Displaying similar documents to “On a class of variational problems defined by polynomial Lagrangians”

Symmetries in finite order variational sequences

Mauro Francaviglia, Marcella Palese, Raffaele Vitolo (2002)

Czechoslovak Mathematical Journal

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We refer to Krupka’s variational sequence, i.e. the quotient of the de Rham sequence on a finite order jet space with respect to a ‘variationally trivial’ subsequence. Among the morphisms of the variational sequence there are the Euler-Lagrange operator and the Helmholtz operator. In this note we show that the Lie derivative operator passes to the quotient in the variational sequence. Then we define the variational Lie derivative as an operator on the sheaves of the variational sequence....

On the invariant variational sequences in mechanics

Šeděnková, Jana

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Summary: The r -th order variational sequence is the quotient sequence of the De Rham sequence on the r th jet prolongation of a fibered manifold, factored through its contact subsequence.In this paper, the first order variational sequence on a fibered manifold with one-dimensional base is considered. A new representation of all quotient spaces as some spaces of (global) forms is given. The factorization procedure is based on a modification of the interior Euler operator, used in the theory...

Jets and the variational calculus

David J. Saunders (2021)

Communications in Mathematics

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We review the approach to the calculus of variations using Ehresmann's theory of jets. We describe different types of jet manifold, different types of variational problem and different cohomological structures associated with such problems.

Generalized Jacobi morphisms in variational sequences

Francaviglia, Mauro, Palese, Marcella

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Summary: We provide a geometric interpretation of generalized Jacobi morphisms in the framework of finite order variational sequences. Jacobi morphisms arise classically as an outcome of an invariant decomposition of the second variation of a Lagrangian. Here they are characterized in the context of generalized Lagrangian symmetries in terms of variational Lie derivatives of generalized Euler-Lagrange morphisms. We introduce the variational vertical derivative and stress its link with...