The paper is devoted to the interior Euler-Lagrange operator in field theory, representing an important tool for constructing the variational sequence. We give a new invariant definition of this operator by means of a natural decomposition of spaces of differential forms, appearing in the sequence, which defines its basic properties. Our definition extends the well-known cases of the Euler-Lagrange class (Euler-Lagrange form) and the Helmholtz class (Helmholtz form). This linear operator has the property of a projector, and its kernel consists of contact forms. The result generalizes an analogous theorem valid for variational sequences over 1-dimensional manifolds and completes the known heuristic expressions by explicit characterizations and proofs.

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Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given non-variational system, conditions assuring variationality (the Helmholtz conditions) of the induced system with respect to a submanifold of a Euclidean space are studied, and the problem of existence of these "variational submanifolds" is formulated in general and solved for second-order systems. The variational sequence theory on sheaves of differential forms is employed as a main tool for analysis of local and global aspects (variationality and variational triviality). The theory is illustrated by examples of holonomic constraints (submanifolds of a configuration Euclidean space) which are variational submanifolds in geometry and mechanics.

Locally variational systems of differential equations on smooth manifolds, having certain de Rham cohomology group trivial, automatically possess a global Lagrangian. This important result due to Takens is, however, of sheaf-theoretic nature. A new constructive method of finding a global Lagrangian for second-order ODEs on 2-manifolds is described on the basis of solvability of exactness equation for Lepage 2-forms, and the top-cohomology theorems. Examples from geometry and mechanics are discussed.

The aim of this paper is to give a survey of recent developments in global variational geometry, and in particular, to complete the results on the construction of classes (terms) in the variational sequences related to higher-order variational problems on fibred spaces. Explicit description of the first order variational sequences is given as an example.