{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "8REBFQD6E0"
    },
    {
      "field": "uses",
      "target": "CBT8F8S8JT"
    },
    {
      "field": "uses",
      "target": "FABJSZ9D3S"
    }
  ],
  "body": "\n## Problem\n\nLet $a \u003c b$ and $A, B, \\ell \\in \\R$, and let $L, G \\colon [a, b] \\times \\R \\times \\R \\to \\R$, written $L(x, z, p)$ and $G(x, z, p)$, be of class $C^2$.\nSet $J(y) = \\int_a^b L(x, y, y') \\, dx$ and $K(y) = \\int_a^b G(x, y, y') \\, dx$, two [[integral functionals|JRQZ803Q5Y]].\nThe *isoperimetric problem* asks for a minimiser of $J$ among the [[admissible functions|G7T008BEWY]] $y \\in C^1([a, b])$ with $y(a) = A$, $y(b) = B$ and the integral constraint $K(y) = \\ell$.\n\n## Remarks\n\nThe name comes from the classical problem of the largest area enclosed by a curve of given length, the case where $K$ is arc length.\n[[Dido's problem|FABJSZ9D3S]] is a version of it.\n\nThe basic necessary condition is Euler's multiplier rule.\nLet $y_0 \\in C^2([a, b])$ be a minimiser, and suppose that $y_0$ does not solve the [[Euler–Lagrange equation|PS0VV149ET]] of $K$.\nThen there is $\\lambda \\in \\R$ such that $y_0$ solves the Euler–Lagrange equation of $L - \\lambda G$:\n$$\\frac{d}{dx} \\big( L_p - \\lambda G_p \\big) = L_z - \\lambda G_z \\quad \\text{on } [a, b].$$\nIt follows from [[Lagrange multipliers on Banach spaces|Q7QK3PF806]] with constraint space $\\R$.\nIf $y_0$ solves the Euler–Lagrange equation of $K$, the rule must be replaced by one with multipliers $(\\lambda_0, \\lambda) \\ne (0, 0)$ for $\\lambda_0 L - \\lambda G$, where $\\lambda_0 = 0$ is possible.\n\nThe [[catenary problem|8REBFQD6E0]] is an isoperimetric problem: it minimises potential energy at fixed length.\n\n## Explanations\n\n### L1 human\n\nMany problems ask for the best curve when some total amount is fixed: the length of a fence, the mass of a chain, the area under a graph.\nThe fixed amount is written as a second integral, and the problem is to minimise the first integral while the second keeps its value.\nAs for constrained problems in calculus, the answer is found by adding a multiple of the constraint to the functional, with an unknown multiplier.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "PQ25NHV0KS"
    ],
    "history": {
      "first_stated": "1697",
      "posed_by": "Jacob Bernoulli"
    },
    "kind": "problem",
    "msc": [
      "49K05",
      "49K21"
    ],
    "name": "Isoperimetric problem of the calculus of variations",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Calculus_of_variations"
      }
    ],
    "tag": "6FY0KHMK5A",
    "uses": [
      "JRQZ803Q5Y",
      "G7T008BEWY",
      "PS0VV149ET",
      "Q7QK3PF806"
    ]
  },
  "kind": "problem",
  "link": "https://tamnd.github.io/math2/node/6FY0KHMK5A/",
  "links": [
    {
      "field": "about",
      "target": "PQ25NHV0KS"
    },
    {
      "field": "uses",
      "target": "JRQZ803Q5Y"
    },
    {
      "field": "uses",
      "target": "G7T008BEWY"
    },
    {
      "field": "uses",
      "target": "PS0VV149ET"
    },
    {
      "field": "uses",
      "target": "Q7QK3PF806"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6FY0KHMK5A.md",
  "msc": [
    "49K05",
    "49K21"
  ],
  "name": "Isoperimetric problem of the calculus of variations",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6FY/6FY0KHMK5A.md",
  "status": null,
  "tag": "6FY0KHMK5A",
  "uri": "m2://6FY0KHMK5A",
  "url": "https://tamnd.github.io/math2/wiki/isoperimetric-problem-of-the-calculus-of-variations/",
  "writing": {
    "tag": "6FY0KHMK5A",
    "kind": "problem",
    "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
      }
    ]
  }
}