What role does Equity theory play in the new system?

What role does category theory play in the Lagrangian formalism of classical mechanics?

• I believe it has something to do with transforming (i.e. a natural transformations) the generalized coordinate system, but I am not too clear on the details.

• Answer:

  What role does it play? None. The major theorems about the Lagrangian were discovered and stated quite clearly decades before category theory was invented. Now to be sure, a few results of the Lagrangian formalism can be stated more cleanly with the language of category theory (search for "symplectic category" or better yet, "symplectic geometry") but I have never heard of any new results in the former that were revealed by using the latter. On the other hand, the AMS Bulletin on Symplectic Geometry at http://www.ams.org/journals/bull/1981-05-01/S0273-0979-1981-14911-9/S0273-0979-1981-14911-9.pdf seems  particularly good at explaining why the Lagrangian formalism led to the idea of a symplectic manifold and a category in which these are the morphisms. Symplectic manifolds have themselves been interesting objects to study on their own.