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Imagine you have Newton's Law of gravitation. How would you solve this differential equation. $$ \frac{\mathrm d^2 r(t)}{\mathrm d t^2 } =-\frac{GM}{r(t)^2} $$

I know the solution of this equation is a conic section, but how would you arrive to that? Also, I know that the final result is a function $r(\theta)$ not $r(t)$, but why? I've tried to search on the internet but I feel like they don't explain too much the process. I would like, if possible, a way of understanding each of the steps of the derivation.

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  • $\begingroup$ This video may answer some of your questions. $\endgroup$
    – M. Enns
    Apr 7, 2022 at 13:35
  • $\begingroup$ Check the solutions here: physics.stackexchange.com/q/14700 $\endgroup$
    – Mauricio
    Apr 7, 2022 at 13:44
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    $\begingroup$ Solve for what? Orbits or free fall? $\endgroup$
    – Qmechanic ♦
    Apr 7, 2022 at 15:19
  • $\begingroup$ See Sec. 4.1 here, but you may need to read earlier sections as well. $\endgroup$
    – J.G.
    Apr 7, 2022 at 16:13

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