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Examples from the calculus of variations. III: Legendre and Jacobi conditions. (English) Zbl 0980.49024

Summary: We deal with a new geometrical interpretation of the classical Legendre and Jacobi conditions: they are represented by the rate and the magnitude of rotation of certain linear subspaces of the tangent space around the tangents to the extremals. (The linear subspaces can be replaced by conical subsets of the tangent space.) This interpretation can be carried over to nondegenerate Lagrange problems but applies also to the degenerate variational integrals mentioned in the preceding Part II [Math. Bohem. 125, No. 2, 187-197 (2000; Zbl 0968.49002)].

MSC:

49K27 Optimality conditions for problems in abstract spaces
58A10 Differential forms in global analysis
58E30 Variational principles in infinite-dimensional spaces
49K10 Optimality conditions for free problems in two or more independent variables
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