Symplectic Structures

The Symplectic Form

Symplectic Manifold

Let $M$ be an even dimensional smooth manifold. A symplectic structure on $M$ is a closed non-degenerate differential 2-form $\omega \in {\Omega }^2(M)$, called the symplectic form, such that \newcommand{\d}{\mathrm{d}}\newcommand{\R}{\mathbb{R}}\d\omega=0, and for all nonzero $v\in T_{p}M$, there exists $u\in T_{p}M$ so that $\omega (v,u)\ne 0$. The pair $(M,\omega )$ is called a symplectic manifold. We call a symplectic manifold exact if the symplectic form is exact.

e.g.

• Suppose $(x^1,\dots ,x^n,y^1,\dots ,y^n)$ is a coordinate of ${\mathbb{R}}^{2n}$. Then $\omega ={\sum }_i\text{d}{x}^i\wedge \text{d}{y}^i$ is the standard symplectic form on ${\mathbb{R}}^{2n}$, with exact form $\lambda ={\sum }_i{x}^i\text{d}{y}^i$. Clearly, $\text{d}\omega =0$ by definition of the exterior derivative.
• The cotangent bundle $T^{\ast }M$ of a smooth manifold $M$ has a natural exact symplectic structure. In fact, there is a canonical 1-form on $T^{\ast }M$, which can be defined as follows: suppose $\pi :T^{\ast }M\to M$ is the natural projection that maps any covector to the base point where it is defined. Then the canonical differential 1-form $\lambda \in {\Omega }^1(T^{\ast }M)$ is defined by ${\lambda }_{\alpha }(\xi ):=\alpha (\text{d}\pi (\xi )),\text{for all }\alpha \in T^{\ast }M,\xi \in T_{\alpha }(T^{\ast }M).$It turns out that $\text{d}\lambda$ is a symplectic form on $T^{\ast }M$. In fact, the cotangent bundle is identified as the phase space in classical mechanics.

Symplectomorphism

A symplectomorphism is a diffeomorphism $f:M\to N$ between symplectic manifolds $(M,{\omega }_M)$ and $(N,{\omega }_N)$ such that $f^{\ast }{\omega }_N={\omega }_M$.

Proposition

Given a diffeomorphism $f:M\to N$, there is a canonical diffeomorphism $F:T^{\ast }M\to T^{\ast }N$ such that $F^{\ast }{\lambda }_N={\lambda }_M$, where ${\lambda }_M$ and ${\lambda }_N$ are the canonical 1-forms on $T^{\ast }M$ and $T^{\ast }N$ respectively. Thus $F$ is a symplectomorphism.

Classification of Equivalence for Symplectic Structure

1. $(M,{\omega }_0)$ and $(M,{\omega }_1)$ are strongly isotopic if there is an isotopy ${\rho }_t:M\to M$ such that ${\rho }_1^{\ast }{\omega }_1={\omega }_0;$
2. $(M,{\omega }_0)$ and $(M,{\omega }_1)$ are deformation-equivalent if there is a smooth family ${\omega }_t$ of symplectic forms joining ${\omega }_0$ to ${\omega }_1;$
3. $(M,{\omega }_0)$ and $(M,{\omega }_1)$ are isotopic if they are deformation-equivalent with $[{\omega }_t]$ independent of $t$.

Proposition

Moser Theorem states that on compact manifold, isotopic is equivalent to strongly isotopic.