{\gamma}^k(t)+\dot{\gamma}^i(t)\dot{\gamma}^j(t)\Gamma^k_{ij}(\gamma(t)) = 0$

where the $\Gamma^k_{ij}$ are the Christoffel symbols. The text says that through any fixed point in the manifold we can find a unique geodesic with a given velocity vector at that point. The text appeals to the existence and uniqueness theorem for ODEs to support this fact. I get that the usual approach would be to make a substitution like $\nu = \dot{\gamma}$ and then apply the theorem, but this technique doesn't seem to get me anywhere since $\Gamma_{ij}^k(\gamma(t))$ depends on $\gamma$. Can someone help me clear up this confusion?

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You can extend Picard's theorem to higher order derivatives quite easily not by just setting $\nu = \dot{\gamma}$ but by $\nu = (\gamma, \dot{\gamma}) \in TM$. See for more details on Picard's theorem in higher order equations – muzzlator Feb 8 '13 at 0:35
I see how it works now. Thanks! – James Miller Feb 8 '13 at 1:20

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