  We show how to construct, for a Lagrangian of arbitrary order, a Lepage
equivalent satisfying the closure property: that the Lepage equivalent vanishes
precisely when the Lagrangian is null. The construction uses a homotopy
operator for the horizontal differential of the variational bicomplex. A choice
of symmetric linear connection on the manifold of independent variables, and a
global homotopy operator constructed using that connection, may then be used to
extend any global Lepage equivalent to one satisfying the closure property. In
the second part of the paper we investigate the r\^ole of vertical
endomorphisms in constructing such Lepage equivalents. These endomorphisms may
be used directly to construct local homotopy operators. Together with a
symmetric linear connection they may also be used to construct global vertical
tensors, and these define infinitesimal nonholonomic projections which in turn
may be used to construct Lepage equivalents. We conjecture that these global
vertical tensors may also be used to define global homotopy operators.
