---
tag: CAN490EPRT
kind: concept
name: Symplectic manifold
msc: [53D05]
specializes: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Symplectic_manifold
same_as:
  wikidata: Q1064500
  wikipedia: Symplectic manifold
---

## Summary

A symplectic manifold is a smooth manifold with a closed nondegenerate $2$-form, its symplectic form.
It is the geometric setting of Hamiltonian mechanics: phase spaces of mechanical systems are symplectic manifolds.
By Darboux's theorem all symplectic manifolds of the same dimension look alike near a point, so the subject is global and close to topology.
Each tangent space is a [[symplectic vector space|PQ7JJY0VPH]], so the dimension is even.

## Explanations

### L1 human

A symplectic form measures oriented area on each little piece of surface in the manifold, rather than lengths.
In the plane with coordinates $(q, p)$, the form $dq \wedge dp$ assigns to a small parallelogram its signed area.
Mechanics uses such a plane: $q$ is the position of a particle on a line and $p$ its momentum.
For several particles one takes many such pairs, and the symplectic form adds up the areas in each pair of coordinates.