{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "5CTHC6612N"
    },
    {
      "field": "uses",
      "target": "6RE2JMYMA0"
    },
    {
      "field": "uses",
      "target": "82YGJ392SJ"
    },
    {
      "field": "uses",
      "target": "C0RDHFD00Y"
    },
    {
      "field": "uses",
      "target": "CGEXQ8BAH5"
    },
    {
      "field": "uses",
      "target": "EPN6C0X93V"
    },
    {
      "field": "uses",
      "target": "EWJK7NY6ZB"
    },
    {
      "field": "uses",
      "target": "FPXZMN0CH5"
    },
    {
      "field": "uses",
      "target": "HYSCKJN3MG"
    },
    {
      "field": "uses",
      "target": "PZZ4X908Y8"
    },
    {
      "field": "uses",
      "target": "QP0MY1E9RK"
    },
    {
      "field": "uses",
      "target": "RDX55HQ3XT"
    },
    {
      "field": "uses",
      "target": "TAKA1DYBNE"
    },
    {
      "field": "uses",
      "target": "W91M4JS396"
    },
    {
      "field": "uses",
      "target": "WBKCGFMT0P"
    },
    {
      "field": "uses",
      "target": "X7ZSDSXYT5"
    },
    {
      "field": "about",
      "target": "82YGJ392SJ"
    },
    {
      "field": "about",
      "target": "RDX55HQ3XT"
    },
    {
      "field": "defines",
      "target": "B5YP8CY8DQ"
    }
  ],
  "body": "\n## Summary\n\nA symplectic manifold is a smooth manifold with a closed nondegenerate $2$-form, its symplectic form.\nIt is the geometric setting of Hamiltonian mechanics: phase spaces of mechanical systems are symplectic manifolds.\nBy Darboux's theorem all symplectic manifolds of the same dimension look alike near a point, so the subject is global and close to topology.\nEach tangent space is a [[symplectic vector space|PQ7JJY0VPH]], so the dimension is even.\n\n## Explanations\n\n### L1 human\n\nA symplectic form measures oriented area on each little piece of surface in the manifold, rather than lengths.\nIn the plane with coordinates $(q, p)$, the form $dq \\wedge dp$ assigns to a small parallelogram its signed area.\nMechanics uses such a plane: $q$ is the position of a particle on a line and $p$ its momentum.\nFor several particles one takes many such pairs, and the symplectic form adds up the areas in each pair of coordinates.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "kind": "concept",
    "msc": [
      "53D05"
    ],
    "name": "Symplectic manifold",
    "same_as": {
      "wikidata": "Q1064500",
      "wikipedia": "Symplectic manifold"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Symplectic_manifold"
      }
    ],
    "specializes": [
      "F45VQ47GPV"
    ],
    "tag": "CAN490EPRT"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/CAN490EPRT/",
  "links": [
    {
      "field": "specializes",
      "target": "F45VQ47GPV"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/CAN490EPRT.md",
  "msc": [
    "53D05"
  ],
  "name": "Symplectic manifold",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CAN/CAN490EPRT.md",
  "status": null,
  "tag": "CAN490EPRT",
  "uri": "m2://CAN490EPRT",
  "url": "https://math2.tamnd.com/wiki/symplectic-manifold/",
  "writing": {
    "tag": "CAN490EPRT",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}