Topics in the calculus of variations: Finite order variational sequences
D. Krupka
Abstract: It is known that there exists a mapping $\Lambda$ assigning
to a first order lagrangian $\lambda$ its Lepagean equivalent $\Lambda
(\lambda)$ in such a way that $d\Lambda (\lambda) = 0$ if and only if the
Euler-Lagrange form $E_\lambda$ vanishes identically, i.e., $E_\lambda =
0$. In this paper we discuss within the theory of finite order variational
sequences an analogue of $\Lambda$ for higher order lagrangians. It
turns out that $\Lambda (\lambda)$ is a class of forms rather than a
differential form defined on the same domain as $\lambda$. The main use
of $\Lambda$ is to define the order of a lagrangian in a more adequate
way than the usual one. We show how the order of a lagrangian can be
determined. We find the order of a variationally trivial lagrangian.
Applying these results to the classification problem of symmetry
transformations we obtain a higher order analogue of the
Noether-Bessel-Hagen equation.
Keywords: Fibered manifold, $r$-jet, lagrangian, contact form,
variational sequence, order of a lagrangian, Euler-Lagrange mapping,
Helmholtz-Sonin mapping, variationally trivial lagrangians.
MS classification: 58E30, 58A10, 58G05, 49F99.