{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Example\n\nLet $a \u003e 0$ and $r \u003e 0$.\nThe *Bunimovich stadium* is the table $Q \\subset \\R^2$ made of the rectangle $[-a, a] \\times [-r, r]$ and two half-discs of radius $r$.\nThe half-discs are centred at $(-a, 0)$ and $(a, 0)$ and sit on the short sides of the rectangle.\nIts boundary consists of two parallel segments of length $2a$ and two semicircles of radius $r$.\nThese pieces have a common tangent at the four junctions, so the boundary is a $C^1$ curve.\nThe table is compact with piecewise smooth boundary, so it carries a [[billiard|XWMKYE038F]] flow.\n\n## Remarks\n\nBunimovich proved that the billiard in the stadium is ergodic for every $a \u003e 0$.\nThis was a surprise, because the boundary has no dispersing part: the semicircles are focusing and the segments are flat, unlike the [[Sinai billiard|GX9Q2PYNZE]].\nThe mechanism is defocusing: a beam focused by a semicircle passes through a focal point and spreads out again before the next reflection, if the free path is long enough.\nFor $a = 0$ the table is a disc and the billiard is integrable, so the length $a \u003e 0$ of the straight parts is essential.\nThe stadium also has a family of periodic orbits bouncing between the two flat sides; they fill a set of zero measure, so they do not contradict ergodicity.\n\n## Explanations\n\n### L1 human\n\nTake a round table, cut it in half, and pull the halves apart, joining them with two straight sides; the result looks like a running track.\nA ball on a round table keeps the same angle forever, but on the stadium the motion becomes chaotic.\nAlmost every shot eventually explores the whole table, spending time in each region in proportion to its area.\nThe stadium is a standard example in the study of quantum chaos.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "instance_of": [
      "XWMKYE038F"
    ],
    "kind": "example",
    "msc": [
      "37C83",
      "37A25"
    ],
    "name": "Bunimovich stadium",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Dynamical_billiards"
      }
    ],
    "tag": "YYB1JP7TWJ",
    "uses": [
      "XWMKYE038F",
      "GX9Q2PYNZE"
    ]
  },
  "kind": "example",
  "link": "https://tamnd.github.io/math2/node/YYB1JP7TWJ/",
  "links": [
    {
      "field": "instance_of",
      "target": "XWMKYE038F"
    },
    {
      "field": "uses",
      "target": "XWMKYE038F"
    },
    {
      "field": "uses",
      "target": "GX9Q2PYNZE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/YYB1JP7TWJ.md",
  "msc": [
    "37C83",
    "37A25"
  ],
  "name": "Bunimovich stadium",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YYB/YYB1JP7TWJ.md",
  "status": null,
  "tag": "YYB1JP7TWJ",
  "uri": "m2://YYB1JP7TWJ",
  "url": "https://tamnd.github.io/math2/wiki/bunimovich-stadium/",
  "writing": {
    "tag": "YYB1JP7TWJ",
    "kind": "example",
    "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
      }
    ]
  }
}