{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c b$ and $A \\in \\R$, and let $L \\colon [a, b] \\times \\R \\times \\R \\to \\R$, written $L(x, z, p)$, be of class $C^1$, with [[partial derivative|09EGC6F8PT]] $L_p$ in $p$.\nConsider the [[integral functional|JRQZ803Q5Y]] $J(y) = \\int_a^b L(x, y, y') \\, dx$ on the class of [[admissible functions|G7T008BEWY]] $y \\in C^1([a, b])$ with $y(a) = A$ and no condition on $y(b)$.\nThe *natural boundary condition* at the free end $b$ is\n$$L_p\\big(b, y(b), y'(b)\\big) = 0.$$\nThe natural boundary condition at a free end $a$ is defined in the same way, with $a$ in place of $b$.\n\n## Remarks\n\nLet $L$ be of class $C^2$ and let $y \\in C^2([a, b])$ minimise $J$ on this class.\nFor $\\eta \\in C^1([a, b])$ with $\\eta(a) = 0$, integration by parts in the [[first variation|5B8NFYB2XQ]] gives\n$$0 = \\delta J(y; \\eta) = \\int_a^b \\Big( L_z - \\frac{d}{dx} L_p \\Big) \\eta \\, dx + L_p\\big(b, y(b), y'(b)\\big) \\, \\eta(b).$$\nVariations with $\\eta(b) = 0$ give the [[Euler–Lagrange equation|PS0VV149ET]], so the integral vanishes; a variation with $\\eta(b) = 1$ then gives the natural boundary condition.\n\nExample: let $J(y) = \\int_0^1 \\big( \\frac{1}{2} y'^2 - y \\big) \\, dx$ with $y(0) = 0$ and $y(1)$ free.\nThe Euler–Lagrange equation is $y'' = -1$ and the natural boundary condition is $y'(1) = 0$.\nThe function $y(x) = x - x^2/2$ satisfies both: $y'' = -1$, $y(0) = 0$ and $y'(1) = 1 - 1 = 0$.\nNon-example: $y(x) = x$ has $y(0) = 0$, but $L_p = y' = 1$ at $x = 1$, so it fails the natural boundary condition.\n\nIn several variables, for $J(u) = \\int_\\Omega \\big( \\frac{1}{2} \\abs{\\nabla u}^2 - f u \\big) \\, dx$ with no condition on $\\partial \\Omega$, the natural boundary condition is $\\partial u / \\partial \\nu = 0$, where $\\nu$ is the outer normal.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "QTRPNFDFJ8",
    "kind": "definition",
    "msc": [
      "49K05",
      "49K10",
      "35A15"
    ],
    "name": "Natural boundary condition",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Calculus_of_variations"
      }
    ],
    "tag": "9NPY7BF6ZJ",
    "uses": [
      "JRQZ803Q5Y",
      "G7T008BEWY",
      "5B8NFYB2XQ",
      "PS0VV149ET",
      "09EGC6F8PT"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/9NPY7BF6ZJ/",
  "links": [
    {
      "field": "defines",
      "target": "QTRPNFDFJ8"
    },
    {
      "field": "uses",
      "target": "JRQZ803Q5Y"
    },
    {
      "field": "uses",
      "target": "G7T008BEWY"
    },
    {
      "field": "uses",
      "target": "5B8NFYB2XQ"
    },
    {
      "field": "uses",
      "target": "PS0VV149ET"
    },
    {
      "field": "uses",
      "target": "09EGC6F8PT"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/9NPY7BF6ZJ.md",
  "msc": [
    "49K05",
    "49K10",
    "35A15"
  ],
  "name": "Natural boundary condition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/9NP/9NPY7BF6ZJ.md",
  "status": null,
  "tag": "9NPY7BF6ZJ",
  "uri": "m2://9NPY7BF6ZJ",
  "url": "https://tamnd.github.io/math2/wiki/natural-boundary-condition-definition/",
  "writing": {
    "tag": "9NPY7BF6ZJ",
    "kind": "definition",
    "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
      }
    ]
  }
}