Symplectic Geometry is a relatively new branch of this old science, Geometry. What is it good for?
If you are interested in the stability of a newly discovered pair of planets in a solar system far, far away. Then you can use the Symplectic Integrator. Correia et al. did just that (below HD45364). Using Hamilton's version of classical mechanics, they were able to integrate the equations of motion of a pair of big planets in the presence of their star.
Old Newton would've been proud of his baby: Classical Mechanics.
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