Topics in the calculus of variations: Finite order variational sequences. (English) Zbl 0811.58018

Kowalski, O. (ed.) et al., Differential geometry and its applications. Proceedings of the 5th international conference, Opava, Czechoslovakia, August 24-28, 1992. Opava: Open Education and Sciences, Silesian Univ.. Math. Publ. (Opava). 1, 473-495 (1993).
The author discusses the well-known result: a natural mapping \(\Lambda\) exists which assigns the Lepagean equivalent \(\Lambda (\lambda)\) (i.e., the Cartan-Poincaré form in alternative terminology) to every first order Lagrange density \(\lambda\) in such a manner that \(d \Lambda (\lambda) = 0\) if and only if the Euler-Langrange system \(E_ \lambda\) of the density \(\lambda\) is trivial. Then the problem is raised if this result can be generalized to higher order but, even after introduction of heavy tools of jet theory, only very particular and for the most part familiar results are derived. On this occasion a new concept of order of Lagrange densities is introduced.
(Reviewer’s remark: the common jet hierarchy is not adapted to degenerate variational problems. Certain Bäcklund transformations do not preserve the order and even the regularity of variational integrals.).
For the entire collection see [Zbl 0797.00017].
Reviewer: J.Chrastina (Brno)


58E30 Variational principles in infinite-dimensional spaces
58A10 Differential forms in global analysis
58A20 Jets in global analysis