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4
votes
Applications of Hamiltonian formalism to classical mechanics
E.T. Whittaker's A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge, multiple editions from 1904 onwards) provides applications, particularly to celestial mechanics.
R. Abr …
2
votes
Mechanical systems with their configuration space being a Lie group
If one dimensional $n$ body systems fit your definition of a "naturally occurring mechanical system", then the paper by M.A. Olshanetsky and A.M. Perelomov, "Completely Integrable Hamiltonian Systems …