What is the motivation behind studying smooth manifolds with a non-degenerate closed two-form?

The subject certainly originated from physics, but is there a deeper reason for why it is still an active subject of research in mathematics? What lead mathematicians to put so much effort to understand these objects?

whywe continue to study them is deeper. Somewhat, it is just to see the consequences of a simple condition (e.g., there are actual topological consequences to having a symplectic form on a manifold), somewhat it is because that condition appears in surprising places (e.g., co-adjoint orbits carry a symplectic structure), somewhat it is because there are continuing applications to physics (e.g., understanding quantization of classical systems), but there is more out there. $\endgroup$ – Aaron Apr 29 '16 at 8:04