Examples from the calculus of variations. I: Nondegenerate problems. (English) Zbl 0968.49001

Summary: The criteria of extremality for classical variational integrals depending on several functions of one independent variable and their derivatives of arbitrary orders for constrained, isoperimetrical, degenerate, degenerate constrained, and so on, cases are investigated by means of adapted Poincaré-Cartan forms. Without ambitions on a noble generalizing theory, the main part of the article consists of simple illustrative examples within a somewhat naive point of view in order to obtain results resembling the common Euler-Lagrange, Legendre, Jacobi, and Hilbert-Weierstrass conditions whenever possible and to discuss some modifications necessary in the degenerate case. The inverse and the realization problems are mentioned, too.


49J10 Existence theories for free problems in two or more independent variables
49K27 Optimality conditions for problems in abstract spaces
49N45 Inverse problems in optimal control
58E30 Variational principles in infinite-dimensional spaces


Zbl 0968.49002
Full Text: EuDML