For the following metric I've been asked to solve its geodesic equations up to the first integrals. $$g=\begin{bmatrix}1 & 0\\0 & \sin^{2}(\theta)\end{bmatrix}$$ So here's what I've done: By squaring I got $$\Big(\frac{d\theta}{ds}\Big)^{2}+\sin^{2}(\theta)\Big(\frac{d\phi}{ds}\Big)^{2}=1$$ which I think is one first integral. Then I calculated the only non vanishing Christoffel symbols which are: $$\Gamma_{\phi\phi}^{\theta}=-\sin(\theta)\cos(\theta)$$ and $$\Gamma_{\theta\phi}^{\phi}=\cot(\theta)$$ which result in the equations:$$\frac{d^{2}\theta}{ds^{2}}-\sin(\theta)\cos(\theta)\Big(\frac{d\phi}{ds}\Big)^{2}=0$$ and $$\frac{d^{2}\phi}{ds^{2}}+\cot(\theta)\frac{d\theta}{ds}\frac{d\phi}{ds}=0$$ Now, if I understand correctly, I have to write the second equation in the form $$\frac{d}{ds}[...]=0 \Leftrightarrow [...]=constant$$ but I can't find a way to do that. $${}$$ So my question is firstly, if I'm going about the right way and, secondly, if I am, how should I proceed? Any help/hint is greatly appreciated.
First integrals for geodesic equation
1
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differential-geometry
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0For one, consider $$\frac{1}{\sin(\theta)} \frac{d}{ds}(\sin(\theta) \frac{d\phi}{ds})=0$$ – 2017-01-09
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0So if I write $sin(\theta)\frac{d\phi}{ds}=R_{0}$, then that's the second integral for the geodaisic and there aren't any others to extract? – 2017-01-09
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1Would [math.se] be a better home for this question? – 2017-01-09