Displaying similar documents to “Implicit Differential Equations From the Singularity Theory Viewpoint”

Examples from the calculus of variations. III. Legendre and Jacobi conditions

Jan Chrastina (2001)

Mathematica Bohemica

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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...

Felix Klein's paper on real flexes vindicated

Felice Ronga (1998)

Banach Center Publications

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In a paper written in 1876 [4], Felix Klein gave a formula relating the number of real flexes of a generic real plane projective curve to the number of real bitangents at non-real points and the degree, which shows in particular that the number of real flexes cannot exceed one third of the total number of flexes. We show that Klein's arguments can be made rigorous using a little of the theory of singularities of maps, justifying in particular his resort to explicit examples. ...