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  "body": "\n## Definition\n\nLet $L(x, y, p)$ be a $C^2$ function on $\\R \\times \\R \\times \\R$, and let $J(y) = \\int_a^b L(x, y, y') \\, dx$.\nLet $p(x, y)$ be the slope function of a [[field of extremals|K4DFT3H9BJ]] of $J$ on a simply connected open set $D \\subseteq \\R^2$.\nFor a piecewise $C^1$ curve $\\gamma$ in $D$, *Hilbert's invariant integral* is the [[line integral|0H5MR9B6GT]]\n$$I^*(\\gamma) = \\int_\\gamma \\Big( L(x, y, p) - p \\, L_p(x, y, p) \\Big) \\, dx + L_p(x, y, p) \\, dy,$$\nwhere $p = p(x, y)$ inside the integrand.\n\n## Remarks\n\nThe integrand is a [[closed differential form|0ZV14XRNF0]] on $D$ because the field consists of extremals.\nSince $D$ is simply connected, the form is exact, so $I^*(\\gamma)$ depends only on the end points of $\\gamma$.\n\nAlong a curve of the field one has $dy = p \\, dx$, so $I^*(\\gamma) = \\int L(x, y, p) \\, dx = J$ on that curve.\nFor the graph of any piecewise $C^1$ function $y$ in $D$, subtracting $I^*$ from $J$ gives\n$$J(y) - I^*(y) = \\int_a^b E\\big(x, y(x), p(x, y(x)), y'(x)\\big) \\, dx,$$\nwith $E$ the [[Weierstrass excess function|R8TWRPSPNV]].\nIf $y_0$ is a field curve and $y$ has the same end points, then $I^*(y) = J(y_0)$, which gives the Weierstrass formula $J(y) - J(y_0) = \\int_a^b E \\, dx$.\n\nExample: for $L = p^2$ and the field $y = x + c$ with $p = 1$, the integrand is $(1 - 2) \\, dx + 2 \\, dy = d(2y - x)$, which is exact.\n\nNon-example: with the slope function $p = 1$ and $L = p^2 + y^2$, the curves $y = x + c$ are not extremals, and the form $(y^2 - 1) \\, dx + 2 \\, dy$ is not closed, since $\\partial_y (y^2 - 1) = 2y$ is not identically equal to $\\partial_x 2 = 0$.\n",
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