  Lepage equivalents of Lagrangians are a higher order, field-theoretical
generalization of the notion of Poincare-Cartan form from mechanics and play a
similar role: they give rise to a geometric formulation (and to a geometric
understanding) of the calculus of variations. A long-standing open problem is
the determination, for field-theoretical Lagrangians of order greater than one,
of a Lepage equivalent with the so-called closure property: the Lepage
equivalent is a closed differential form if and only if the Lagrangian has
vanishing Euler-Lagrange expressions. The present paper proposes a solution to
this problem, for general Lagrangians, of any order. The construction is a
local one; yet, we show that in most of the cases of interest for physical
applications, the obtained Lepage equivalent is actually globally defined. A
variant of this construction, which is advantageous for reducible Lagrangians,
is also introduced. In particular, for reducible Lagrangians of order two, the
obtained Lepage equivalents are of order one.
