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 h$\ddot{o}$rmander's classic book ALPDO(The analysis of partial differential operatorsⅠ-Ⅳ )，he wroted：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 ⅩⅩⅠto 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.