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Placeholders in Riemannian manifolds
In non-positive curvature, the path metric and the geodesic equation look like this:
$ds^2 = dr^2 + f^2(r)ds^2_{M}$
$\ddot{r} + f^2(r) \dot{r} = 0$
Is there a way to derive the Riemannian metric from these equations? It looks like the metric is $g = dr^2 + f^2(r)ds_{M}^2$ but I don’t know how to get to the’metric part’ from this. I.e. I don’t know how to get to $\delta = \sqrt{g(\delta,\delta)}$.
It also seems like it should follow that
$$ \dot{r} = \sqrt{\delta(\dot{r},\dot{r})} $$
A:
I was able to answer this by taking the first few steps of the derivation of the geodesic equation above.
It’s true that if the second equation is multiplied by $\dot{r}$ and integrated, we get:
$$ \int \left( \dot{r} \dot{r} + f^2(r) \dot{r}^2 \right) dr = 0 $$
And so, if we take the square root of both sides, we get:
$$ \sqrt{ \dot{r}^2 + f^2(r) \dot{r}^2 } = C$$
However, I cannot see why we should make the change of variables
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What’s New In Right-Click ReplaceThis?
# ReplaceThis is a Windows File Replacer. It allows you to replace files
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