Second order variations in variational sequences. (English) Zbl 0977.58019

Francaviglia, Mauro; Palese, Marcella

Kozma, L. (ed.) et al., Steps in differential geometry. Proceedings of the colloquium on differential geometry, Debrecen, Hungary, July 25-30, 2000. Debrecen: Univ. Debrecen, Institute of Mathematics and Informatics, 119-130 (2001).

Summary: We provide a geometrical characterization of the second order variation of a generalized Lagrangian in the framework of variational sequences.
We define the variational vertical derivative as an operator on the sheaves of the variational sequence and stress its link with the classical concept of variation. The main result is the intrinsic formulation of a theorem which states the relation between the variational vertical derivative of the Euler-Lagrange morphism of a generalized Lagrangian and the Euler-Lagrange morphism of the variational vertical derivative of the Lagrangian itself.
For the entire collection see [Zbl 0966.00031].

MSC:

58E30	Variational principles in infinite-dimensional spaces
55R10	Fiber bundles in algebraic topology
58A12	de Rham theory in global analysis
58A20	Jets in global analysis

Keywords:

fibered manifold; jet space; variational sequence; second order variation; variational vertical derivative; Euler-Lagrange morphism