Hamiltonian formalisms for multidimensional calculus of variations and perturbation theory

Abstract

In a first part we propose an introduction to multisymplectic formalisms, which are generalisations of Hamilton's formulation of Mechanics to the calculus of variations with several variables: we give some physical motivations, related to the quantum field theory, and expound the simplest example, based on a theory due to T. de Donder and H. Weyl. In a second part we explain quickly a work in collaboration with J. Kouneiher (math-ph/0211046) on generalizations of the de Donder--Weyl theory (known as Lepage theories). Lastly we show that in this framework a perturbative classical field theory (analog of the perturbative quantum field theory) can be constructed.