---
tag: A2KQ49Y4NF
kind: concept
name: Poisson manifold
msc: [53D17, 37J39]
specializes: [F45VQ47GPV]
sources:
  - url: https://en.wikipedia.org/wiki/Poisson_manifold
same_as:
  wikidata: Q1464936
  wikipedia: Poisson manifold
---

## Summary

A Poisson manifold is a [[smooth manifold|F45VQ47GPV]] with a bracket on its smooth functions.
The bracket has the algebraic properties of the [[Poisson bracket|2S0MXDT4YH]] of a [[symplectic manifold|CAN490EPRT]].
Every symplectic manifold is a Poisson manifold, but the bracket of a Poisson manifold may be degenerate.
Poisson manifolds are the phase spaces of mechanics with constraints or symmetry, and they are the classical limits that deformation quantization starts from.

## Explanations

### L1 human

In mechanics the Poisson bracket of two observables tells how one changes along the motion generated by the other.
A Poisson manifold keeps only the rules this bracket obeys: it is antisymmetric, satisfies the Jacobi identity, and behaves like a derivative in each slot.
Such a manifold splits into pieces, called symplectic leaves, each of which is an ordinary symplectic manifold.
For example, on $\R^3$ the bracket with $\{x, y\} = z$, $\{y, z\} = x$, $\{z, x\} = y$ comes from the rotation group, and its leaves are the spheres centered at the origin and the origin itself.