I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?



a question about geometric quantization background
What is the classical phase space (by phase space I'll, atleast in this post, mean *classical* phase space)? It is simply the cotangent bundle $\mathrm{T}^{\*}(\mathscr{C})$ of the configuration space $\mathscr{C}$. "... in a lecture Thurston mentioned "Any manifold can be seen as the configuration space of some physical system." (cf. [MO/16460](https://mathoverflow.net/questions/16460/how-to-see-the-phase-space-of-a-physical-system-as-the-cotangent-bundle))". It can be seen as the cotangent bundle because of the Poisson bracket - see [the answer to the above linked question](https://mathoverflow.net/questions/16460/how-to-see-the-phase-space-of-a-physical-system-as-the-cotangent-bundle/16462#16462). Any cotangent bundle of a finite-dimensional smooth manifold (the configuration space, $\mathscr{C}$, is finite-dimensional and smooth) is a symplectic manifold. For a proof, see [this essay, page $7$](http://people.mpim-bonn.mpg.de/haugseng/partIIIessay.pdf). Hence, the classical phase space $\mathrm{T}^{\*}(\mathscr{C})$ is *always* a symplectic manifold.


