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

Chrastina, Jan

Math. Bohem. 125, No. 1, 55-76 (2000).

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.

MSC:

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

Keywords:

variational integral; critical curve; adjoint module; initial form; Poincaré-Cartan form; Lagrange problem; Mayer field; Weierstrass function

Citations:

Zbl 0968.49002