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  "body": "\n## Problem\n\nHilbert's twenty-third problem asks for the further development of the methods of the [[calculus of variations|PQ25NHV0KS]].\nUnlike most of Hilbert's problems, it does not ask a single question with a definite answer.\nIt is a call to extend and deepen the theory of necessary and sufficient conditions for extrema of functionals.\n\n## Remarks\n\nIn the text of the problem Hilbert presented his own contribution, the [[invariant integral|D1EBAAFEWV]], which gives a short route to sufficient conditions for a minimum in the Weierstrass theory.\nBecause it is a programme rather than a statement, the problem is not regarded as solved or unsolved.\nThe closely related [[twentieth problem|187NP5Z002]] asks for the existence of solutions of variational boundary value problems.\n\n## Explanations\n\n### L1 human\n\nThe last problem on Hilbert's famous list of 1900 is not a precise question.\nIt asks mathematicians to keep developing the calculus of variations, the theory of finding curves and surfaces that make a quantity smallest or largest.\nThat theory has grown a great deal since then, in directions that Hilbert could not foresee.\n",
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      "first_stated": "1900",
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  "msc": [
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