Symplectic manifold

A symplectic manifold is a pair $(M,\omega)$ where $M$ is a smooth manifold and $\omega \in \Omega^2(M)$ is a differential $2$-form such that:

1. Nondegeneracy: for every $p\in M$, the bilinear form $\omega_p:T_pM\times T_pM\to \mathbb{R}$ is nondegenerate. Equivalently,

   $$\iota_v \omega_p = 0 \quad\Longrightarrow\quad v=0 \quad (v\in T_pM),$$

   or equivalently the map $T_pM \to T_p^*M$, $v\mapsto \iota_v\omega_p$, is an isomorphism.

2. Closedness:

   $$d\omega = 0.$$

These conditions force $\dim M$ to be even, say $\dim M = 2n$, and $\omega^n$ is nowhere vanishing. In particular, $\omega^n/n!$ is a canonical volume form and orients $M$:

Darboux theorem (local normal form)

A fundamental feature of symplectic geometry is that symplectic forms have no local invariants: for each $p\in M$ there exist local coordinates $(q_1,\dots,q_n,p_1,\dots,p_n)$ near $p$ such that

Hamiltonian vector fields

Given a smooth function $H\in C^\infty(M)$ (a Hamiltonian), the Hamiltonian vector field $X_H$ is defined by

Nondegeneracy guarantees $X_H$ exists and is unique. The flow of $X_H$ preserves $\omega$ (it is a symplectomorphism flow).

Closedness and Stokes

Because $d\omega=0$, Stokes’ theorem implies that $\omega$ integrates to zero over boundaries: for any suitable $3$-chain $C$,

See Stokes' theorem .