Displaying similar documents to “Examples from the calculus of variations. I. Nondegenerate problems”

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...

On the Lagrange-Souriau form in classical field theory

D. R. Grigore, Octavian T. Popp (1998)

Mathematica Bohemica

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The Euler-Lagrange equations are given in a geometrized framework using a differential form related to the Poincare-Cartan form. This new differential form is intrinsically characterized; the present approach does not suppose a distinction between the field and the space-time variables (i.e. a fibration). In connection with this problem we give another proof describing the most general Lagrangian leading to identically vanishing Euler-Lagrange equations. This gives the possibility to...