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  "body": "\n## Summary\n\nHilbert's invariant integral is a line integral attached to a [[field of extremals|K4DFT3H9BJ]] of a variational problem.\nIts value depends only on the end points of the path, not on the path inside the field.\nAlong curves of the field it equals the functional itself.\nComparing it with the functional on an arbitrary curve gives the Weierstrass formula, which expresses $J(y) - J(y_0)$ as an integral of the [[Weierstrass excess function|R8TWRPSPNV]].\n\n## Explanations\n\n### L1 human\n\nInside a field of extremals, each point comes with a preferred slope, the slope of the field curve through it.\nHilbert built from these slopes a line integral that does not care which path one takes between two points.\nOn a field curve it gives exactly the value of the functional.\nSo to compare a competitor curve with the field curve, one compares the functional on the competitor with this path-independent integral.\nThe difference is an integral of the excess function, and if the excess is nonnegative, the field curve wins.\n",
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