HAMILTONIAN FIELD THEORY REVISITED: A GEOMETRIC APPROACH TO REGULARITY

Abstract

The aim of the paper is to announce some recent results concerning Hamiltonian theory for higher order variational problems on fibered manifolds. A reformulation, generalization and extension of basic concepts such as Hamiltonian system, Hamilton equations, regularity, and Legendre transformation, is presented. The theory is based on the concept of Lepagean (n+1)-form (where n is the dimension of the base manifold). Contrary to the classical approach, where Hamiltonian theory is related to a single Lagrangian, within the present setting a Hamiltonian system is associated with an Euler– Lagrange form, i.e., with the class of all equivalent Lagrangians. Hamilton equations are introduced to be equations for integral sections of an exterior differential system, defined by a Lepagean (n + 1)-form. Relations between extremals and solutions of Hamilton equations are studied in detail. A revision of the concepts of regularity and Legendre transformation is proposed, reflecting geometric properties of the related exterior differential system. The new look is shown to lead to new regularity conditions and Legendre transformation formulas, and provides a procedure of regularization of variational problems. Relations to standard Hamilton–De Donder theory, as well as to multisymplectic geometry are studied. Examples of physically interesting Lagrangian systems which are traditionally singular, but regular in this revised sense, are discussed.