2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/56463We discuss intrinsic aspects of Krupka's approach to finite-order variational sequences. We give intrinsic isomorphisms of the quotient subsheaves of the short finite-order variational sequence with sheaves of forms on jet spaces of suitable order, obtaining a new finite-order (short exact) variational sequence which is made by sheaves of polynomial differential operators. Moreover, we present an intrinsic formulation for the Helmholtz condition of local variationality using a technique introduced by Kolar that we have adapted to our context. Finally, we provide the minimal order solution to the inverse problem of the calculus of variations and a solution of the problem of the variationally trivial Lagrangian.LaTeX2e with Paul Taylor's diagrams, 38 pages, no figures. Long version of a paper appeared in Math. Proc. Camb. Phil. Soc. 125 (1999), 321-333Mathematical PhysicsHigh Energy Physics - TheoryDifferential Geometry58G05;58A12,58A20,58E30text