{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "EEE4MM9N1G"
    },
    {
      "field": "defines",
      "target": "FBTWQEMTPS"
    }
  ],
  "body": "\n## Summary\n\nThe contact structure of a [[jet bundle|HC3VBBCJQP]] is the family of contact forms: the $1$-forms that vanish on every [[prolonged section|BMCCQ8JXNE]].\nTheir common kernel is a distribution, called the contact or Cartan distribution.\nThe contact forms tell the holonomic sections apart from all others.\nA section of the jet bundle is a prolongation exactly when it pulls back every contact form to zero.\nIn the simplest case, the first jet bundle of functions of one variable, it is the standard [[contact structure|HBQ8BZ7XB5]] on $\\R^3$.\n\n## Explanations\n\n### L1 human\n\nIn the jet space with coordinates $(x, u, p)$, the letter $p$ is meant to be the slope $du/dx$.\nThe contact form $du - p\\, dx$ expresses this wish.\nAlong the curve $x \\mapsto (x, s(x), s'(x))$ of a function $s$, this form is zero, since $du = s'(x)\\, dx = p\\, dx$ there.\nAlong a curve where $p$ is not the slope, the form is not zero.\nSo the contact form is a test that a curve in jet space really comes from a function.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58A20",
      "58A30",
      "35A30"
    ],
    "name": "Contact structure on a jet bundle",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Jet_bundle"
      }
    ],
    "tag": "GCS702QT54"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/GCS702QT54/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/GCS702QT54.md",
  "msc": [
    "58A20",
    "58A30",
    "35A30"
  ],
  "name": "Contact structure on a jet bundle",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GCS/GCS702QT54.md",
  "status": null,
  "tag": "GCS702QT54",
  "uri": "m2://GCS702QT54",
  "url": "https://tamnd.github.io/math2/wiki/contact-structure-on-a-jet-bundle/",
  "writing": {
    "tag": "GCS702QT54",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}