Math 2.0

Calculus of variations and optimal control; optimization Classical problems Problem

Hilbert's twenty-third problem

Builds on Calculus of variationsHilbert's invariant integralHilbert's twentieth problem

Problem

Hilbert's twenty-third problem asks for the further development of the methods of the calculus of variations. 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, 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 asks for the existence of solutions of variational boundary value problems.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Boundary value problem, concept, no evidence levelBoundary value problemconceptCalculus of variations, concept, no evidence levelCalculus of variationsconceptDirect method in the calculus of variations, concept, no evidence levelDirect method in the …conceptHilbert's invariant integral, concept, no evidence levelHilbert's invariant i…conceptHilbert's twentieth problem, problem, no evidence levelHilbert's twentieth p…problemHilbert's twenty-third problem, problem, no evidence levelHilbert's twenty-thir…problem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Boundary value problem concept
  2. Calculus of variations concept
  3. Direct method in the calculus of variations concept
  4. Hilbert's invariant integral concept
  5. Hilbert's twentieth problem problem
  6. Hilbert's twenty-third problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_twenty-third_problem
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AboutCalculus of variations
UsesCalculus of variations, Hilbert's invariant integral, Hilbert's twentieth problem
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