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  "body": "\n## Statement\n\nLet $\\Omega \\subseteq \\R^n$ be open, and consider maps $u \\colon \\Omega \\to \\R^m$.\nLet $J(u) = \\int_\\Omega L(x, u, \\partial u, \\dots, \\partial^k u) \\, dx$ be an [[integral functional|JRQZ803Q5Y]] with smooth Lagrangian $L$.\nLet $E_1(u), \\dots, E_m(u)$ be its [[Euler–Lagrange expressions|PS0VV149ET]], so that the Euler–Lagrange equations read $E(u) = 0$.\nAssume that $J$ has a symmetry depending on $r$ arbitrary smooth functions $g_1, \\dots, g_r$ of $x$.\nIts infinitesimal variations are\n$$\\delta u_\\alpha = \\sum_{j=1}^{r} \\mathcal{D}_{\\alpha j} \\, g_j, \\quad \\alpha = 1, \\dots, m,$$\nwhere each $\\mathcal{D}_{\\alpha j}$ is a linear differential operator whose coefficients depend on $x$, $u$ and derivatives of $u$.\nAssume that each such variation leaves $L$ invariant up to a total divergence.\nThen the Euler–Lagrange expressions satisfy $r$ identities\n$$\\sum_{\\alpha=1}^{m} \\mathcal{D}_{\\alpha j}^{*} \\, E_\\alpha(u) = 0, \\quad j = 1, \\dots, r,$$\nfor every smooth $u$, where $\\mathcal{D}_{\\alpha j}^{*}$ is the formal adjoint of $\\mathcal{D}_{\\alpha j}$.\n\n## Remarks\n\nInvariance up to a divergence gives $\\sum_\\alpha E_\\alpha(u) \\, \\delta u_\\alpha = \\operatorname{div} S$ for some expression $S$ that is linear in the $g_j$ and their derivatives.\nIntegrating by parts moves the derivatives from $g_j$ onto $E_\\alpha$, so $\\sum_j g_j \\sum_\\alpha \\mathcal{D}^*_{\\alpha j} E_\\alpha$ is a divergence too.\nAn expression of the form $\\sum_j g_j F_j$ that is a divergence for all compactly supported $g_j$ must have $F_j = 0$.\n\nExample: the length $J(x, y) = \\int_a^b \\sqrt{\\dot x^2 + \\dot y^2} \\, dt$ of a regular plane curve, one with $(\\dot x, \\dot y) \\ne 0$ everywhere, is invariant under reparametrisations $t \\mapsto t + \\varepsilon g(t)$.\nThe variations are $\\delta x = \\dot x \\, g$ and $\\delta y = \\dot y \\, g$, up to sign, so here $\\mathcal{D}$ is multiplication by $\\dot x$ and $\\dot y$.\nThe Euler–Lagrange expressions are $E_x = -\\frac{d}{dt} T_1$ and $E_y = -\\frac{d}{dt} T_2$, with $T = (\\dot x, \\dot y) / \\abs{(\\dot x, \\dot y)}$.\nThe identity is $\\dot x \\, E_x + \\dot y \\, E_y = 0$ for every regular curve; it holds because $T \\cdot \\dot T = 0$.\n\nUnlike the first [[Noether's theorem|XHPDM5T2JG]], the result gives identities that hold for all $u$, not conservation laws along solutions.\nSo the Euler–Lagrange equations are not independent, and their solutions are not determined by initial data alone.\n\n## Explanations\n\n### L1 human\n\nSome symmetries are not a single motion but a whole family that can vary from point to point, like a change of time scale that may differ at each moment.\nNoether's second theorem says that such a large symmetry forces relations among the equations of motion themselves.\nThese relations hold for every function, not only for solutions.\nThey show that the equations do not fix the solution uniquely, because the symmetry can still move it.\n\n### L2 human\n\nThis is the theorem behind gauge theories and general relativity.\nIn electromagnetism, gauge invariance gives the identity that the divergence of the Maxwell equations vanishes identically, which is linked to charge conservation.\nIn general relativity, invariance under diffeomorphisms gives the contracted Bianchi identities.\nThe identities explain why such equations are underdetermined and need gauge fixing before one can solve an initial value problem.\n",
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