On different geometric formulations of Lagrangian formalism (Q1303767)

We consider two geometric formulations of Lagrangian formalism on fibred manifolds: Krupka's theory of finite order variational sequences, and Vinogradov's infinite order variational sequence associated with the \(C\)-spectral sequence. On the one hand, we show that the direct limit of Krupka's variational bicomplex is a new infinite order variational bicomplex which yields a new infinite order variational sequence. On the other hand, by means of Vinogradov's \(C\)-spectral sequence, we provide a new finite order variational sequence whose direct limit turns out to be Vinogradov's infinite order variational sequence. Finally, we provide an equivalence of the two finite order and infinite order variational sequences up to the space of Euler-Lagrange morphisms.