Knowlpedia

Symplectic manifold

A smooth manifold equipped with a closed, nondegenerate 2-form.

A symplectic manifold is a pair (M,ω)(M,\omega) where MM is a smooth manifold and ω∈Ω2(M)\omega \in \Omega^2(M) is a differential 22-form such that:

  1. Nondegeneracy: for every p∈Mp\in M, the bilinear form ωp:TpM×TpM→R\omega_p:T_pM\times T_pM\to \mathbb{R} is nondegenerate. Equivalently,
    ιvωp=0⟹v=0(v∈TpM),\iota_v \omega_p = 0 \quad\Longrightarrow\quad v=0 \quad (v\in T_pM),
    or equivalently the map TpM→Tp∗MT_pM \to T_p^*M, v↦ιvωpv\mapsto \iota_v\omega_p, is an isomorphism.
  1. Closedness:
    dω=0.d\omega = 0.
Remarks

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

1n! ω∧n≠0everywhere on M.\frac{1}{n!}\,\omega^{\wedge n} \neq 0 \quad \text{everywhere on } M.

Darboux theorem (local normal form)

A fundamental feature of symplectic geometry is that symplectic forms have no local invariants: for each p∈Mp\in M there exist local coordinates (q1,…,qn,p1,…,pn)(q_1,\dots,q_n,p_1,\dots,p_n) near pp such that

ω=∑i=1ndqi∧dpi.\omega = \sum_{i=1}^n dq_i \wedge dp_i .

Hamiltonian vector fields

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

ιXHω=dH.\iota_{X_H}\omega = dH.

Nondegeneracy guarantees XHX_H exists and is unique. The flow of XHX_H preserves ω\omega (it is a flow).

Closedness and Stokes

Because dω=0d\omega=0, Stokes' theorem implies that ω\omega integrates to zero over boundaries: for any suitable 33-chain CC,

∫∂Cω=∫Cdω=0.\int_{\partial C}\omega = \int_C d\omega = 0.

See .