On finite order variational sequences

We 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 Kolář 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.