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Mathematics of classical mechanics, including Hamiltonian mechanics, Lagrangian mechanics, applications of symplectic geometry to mechanics, deterministic chaos, resonance etc.
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Stable equilibria of points on the 2-sphere
This is the famous Thomson problem. You can find a list of optimal configurations and many references on the Wikipedia page. Your intuitions for $n=7, 8, 9, 20$ are wrong, and $n=5$ is not that obvi …