---
tag: A8TS2SNX9Z
kind: concept
name: Symplectomorphism
msc: [53D05, 53D22]
specializes: [0KZZ70ZHXP]
sources:
  - url: https://en.wikipedia.org/wiki/Symplectomorphism
same_as:
  wikidata: Q1049064
  wikipedia: Symplectomorphism
---

## Summary

A symplectomorphism is a diffeomorphism between symplectic manifolds that carries one symplectic form to the other.
These maps are the isomorphisms of symplectic geometry.
In mechanics they are the canonical transformations, and the time-$t$ maps of Hamiltonian flows are symplectomorphisms.
Unlike isometries of a Riemannian manifold, the symplectomorphisms of a symplectic manifold form an infinite-dimensional group.

## Explanations

### L1 human

In the plane, a symplectomorphism is a smooth change of shape that keeps every area the same, with orientation.
A rotation keeps areas, and so does a shear that slides each vertical line up or down.
Stretching by $2$ in one direction and keeping the other direction fixed doubles areas, so it is not a symplectomorphism.
In higher dimensions a symplectomorphism keeps a whole family of areas, one for each pair of coordinates, not only volume.