Recent results in variational sequence theory (Q2733937)

The article presents an extensive survey of the author results concerning formal interrelations between Euler-Lagrange operators and the theory of variational sequences. Multidimensional generalizations of the Poincaré-Cartan form, the so-called Lepage equivalent, serve as the main tool. Only rather illustrative applications are involved and all connections to the topical investigations (Hamiltonian theory, the inverse and equivalence problems, extremality, Dirac structures, and so like) are tacitly passed over. NEWLINENEWLINENEWLINEContents: Fundamental concepts of jet theory on fibered manifolds \(\pi:Y\to X\), jet prolongations \(J^rY\), horizontal and vertical objects, contact forms, finite order variational sequences (a factor of the common de Rham sequence), variational functionals \(\int_\Omega J^r\mu^* \rho\) (where \(\Omega\subset X\) is a compact submanifold, \(\mu\) a cross-section of \(J^rY\), \(\rho\) an \(n\)-form in the space \(J^rY)\), first variation, extremals, Lepage forms (defined by using coordinate charts), Euler-Lagrange and Helmholtz-Sonin operator (which appear thanks to the rigid fibered structure), variational sequences with localization of Euler-Lagrange operators.NEWLINENEWLINENEWLINE[See also the following paper of \textit{M. Krbek}, \textit{J. Musilova} and \textit{J. Kašparová}, ibid., 147-160 (2001; Zbl 1003.58017) for a slight extension of the underlying results].NEWLINENEWLINEFor the entire collection see [Zbl 0966.00031].