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Legendre condition; Jacobi condition; Poincaré-Cartan form; Lagrange problem; degenerate variational integral
We will 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.
[1] J. Chrastina: Examples from the calculus of variations I. Nondegenerate problems. Math. Bohem. 125 (2000), 55–76. MR 1752079 | Zbl 0968.49001
[2] W. Fulton, J. Harris: Representation Theory. Graduate Texts in Mathematics 129, Springer, 1996. MR 1153249
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