What is Symplectic geometry: Definition and 12 Discussions

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.

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1. A Hamiltonian formulation of classical mechanics as symplectic manifold

Hi, in the Hamiltonian formulation of classical mechanics, the phase space is a symplectic manifold. Namely there is a closed non-degenerate 2-form ##\omega## that assign a symplectic structure to the ##2m## even dimensional manifold (the phase space). As explained here Darboux's theorem since...
2. I Integrability of the tautological 1-form

Apologies for potentially being imprecise and clunky, but I'm trying understand integrability of the following Hamiltonian $$H(x,p)=\langle p,f(x) \rangle$$ on a 2n dimensional vector space $$T^{\ast}\mathcal{M} =\mathbb{R}^{2n}.$$ Clearly this is just the 1-form \theta_{(x,p)} =...
3. I Darboux theorem for symplectic manifold

Hi, I am missing the point about the application of Darboux theorem to symplectic manifold case as explained here Darboux Theorem. We start from a symplectic manifold of even dimension ##n=2m## with a symplectic differential 2-form ##w## defined on it. Since by definition the symplectic 2-form...
4. Canonical invariance vs. Lorentz invariance

Homework Statement I have an assignment to prove that specific intensity over frequency cubed is Lorentz invariant. One of the main tasks there is to prove the invariance of phase space d^3q \ d^3p and I am trying to prove it with symplectic geometry. I am following Jorge V. Jose and Eugene J...
5. A Lorentz Invariant Phase Space: Symplectic Geometry

I have an assignment to show that specific intensity over frequency cubed \frac{I}{\nu^3}, is Lorentz invariant and one of the main topics there is to show that the phase space is Lorentz invariant. I did it by following J. Goodman paper, but my professor wants me to show this in another way...
6. I Generalized Momentum is a linear functional of Velocity?

Generalized momentum is covariant while velocity is contravariant in coordinate transformation on configuration space, thus they are defined in the tangent bundle and cotangent bundle respectively. Question: Is that means the momentum is a linear functional of velocity? If so, the way to...
7. A Symplectic geometry of phase space

What is a symplectic manifold or symplectic geometry? (In intuitive terms please) I have a vague understanding that it involves some metric that assigns an area to a position and conjugate momentum that happens to be preserved. What is 'special' about Hamilton's formulation that makes it more...
8. [Symplectic geometry] Show that a submanifold is Lagrangian

Homework Statement Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to...
9. Symplectic Geometry: Learn, Understand & Expert Articles/Books

I am interesting in symplectic geometry now. But I have only little knowledge about it. Can someone show me some materials or courses to learn or understand this subject. I want to know the classic articles and books about symplectic geometry and who are the experts in this field. Thank you .
10. Applications of Symplectic Geometry

I am a mathematics graduate student who is doing research in symplectic geometry (specifically symplectic toric orbifolds, symplectic reduction, Hamiltonian actions of tori in the symplectic category, etc). I often have tried to convince others of the importance of symplectic geometry, so I...
11. Symplectic geometry. What's this?

My training is in mathematics. But during my free time I also try to understand fundamental physics. Recently I came across a material which said that the geometry of classical mechanics is symplectic. I'm not sure of the meaning. It was relating to the Hamiltonian which I'm also not...
12. Symplectic Geometry in Physics: String Theory & Beyond

How crucial is symplectic geometry to fundamental physics? Any examples? I know it is related to string theory.