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.