The Fundamental Lepage Form in Two Independent Variables: A Generalization Using Order-Reducibility

A second-order generalization of the fundamental Lepage form geometric calculus variations over fibered manifolds with 2-dimensional base is described by means insisting on (i) an equivalence relation “Lepage differential 2-form closed if and only associated Lagrangian trivial” (ii) principal component form, extending well-known Poincaré–Cartan preserving order prescribed a given Lagrangian. This approach completes several attempts finding equivalent possessing condition (i), which for first-order Lagrangians in field theory due to Krupka Betounes.