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  "body": "\n## Statement\n\nLet $L \\colon \\R^n \\times \\R^n \\to \\R$, written $L(q, v)$, be of class $C^2$.\nLet $(h^s)_{s \\in \\R}$ be the [[flow|06738GGWWK]] of a complete $C^1$ vector field $X$ on $\\R^n$, so that each $h^s$ is a [[diffeomorphism|0KZZ70ZHXP]] and $X(q) = \\frac{d}{ds} h^s(q) \\big|_{s = 0}$.\nAssume that $L$ is invariant under the flow: $L\\big(h^s(q), Dh^s(q) \\, v\\big) = L(q, v)$ for all $s \\in \\R$, $q \\in \\R^n$ and $v \\in \\R^n$.\nThen the function\n$$I(q, v) = \\sum_{i=1}^n \\frac{\\partial L}{\\partial v_i}(q, v) \\, X_i(q)$$\nis a [[first integral|0262JP0NV9]] of the [[Euler–Lagrange equations|PS0VV149ET]] $\\frac{d}{dt} \\frac{\\partial L}{\\partial v}(q, \\dot q) = \\frac{\\partial L}{\\partial q}(q, \\dot q)$.\nThat is, $t \\mapsto I(q(t), \\dot q(t))$ is constant for every $C^2$ solution $q(t)$.\n\n## Remarks\n\nAlong a solution, the Euler–Lagrange equations give $\\frac{d}{dt} I = \\frac{\\partial L}{\\partial q} \\cdot X(q) + \\frac{\\partial L}{\\partial v} \\cdot DX(q) \\, \\dot q$.\nThe right side is the derivative in $s$ at $s = 0$ of $L(h^s(q), Dh^s(q) \\, \\dot q)$, which is $0$ by invariance.\n\nExample: in the plane, let $L(q, v) = \\frac{1}{2} \\abs{v}^2 - U(\\abs{q}^2)$ for a $C^2$ function $U \\colon \\R \\to \\R$.\nIt is invariant under the rotations $h^s$, with $X(q) = (-q_2, q_1)$.\nThe first integral is the angular momentum $I = q_1 v_2 - q_2 v_1$, which is conserved in every central force motion.\n\nA related fact covers time: if $L$ does not depend on $t$, the energy $\\dot q \\cdot \\frac{\\partial L}{\\partial v} - L$ is conserved.\nThe field-theory version says that each continuous symmetry of an action $\\int L(x, u, \\nabla u) \\, dx$ gives a [[conservation law|8RA7AMWSAF]] in divergence form.\n\n## Explanations\n\n### L1 human\n\nNoether's theorem links symmetries with conserved quantities.\nIf a mechanical system looks the same after you rotate it, then its angular momentum stays constant.\nIf it looks the same after you shift it in space, its momentum stays constant.\nIf its laws do not change in time, its energy stays constant.\n\n### L2 human\n\nThe theorem turns a geometric fact, invariance of the Lagrangian under a one-parameter group, into a first integral that can be written down explicitly.\nFirst integrals lower the order of the equations of motion, and in Hamiltonian terms the conserved quantity generates the symmetry through Poisson brackets.\nApplied to a geodesic problem with a Killing field, it gives Clairaut's relation on surfaces of revolution.\nIn field theory, the same argument produces conserved currents, such as the energy–momentum tensor from translations.\n",
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