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What on Earth does the symplectic form have to do with the physics, or the motion, of such a system?

I was reading in Singers "Symmetry in mechanics" and she wrote about a one dimensional motion, such that the cotangent bundle was a two-dimensional symplectic manifold. She did this by showing that there exists an "area form" which mixes position coordinates and momentum coordinates. But whyyyy?!?! The area form, though it may be a symplectic form, tells me nothing about how such a one-dimensional mechanics problem will turn out/time-develop?

Very angry, please help me :)