{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "4HZX2EN319"
    },
    {
      "field": "about",
      "target": "JMVHZ6HDNZ"
    },
    {
      "field": "about",
      "target": "ZRV9T8DD05"
    },
    {
      "field": "uses",
      "target": "4HZX2EN319"
    },
    {
      "field": "uses",
      "target": "JMST03Q37Q"
    },
    {
      "field": "uses",
      "target": "JMVHZ6HDNZ"
    },
    {
      "field": "uses",
      "target": "ZRV9T8DD05"
    },
    {
      "field": "defines",
      "target": "RS9VHTWR4K"
    }
  ],
  "body": "\n## Summary\n\nThe second variation of a [[functional|1EQM0DJ4NC]] at a function $y_0$ is the second derivative of $\\varepsilon \\mapsto J(y_0 + \\varepsilon h)$ at $\\varepsilon = 0$.\nIt plays the role of the second derivative test of ordinary calculus.\nAt a minimiser the [[first variation|5B8NFYB2XQ]] vanishes and the second variation is nonnegative in every direction $h$.\nFor integral functionals it is a quadratic functional in $h$, and its study leads to the Legendre condition, the Jacobi equation and conjugate points.\n\n## Explanations\n\n### L1 human\n\nAt the bottom of a valley the ground is flat, and it curves upward in every direction.\nThe first variation measures the slope of a functional, and the second variation measures its curvature.\nFor $J(y) = \\int_0^1 y'(x)^2 \\, dx$ one gets $J(y_0 + \\varepsilon h) = J(y_0) + 2\\varepsilon \\int_0^1 y_0' h' \\, dx + \\varepsilon^2 \\int_0^1 h'^2 \\, dx$.\nSo the second variation is $2\\int_0^1 h'(x)^2 \\, dx$, which is never negative.\nThis fits the fact that straight lines minimise $J$ among functions with fixed end values.\nThe same idea in Riemannian geometry gives the [[second variation of arc length|ZB91TH6FKV]].\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "49K05",
      "49K10"
    ],
    "name": "Second variation",
    "same_as": {
      "wikidata": "Q137710193",
      "wikipedia": "Second variation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Second_variation"
      }
    ],
    "tag": "4C3BANJ8AH"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/4C3BANJ8AH/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/4C3BANJ8AH.md",
  "msc": [
    "49K05",
    "49K10"
  ],
  "name": "Second variation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/4C3/4C3BANJ8AH.md",
  "status": null,
  "tag": "4C3BANJ8AH",
  "uri": "m2://4C3BANJ8AH",
  "url": "https://tamnd.github.io/math2/wiki/second-variation/",
  "writing": {
    "tag": "4C3BANJ8AH",
    "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
      }
    ]
  }
}