  The Lagrangian formalism on a arbitrary non-fibrating manifold is considered.
The kinematical description of this generic situation is based on the concept
of (higher-order) Grassmann manifolds which is the factorization of the regular
velocity manifold to the action of the differential group. Here we introduce in
this context the basic concepts of the Lagrangian formalism as Lagrange,
Euler-Lagrange and Helmholtz-Sonin forms. These objects come in pairs, namely
we have homogeneous objects (defined on the regular velocity manifold) and
non-homogeneous objects (defined on the Grassmann manifold). We will establish
the connection between the homogeneous objects and their non-homogeneous
counterparts. As a result we will conclude that the generic expressions for a
variationally trivial Lagrangian and for a locally variational differential
equation remain the same as in the fibrating case.
