---
tag: YCKDR0S7V1
kind: problem
name: Hilbert's twenty-third problem
msc: [49-03, 49K05, 49-02]
uses: [PQ25NHV0KS, D1EBAAFEWV, 187NP5Z002]
about: [PQ25NHV0KS]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%27s_twenty-third_problem
same_as:
  wikidata: Q5761180
  wikipedia: Hilbert's twenty-third problem
history:
  posed_by: "David Hilbert"
  first_stated: "1900"
---

## Problem

Hilbert's twenty-third problem asks for the further development of the methods of the [[calculus of variations|PQ25NHV0KS]].
Unlike most of Hilbert's problems, it does not ask a single question with a definite answer.
It is a call to extend and deepen the theory of necessary and sufficient conditions for extrema of functionals.

## Remarks

In 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.
Because it is a programme rather than a statement, the problem is not regarded as solved or unsolved.
The closely related [[twentieth problem|187NP5Z002]] asks for the existence of solutions of variational boundary value problems.

## Explanations

### L1 human

The last problem on Hilbert's famous list of 1900 is not a precise question.
It asks mathematicians to keep developing the calculus of variations, the theory of finding curves and surfaces that make a quantity smallest or largest.
That theory has grown a great deal since then, in directions that Hilbert could not foresee.