Lagrangian formalism and the intrinsic geometry of PDEs

This report is an attempt to answer the following question. Where exactly does a differential equation contain information about its variational nature? Apparently, in the general case, the concept of a presymplectic structure as a closed variational 2-form may not be sufficient to describe variational principles in terms of intrinsic geometry. I will introduce the concept of an internal Lagrangian and relate it to the Vinogradov C-spectral sequence.