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### Proving the Stability of the Stable Lagrange Points

I'm working on a problem in Kibble's Classical Mechanics (Chapter 11, Problem 12) and came upon a method for exploring the stability of an equilibrium point that I wasn't familiar with. The question ...
### How wide is the $L_4$ and $L_5$ Moon-Earth Lagrangian point?
How far would an object need to be displaced from the $L_4$ and $L_5$ Lagrangian points so that it can escape returning to those points?