TITLE: Why is symplectic geometry so important in modern PDE ?
QUESTION [38 upvotes]: First, we recall that symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form, $\omega$, called the symplectic form. The study of symplectic manifolds is called symplectic geometry. In Hörmander's classic book ALPDO (The analysis of partial differential operators Ⅰ-Ⅳ) he wrote: symplectic geometry pervades a large part of the modern theory of linear partial differential operators with variable coefficients. And he had devoted the entire chapter ⅩⅩⅠ to discuss it. 
Now, with some basic background (its origins in the Hamiltonian formulation of classical mechanics), I want to know further that why it would make such a important role in modern analysis (such as fourier integral operators)?
Thanks in advance.

REPLY [4 votes]: As the cotangent space has a natural symplectic structure, we can also phrase the question as "Why is cotangent space so important in modern pde?". A nice answer for this question is given in http://math.berkeley.edu/~mjv/WhereforeCot.pdf.