  The exactness equation for Lepage 2-forms, associated with variational
systems of ordinary differential equations on smooth manifolds, is analyzed
with the aim to construct a concrete global variational principle. It is shown
that locally variational systems defined by homogeneous functions of degree $c
\neq 0, 1$ are automatically globally variational. A new constructive method of
finding a global Lagrangian is described for these systems, which include for
instance the geodesic equations in Riemann and Finsler geometry.
