How Lagrange points solved one of physics' biggest problems

The Lagrange points aren't as simple to derive as one, like me, would think. Although a FBD (Free Body Diagram) is the heart of the solution, there are more variables than one would think for something that looks straight forward.

Here is the math, per my little search. The solution will still require, surprisingly, the iteration process for improved accuracy, or is there a solution without iteration tweaks?


"Science begets knowledge, opinion ignorance.
Helio, of course we are dealing with the infinite number of body problem, but with almost all of these having negligible effect (during our lifetimes and maybe a few billion years beyond). However, eventually, some will have measurable effect. I was thinking of the system: Sun, Jupiter, Saturn, Uranus, Neptune, Galilean satellites and Titan and Triton (and maybe Centauri system) as an example. Bit of math needed there?

Cat ;) ;) ;)
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