Hilbert's twenty-third problem

Hilbert's twenty-third problem is the last of Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. In contrast with Hilbert's other 22 problems, his 23rd is not so much a specific "problem" as an encouragement towards further development of the calculus of variations. His statement of the problem is a summary of the state-of-the-art (in 1900) of the theory of calculus of variations, with some introductory comments decrying the lack of work that had been done of the theory in the mid to late 19th century.

Original statement

The problem statement begins with the following paragraph:

So far, I have generally mentioned problems as definite and special as possible.... Nevertheless, I should like to close with a general problem, namely with the indication of a branch of mathematics repeatedly mentioned in this lecture-which, in spite of the considerable advancement lately given it by Weierstrass, does not receive the general appreciation which, in my opinion, it is due-I mean the calculus of variations.

Calculus of variations

Main article: Calculus of variations

Calculus of variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. The interest is in extremal functions that make the functional attain a maximum or minimum value – or stationary functions – those where the rate of change of the functional is zero.

References

• Stampacchia, Guido (1976). "Hilbert's Twenty-Third Problem: Extension of the Calculus of Variations". In Felix E. Browder (ed.). Mathematical Developments Arising from Hilbert Problems. Proceedings of Symposia in Pure Mathematics. Vol. XXVIII.2. American Mathematical Society. pp. 611–628. ISBN 0-8218-1428-1.