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I've heard that doing numerical integration on the Solar System, to see if it is stable in the long term, is hard. One ends up artifacts of the numerical integration routine. But there is a special method, Symplectic Integration. Doing a web search on Symplectic only gets me advanced maths that I cannot understand.

This video comes at the issue from a coding perspective. Why does a simple trick make the numerical integration work so much better? Because it preserves the Poincare Invariants! And this does seem to be the Symplectic Integration that I wanted to learn about :-)