  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.
