{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "66G7TEZ3JW"
    },
    {
      "field": "uses",
      "target": "YZ4268SK8G"
    }
  ],
  "body": "\n## Summary\n\nThe energy of a path in a [[Riemannian manifold|7CBT4SM83Y]] is half the integral of the squared speed.\nAmong paths with fixed ends, its critical points are exactly the [[geodesics|NSYMQMHPDN]] with constant speed.\nIn variational work it is preferred to the length.\nIt is not invariant under reparametrization, so its critical points are better behaved and its Hessian is easier to compute.\nIt is the main example of a function to which Morse theory on infinite-dimensional manifolds applies.\n\n## Explanations\n\n### L1 human\n\nThe length of a path does not care how fast it is travelled; the energy does.\nThe energy is smallest, for a given length and travel time, when the speed is constant.\nSo minimizing energy finds the shortest path and also a uniform way to travel along it.\nFor example, in the plane the path $t \\mapsto (t, 0)$ for $0 \\le t \\le 1$ has energy $\\frac{1}{2}$, while $t \\mapsto (t^2, 0)$ covers the same segment with energy $\\frac{2}{3}$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "58E10",
      "53C22"
    ],
    "name": "Energy functional on paths",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Geodesic"
      }
    ],
    "tag": "Q12AB1MJ64"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/Q12AB1MJ64/",
  "links": null,
  "markdown": "https://tamnd.github.io/math2/node/Q12AB1MJ64.md",
  "msc": [
    "58E10",
    "53C22"
  ],
  "name": "Energy functional on paths",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q12/Q12AB1MJ64.md",
  "status": null,
  "tag": "Q12AB1MJ64",
  "uri": "m2://Q12AB1MJ64",
  "url": "https://tamnd.github.io/math2/wiki/energy-functional-on-paths/",
  "writing": {
    "tag": "Q12AB1MJ64",
    "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
      }
    ]
  }
}