Lepage Forms, Closed 2-Forms and Second-Order Ordinary Differential Equations

Lepage 2-forms appear in the variational sequence as representatives of the classes of 2-forms. In the theory of ordinary differential equations on jet bundles they are used to construct exterior differential systems associated with the equations and to study solutions, and help to solve the inverse problem of the calculus of variations: since variational equations are characterized by Lepage 2-forms that are closed. In this paper, a general setting for Lepage forms in the variational sequence is presented, and Lepage 2-forms in the theory of second-order differential equations in general and of variational equations in particular, are investigated in detail.