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Why is the symplectic manifold version of Hamiltonian mechanics used in Newtonian mechanics?
I was sitting around with some friends the other day trying to come up with an example of where the global formulation of Hamilton's Equations is really necessary.
Recall the formulation: Given (M,omega,H) where
- M is a 2n dimensional real manifold
- omega is a symplectic form (closed non-degenerate two form)
- H: M -> RR is a Hamiltonian
Then one finds the Hamiltonian vector field X_H by solving omega(-,X_H) = dH and derives Hamilton's equations by saying that a Hamiltonian flow goes along this vector field.
What is a physical example where this formulation of Hamiltonian Mechanics is really necessary? Did Arnold (or whoever invented this) have some application in mind?
