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MATHEMATICA BOHEMICA, Vol. 126, No. 1, pp. 93-111 (2001)
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# Examples from the calculus of variations III. Legendre and Jacobi conditions

## Jan Chrastina

* Jan Chrastina*, Masarykova Universita, Katedra matematické analyzy, Janackovo nam. 2a, 662 95 Brno, Czech Republic, e-mail: ` mdchr@fce.vutbr.cz`

**Abstract:**
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.

**Keywords:** Legendre condition, Jacobi condition, Poincar/'e-Cartan form, Lagrange problem, degenerate variational integral

**Classification (MSC2000):** 49-01, 49K15, 58A10

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