Article
MSC:
49-01,
49J10,
49K15,
49K27,
49N45,
58A10,
58E30 | MR 1752079 | Zbl 0968.49001 | DOI: 10.21136/MB.2000.126263
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Keywords:
variational integral; critical curve; adjoint module; initial form; Poincaré-Cartan form; Lagrange problem; Mayer field; Weierstrass function; diffiety
variational integral; critical curve; adjoint module; initial form; Poincaré-Cartan form; Lagrange problem; Mayer field; Weierstrass function; diffiety
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 Poincare-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.
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References:
[1] O. Bolza: Vorlesungen über Variationsrechnung. B. G. Teubner, Leipzig, 1906.
[2] C. Carathéodory: Variationsrechnung und partielle Differentialgleichungen erster Ordnung. B. G. Teubner, Leipzig, 1935.
[3] J. Chrastina: Solution of the inverse problem of the calculus of variations. Math. Bohem. 119 (1994), 157-201. MR 1293249 | Zbl 0821.49026
[4] M. Giquanta S. Hildebrandt: Calculus of variations I, II. Grundlehren der Math. Wiss. 310, 311, Springer, 1995.
[5] P. A. Griffiths: Exterior differential systems and the calculus of variations. Birkhäuser, Boston, 1983. MR 0684663 | Zbl 0512.49003
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