{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "EMXK9T0X1Q"
    },
    {
      "field": "uses",
      "target": "RB08381QYR"
    },
    {
      "field": "defines",
      "target": "PPTX47481A"
    }
  ],
  "body": "\n## Summary\n\nA field of extremals is a family of solutions of the [[Euler–Lagrange equation|PS0VV149ET]] that covers a region of the $(x, y)$ plane.\nExactly one curve of the family passes through each point of the region.\nThe slopes of the curves define a slope function $p(x, y)$ on the region.\nEmbedding an extremal in such a field is the key step of the Weierstrass theory of sufficient conditions for a strong minimum.\nIt is the classical counterpart of the method of characteristics for the Hamilton–Jacobi equation.\n\n## Explanations\n\n### L1 human\n\nThink of the lines of a flow that fill a strip without crossing, like parallel straight lines.\nFor the length functional $\\int \\sqrt{1 + y'^2} \\, dx$ the extremals are straight lines, and the lines $y = x + c$ for all real $c$ form a field covering the whole plane.\nEach point lies on exactly one of these lines, and the slope function is the constant $p = 1$.\nA pencil of lines through one point outside a region also forms a field on that region; this is a central field.\nWhen a minimiser is surrounded by such a field, one can compare it with every nearby curve, not only with smooth small perturbations.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "49K05"
    ],
    "name": "Field of extremals",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Calculus_of_variations"
      }
    ],
    "tag": "K4DFT3H9BJ"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/K4DFT3H9BJ/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/K4DFT3H9BJ.md",
  "msc": [
    "49K05"
  ],
  "name": "Field of extremals",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/K4D/K4DFT3H9BJ.md",
  "status": null,
  "tag": "K4DFT3H9BJ",
  "uri": "m2://K4DFT3H9BJ",
  "url": "https://tamnd.github.io/math2/wiki/field-of-extremals/",
  "writing": {
    "tag": "K4DFT3H9BJ",
    "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
      }
    ]
  }
}