An explanation for mathematicians of the three-body problem using a simple example, and the moons of Saturn

I am curious to know a simple example that provide me evidence that there are no solution (maybe chaotic) for the so-called three-body problem, and an explanation from an informative point of view of what means a phrase that I've read in Wikipedia's article dedicated to the Moons of Saturn.

I know from an informative point of view that there is an unsolved problem in physics concerning the motion in a gravitational field of more of two bodies, is the so-called three-body problem, see this Wikipedia. On the other hand Wikipedia's article dedicated to the Moons of Saturn tell us in the first paragraph that Saturn has 62 moons with confirmed orbits.

Question. A) Can you answer as a reference request or with yourself didactic example (I would like to know a simple example, explained from an informative point of view for mathematicians) about an example of motion of three or more bodies where is evidenced that there is no solution, or well it is chaotic?

B) What is the meaning of a confirmed orbit in the quote that I've cited in my second paragraph? It means that with numerical analysis or different evidence (measurement of periods or photographs of such moons) that physics known that a moon has a confirmed orbit or what is my misunderstanding? Many thanks.

I am asking B) because I don't know if the orbits of such moons are solutions of an equation or these are determined from a different way.

• My limited understanding is that in the case of the Saturnian system of moons (or Jovian, or...) the orbits are essentially those of a two-body system (=Saturn + a single moon). The point is that the existing moons have such feeble effects on each other, that those can be handled as minor perturbations to the two-body system orbit. In other words, Saturn is the boss, and the moons just follow the lead. To see chaotic behavior you need 3 bodies of comparable masses. Code/run simulations to see that. The caveat there is that it is not clear how to tell the difference... – Jyrki Lahtonen Jan 30 '18 at 11:43
• (cont'd) between truly chaotic behavior and artefacts produced by approximations you do to make the simulation. I'm afraid I'm not an expert on how to code that well. – Jyrki Lahtonen Jan 30 '18 at 11:45
• Anyway, due to the age of the Solar system we can relatively safely assume that moons indulging in chaotic behavior have escaped, impacted on Saturn, ... whatnot... ages ago. – Jyrki Lahtonen Jan 30 '18 at 11:46
• I don't think I'm worthy to answer this. I was once told here that to make a simulation observing conservation laws (that my simplistic incremental simulations violated when I ran them to the amusement of myself and my son) you would need to use so called symplectic integrators whatever they may be. – Jyrki Lahtonen Jan 30 '18 at 11:55
• It's your decision @JyrkiLahtonen . Now if the moons fall for lack of your explanation it will be your responsibility. – user243301 Jan 30 '18 at 11:57