Mathematical structure of classical mechanic is described by symplectic geometry which is a smooth manifold with a non-degenrate closed 2-form $\omega$.
I understand the requirement that $\omega$ should be non-degenrate and conservation of energy along flow lines implies $\omega$ to be a 2-form. are not these conditions enough to deal with classical mechanics?
What is the physical (not mathematical) motivation for symplectic form to be closed?
