Skip to main content
MathOverflow

Return to Question

Typos fixed
Source Link
David Roberts ♦
David Roberts ♦
  • 35.5k
  • 11
  • 124
  • 349

First,we we recall that symplectic manifold is a smooth manifold, M$M$, 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 In Hörmander's classic book ALPDO  (The analysis of partial differential operatorsⅠoperators Ⅰ-Ⅳ  )，he wroted：symplectic he wrote: symplectic geometry pervades a large part of the modern theory of linear partial differential operators with variable coefficients.And And he had devoted the entire chapter ⅩⅩⅠtoⅩⅩⅠ to discuss it.

Now,with with some basic background ( itsits 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.

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.

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.

Source Link
user23078
user23078
  • 1.6k
  • 16
  • 24

Why is symplectic geometry so important in modern PDE ?

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.