{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $M$ be a smooth manifold, the configuration space of a mechanical system.\nLet $g$ be a Riemannian metric on $M$, so that the kinetic energy at the velocity $v \\in T_qM$ is $T = \\tfrac12 g_q(v, v)$.\nLet $V \\colon M \\to \\R$ be a smooth potential, so that the motions $q(t)$ solve the Euler–Lagrange equations of $L = T - V$.\nFix an energy value $E$ and let $M_E = \\set{q \\in M : V(q) \u003c E}$.\nFor a curve $\\gamma \\colon [a, b] \\to M_E$ define the *abbreviated action*\n$$S_0[\\gamma] = \\int_a^b \\sqrt{2\\,(E - V(\\gamma(s)))}\\; \\norm{\\gamma'(s)}_g \\, ds.$$\nLet $\\gamma$ be a regular curve in $M_E$.\nThen, up to reparametrisation, $\\gamma$ is the path of a motion with total energy $T + V = E$ if and only if the following holds.\nThe curve $\\gamma$ is a critical point of $S_0$ among curves in $M_E$ with the same endpoints.\n\n## Remarks\n\nThe functional $S_0$ is the length functional of the *Jacobi metric* $2(E - V)\\,g$ on $M_E$.\nSo the paths of motion of energy $E$ are the [[geodesics|NSYMQMHPDN]] of the Jacobi metric.\nThis metric is a conformal change of the kinetic metric $g$ of the [[Riemannian manifold|7CBT4SM83Y]] $(M, g)$.\nAlong a true motion $\\norm{\\dot q}_g = \\sqrt{2(E - V)}$, so $S_0$ equals $\\int 2T\\,dt = \\int p \\cdot dq$, the classical form of the abbreviated action.\nThe proof compares the Euler–Lagrange equations of the length of the Jacobi metric with those of $L$, after the change of time $dt = \\norm{\\gamma'}_g / \\sqrt{2(E - V)}\\, ds$.\nUnlike [[Hamilton's principle|FAQ9CH2Z76]], the variation fixes the energy and leaves the travel time free.\nFor a free particle ($V = 0$), the paths are the geodesics of $g$ itself.\nThe hypothesis $V \u003c E$ is needed: where $V = E$ the Jacobi metric degenerates, and these are the turning points of the motion.\n\n## Explanations\n\n### L1 human\n\nA moving body with fixed total energy picks a path between two points that makes a certain quantity stationary.\nThis quantity adds up, along the path, the speed the body would have at each point times the length travelled there, and it ignores how long the trip takes.\nSo the shape of the path can be found without solving for the timing first.\n\n### L2 human\n\nMaupertuis's principle turns a dynamics problem into a geometry problem.\nOrbits of energy $E$ become geodesics of the Jacobi metric $2(E - V)g$, and tools of Riemannian geometry, such as curvature and comparison theorems, then give information on the stability of orbits.\nIt is derived from the [[stationary-action principle|Q104C67P26]] by fixing the energy instead of the time.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "PQ25NHV0KS"
    ],
    "kind": "statement",
    "msc": [
      "70H30",
      "49S05"
    ],
    "name": "Maupertuis's principle",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q644922",
      "wikipedia": "Maupertuis's principle"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Maupertuis%27s_principle"
      }
    ],
    "tag": "ZXY0AYKMXZ",
    "uses": [
      "7CBT4SM83Y",
      "NSYMQMHPDN",
      "PQ25NHV0KS"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/ZXY0AYKMXZ/",
  "links": [
    {
      "field": "about",
      "target": "PQ25NHV0KS"
    },
    {
      "field": "uses",
      "target": "7CBT4SM83Y"
    },
    {
      "field": "uses",
      "target": "NSYMQMHPDN"
    },
    {
      "field": "uses",
      "target": "PQ25NHV0KS"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/ZXY0AYKMXZ.md",
  "msc": [
    "70H30",
    "49S05"
  ],
  "name": "Maupertuis's principle",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZXY/ZXY0AYKMXZ.md",
  "status": null,
  "tag": "ZXY0AYKMXZ",
  "uri": "m2://ZXY0AYKMXZ",
  "url": "https://tamnd.github.io/math2/wiki/maupertuiss-principle/",
  "writing": {
    "tag": "ZXY0AYKMXZ",
    "kind": "statement",
    "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
      }
    ]
  }
}