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Rewriting Central Force Problem of Black Hole Potential
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[QUOTE="stevendaryl, post: 5696450, member: 372855"] I'm not sure about the motivation for going from the equation in terms of [itex]\dot{r}[/itex] to the second-order equation in terms of [itex]u[/itex], but here's how it could be done: First take a time derivative of your equation for [itex]\dot{r}[/itex]. The right side of the equation is a constant, so that gives: [itex]2 \dot{r} \ddot{r} - F(r) \dot{r} = 0[/itex] where [itex]- F(r)[/itex] is the derivative of that potential-like expression on the left side. Dividing through by [itex]\dot{r}[/itex] and rearranging gives you a second-order equation for [itex]r[/itex]: [itex]\ddot{r} = \frac{1}{2} F(r)[/itex] Now, at this point, you can use a trick from nonrelativistic orbital dynamics. You do a simultaneous change of dependent variable from [itex]r[/itex] to [itex]u = \frac{1}{r}[/itex] and a change of independent variable from [itex]t[/itex] to [itex]\phi[/itex]. [/QUOTE]
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Rewriting Central Force Problem of Black Hole Potential
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