{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $Q \\subset \\R^2$ be a [[compact|0TVJZYB6SH]] connected set whose [[boundary|05D5EGMFYV]] $\\partial Q$ is a finite union of smooth curves, called the table.\nA *billiard trajectory* in $Q$ is a path $t \\mapsto q(t)$ in $Q$ with the following properties:\n\n1. Between two times at which $q(t) \\in \\partial Q$, the path is a straight segment run at unit speed.\n2. At a time when $q(t)$ is a smooth point of $\\partial Q$, the velocity after the collision is the mirror image of the velocity before it. The component tangent to $\\partial Q$ is kept and the normal component changes sign.\n\nThe *billiard flow* is the flow on the unit tangent vectors $(q, v)$, $q \\in Q$, $\\abs{v} = 1$, that moves $(q, v)$ along its trajectory.\nThe *billiard map* acts on the pairs $(s, \\theta)$.\nHere $s$ is the arc-length parameter of a point of $\\partial Q$, and $\\theta \\in (0, \\pi)$ is the angle between an inward unit vector and the boundary.\nThe map sends $(s, \\theta)$ to the next collision point and its outgoing angle.\nThese two [[dynamical systems|BBZP1R872N]] are called *dynamical billiards*.\n\n## Remarks\n\nExample: let $Q$ be the unit disc.\nA chord leaving the circle at angle $\\theta$ to the tangent subtends the arc $2\\theta$ and meets the circle again at the same angle $\\theta$.\nSo the billiard map is $(s, \\theta) \\mapsto (s + 2\\theta \\bmod 2\\pi, \\theta)$, and the angle is a conserved quantity.\n\nNon-example: a trajectory that hits a corner of $\\partial Q$, where the boundary has no tangent, is not continued by clause 2.\nSuch trajectories form a set of measure zero and are left out.\n\nThe billiard flow preserves the Liouville measure $dq \\, dv$.\nThe billiard map preserves the [[invariant measure|B232EZ1FMB]] $\\sin \\theta \\, ds \\, d\\theta$.\nThe same definitions make sense for tables in $\\R^n$ and on Riemannian manifolds, such as the flat torus.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "XWMKYE038F",
    "kind": "definition",
    "msc": [
      "37C83",
      "37A05",
      "37D40"
    ],
    "name": "Dynamical billiards",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dynamical_billiards"
      }
    ],
    "tag": "FDWY67RFRT",
    "uses": [
      "0XPR1E85W0",
      "0TVJZYB6SH",
      "05D5EGMFYV",
      "BBZP1R872N",
      "B232EZ1FMB"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/FDWY67RFRT/",
  "links": [
    {
      "field": "defines",
      "target": "XWMKYE038F"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    },
    {
      "field": "uses",
      "target": "0TVJZYB6SH"
    },
    {
      "field": "uses",
      "target": "05D5EGMFYV"
    },
    {
      "field": "uses",
      "target": "BBZP1R872N"
    },
    {
      "field": "uses",
      "target": "B232EZ1FMB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/FDWY67RFRT.md",
  "msc": [
    "37C83",
    "37A05",
    "37D40"
  ],
  "name": "Dynamical billiards",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FDW/FDWY67RFRT.md",
  "status": null,
  "tag": "FDWY67RFRT",
  "uri": "m2://FDWY67RFRT",
  "url": "https://tamnd.github.io/math2/wiki/dynamical-billiards-definition/",
  "writing": {
    "tag": "FDWY67RFRT",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}